# The Enigmatic Brown Dwarf WISEA J153429.75$-$104303.3 (aka "The Accident")

J. Davy Kirkpatrick IPAC, Mail Code 100-22, Caltech, 1200 E. California Blvd.,
Pasadena, CA 91125, USA<EMAIL_ADDRESS>Federico Marocco IPAC, Mail
Code 100-22, Caltech, 1200 E. California Blvd., Pasadena, CA 91125, USA  Dan
Caselden Gigamon Applied Threat Research, 619 Western Avenue, Suite 200,
Seattle, WA 98104, USA  Aaron M. Meisner NSF’s National Optical-Infrared
Astronomy Research Laboratory, 950 N. Cherry Ave., Tucson, AZ 85719, USA
Jacqueline K. Faherty Department of Astrophysics, American Museum of Natural
History, Central Park West at 79th Street, New York, NY 10034, USA  Adam C.
Schneider US Naval Observatory, Flagstaff Station, P.O.Box 1149, Flagstaff, AZ
86002, USA  Department of Physics and Astronomy, George Mason University,
MS3F3, 4400 University Drive, Fairfax, VA 22030, USA  Marc J. Kuchner NASA
Goddard Space Flight Center, Exoplanets and Stellar Astrophysics Laboratory,
Code 667, Greenbelt, MD 20771, USA  S. L. Casewell School of Physics and
Astronomy, University of Leicester, University Road, Leicester LE1 7RH, UK
Christopher R. Gelino  IPAC, Mail Code 100-22, Caltech, 1200 E. California
Blvd., Pasadena, CA 91125, USA  Michael C. Cushing The University of Toledo,
2801 West Bancroft Street, Mailstop 111, Toledo, OH 43606, USA  Peter R.
Eisenhardt  Jet Propulsion Laboratory, California Institute of Technology, MS
169-237, 4800 Oak Grove Drive, Pasadena, CA 91109, USA  Edward L. Wright
Department of Physics and Astronomy, University of California Los Angeles, 430
Portola Plaza, Box 951547, Los Angeles, CA, 90095-1547, USA  Steven D. Schurr
IPAC, Mail Code 100-22, Caltech, 1200 E. California Blvd., Pasadena, CA 91125,
USA

(Received 2021 May 10; Revised 2021 May 18; Accepted 2021 May 21)

###### Abstract

Continued follow-up of WISEA J153429.75$-$104303.3, announced in Meisner et
al. (2020), has proven it to have an unusual set of properties. New imaging
data from Keck/MOSFIRE and HST/WFC3 show that this object is one of the few
faint proper motion sources known with $J-$ch2 $>$ 8 mag, indicating a very
cold temperature consistent with the latest known Y dwarfs. Despite this, it
has W1$-$W2 and ch1$-$ch2 colors $\sim$1.6 mag bluer than a typical Y dwarf. A
new trigonometric parallax measurement from a combination of WISE, Spitzer,
and HST astrometry confirms a nearby distance of $16.3^{+1.4}_{-1.2}$ pc and a
large transverse velocity of $207.4{\pm}15.9$ km s-1. The absolute $J$, W2,
and ch2 magnitudes are in line with the coldest known Y dwarfs, despite the
highly discrepant W1$-$W2 and ch1$-$ch2 colors. We explore possible reasons
for the unique traits of this object and conclude that it is most likely an
old, metal-poor brown dwarf and possibly the first Y subdwarf. Given that the
object has an HST F110W magnitude of 24.7 mag, broad-band spectroscopy and
photometry from JWST are the best options for testing this hypothesis.

Stellar types (1634), T dwarfs (1679), Y dwarfs (1827), Proper motions (1295),
Metallicity (1031)

††journal: The Astrophysical Journal Letters††facilities: HST (WFC3), Keck:I
(MOSFIRE), WISE, NEOWISE††software: WiseView (Caselden et al. 2018).

##  1 Introduction

The coldest brown dwarfs are a difficult population to characterize. What
little flux they emit is concentrated near 5 $\mu$m, and, because this region
is largely unobservable from the ground, space-based missions offer the best
chance of discovery and follow-up. Studies at other wavelengths are hampered
by intrinsically faint magnitudes. For known Y dwarfs – the coldest brown
dwarfs with effective temperatures below $\sim$450K (Cushing et al. 2011) –
absolute $J$-band magnitudes range from 19.4 to 28.2 mag (Kirkpatrick et al.
2021).

Despite these difficulties, the number of known, very cold brown dwarfs is
slowly growing, with fifty confirmed or suspected Y dwarfs now cataloged
(Kirkpatrick et al. 2021). Nonetheless, uncovering clear trends in colors and
absolute magnitudes is proving elusive. WISE J085510.83$-$071442.5 (hereafter,
WISE 0855$-$0714) is the fourth closest (sub)stellar system to the Sun and the
coldest brown dwarf known ($\sim$250K; Luhman 2014). Despite estimates that
$15^{+20}_{-11}$ such frigid objects (Wright et al. 2014) have been imaged in
data sets by the Wide-Field Infrared Survey Explorer (WISE; Wright et al.
2010; Mainzer et al. 2014), it is the only brown dwarf recognized to have
$T_{\rm eff}<350K$ (Kirkpatrick et al. 2021), although two others may also
fall in this temperature zone – WISEA J083011.95+283716.0 (hereafter, WISE
0830+2837; Bardalez Gagliuffi et al. 2020) and WISEPA J182831.08+265037.8
(hereafter, WISE 1828+2650; Cushing et al. 2011). WISE 1828+2650, in fact,
stands out on color-mag plots as 1.5-2.0 magnitudes more luminous at $J$ and
ch2 than Y dwarfs of similar $J-$ch2 color (Kirkpatrick et al. 2021), and it
has a unique near-infrared spectrum (Cushing et al. 2011, 2021).

WISE 1828+2650 may be hinting at the variety present among the coldest brown
dwarfs, and this diversity is becoming more evident as we obtain detailed
characterization of the larger Y dwarf sample. The lack of other discoveries
as cold as WISE 0855$-$0714 may imply that it is not a typical member of the
ultra-cold brown dwarf population. As discussed in section 9.4 of Kirkpatrick
et al. (2021), objects at these coldest temperatures are expected to span a
wider range of ages than warmer brown dwarfs, suggesting a wider range of
metallicities and photometric properties.

In this Letter, we present additional follow-up of WISEA J153429.75$-$104303.3
(hereafter, WISE 1534$-$1043), nicknamed "The Accident" by its discoverer D.
Caselden, who serendipitously spotted it as a high-motion object in a field
containing an unrelated motion candidate. New data provide a robust parallax
and extremely red $J-$ch2 color. The parallax places it securely within 20 pc,
and the colors and absolute magnitudes are unlike those of any brown dwarf
known. Thus, WISE 1534$-$1043 appears to be the most enigmatic cold brown
dwarf yet identified and bolsters the idea that such objects span a wide range
of observational properties.

Figure 1: Images, each $\sim$1 arcminute on a side, showing the proper motion
of WISE 1534$-$1043 (circled in blue). The top two rows show WISE W1 and W2
images covering a five-year time baseline. The bottom two rows show more
recent images from Spitzer, Keck/MOSFIRE, and HST/WFC3. The source was not
detected in the Keck image, so the position marked is inferred from the
astrometric solution. The HST image includes a $5{\arcsec}{\times}5{\arcsec}$
inset highlighting the detection. North is up and east is to the left.

##  2 New Data

Meisner et al. (2020) showed that the $J-$ ch2 color limit and presumed high
velocity of WISE 1534$-$1043 placed it amongst a small set of mid- to late-T
subdwarfs. Kirkpatrick et al. (2021) used available WISE and Spitzer data to
measure a parallax, albeit with a large (30%) uncertainty. Here we present two
new data sets that produce a robust color and parallax reinforcing the
uniqueness of this object.

First, we have obtained a deep $J_{\rm MKO}$-band observation from MOSFIRE
(McLean et al. 2010, 2012) at the W. M. Keck Observatory. These data, obtained
in $\sim 1{\farcs}1$ seeing on 2020 Jun 08 (UT) using 60 $\times$ 60s frames
in a repeated 5-point dither pattern, were combined with IDL procedures that
use bright stars for re-registration and alignment. The result is shown in
Figure 1 and compared to earlier unWISE (Lang 2014; Meisner et al. 2018a, b)
and Spitzer images. Using the preliminary proper motion value from Kirkpatrick
et al. (2021) to propagate the predicted position to that epoch, we find no
detection down to a 3$\sigma$ limit of $J_{\rm MKO}$ = 23.8 mag by calibrating
to bright, unsaturated stars whose 2MASS $J$ magnitudes were converted to the
MKO system111
https://irsa.ipac.caltech.edu/data/2MASS/docs/releases/allsky/doc/sec6_4b.html.
This limit and other pertinent data on WISE 1534$-$1043 are listed in Table 1.

Second, we have obtained still deeper imaging at a similar wavelength (F110W)
using the Wide Field Camera 3 (WFC3) on the Hubble Space Telescope (HST) as
part of program 16243 (PI: Marocco). WISE 1534$-$1043 was observed on 2021 Feb
11 (UT) in MULTIACCUM mode with STEP100 sampling for a total exposure time of
2396.92 s. The total exposure was broken into four exposures of 599.23 s each
and dithered using the WFC3-IR-DITHER-BOX-MIN pattern, with point spacing of
0$\farcs$572 and line spacing of 0$\farcs$365.

WISE 1534$-$1043 is detected at S/N $\approx$ 5 per exposure with an F110W
magnitude of 24.695$\pm$0.083 mag (Table 1). As shown in Figure 1, it falls in
an uncontaminated region just north of a much brighter object. For astrometry
we used the HST pipeline’s calibrated exposures (flt files). In each exposure,
we measured the (x,y) positions for all stars via PSF-fitting using the
img2xym program222https://www.stsci.edu/~jayander/WFC3/WFC3IR_PSFs/, applied
the geometric distortion correction provided by STScI, and used the native WCS
to compute initial guesses for RA and Dec. To place the observations onto an
absolute reference frame, we used these initial positions to find Gaia Early
Data Release 3 (eDR3) sources – nineteen total – within the ${\sim}2{\times}2$
arcmin field of view. We used the eDR3 proper motions and parallaxes to update
the reference star coordinates from the 2016.0 Gaia epoch to the 2021.1 HST
epoch. Finally, we determined the standard (x,y) to (RA,Dec) transformation
(see, e.g., equations 1-4 in Bedin & Fontanive 2018) via least-square fitting.
The typical RMS of the fit is 2.5-3.5 mas. The RA and Dec of the nineteen
reference stars across the four exposures have a standard deviation of
$\sim$1.7 mas.

The (x,y) position of WISE 1534$-$1043 in each exposure was measured as
described above. Due to the relatively low S/N of the detection, its
positional measurements have a standard deviation of 22.0 mas in RA and 16.4
mas in Dec. We take these as the empirical uncertainties of our PSF fit, add
the RMS of the astrometric calibration in quadrature to compute the RA and Dec
uncertainties in each exposure, and take the weighted mean of the four
measurements and its uncertainty as the final positional measurement of WISE
1534$-$1043.

This position was combined with our published unWISE and Spitzer astrometry
and fed into the code of Kirkpatrick et al. (2021) to produce a new
astrometric solution (Figure 2), resulting in an absolute parallax of
61.4$\pm$4.7 mas and a proper motion of 2687.1$\pm$11.3 mas yr-1 (Table 1),
the high value of the latter having been originally noted by Meisner et al.
(2020).

Table 1: Compiled Data for WISE 1534$-$1043

Parameter | ValueaaMagnitudes are given in the Vega system. Conversions between Vega and AB magnitudes for HST/WFC3 filters can be found in https://www.stsci.edu/hst/instrumentation/wfc3/data-analysis/photometric-calibration/ir-photometric-calibration. | Ref.
---|---|---
(1) | (2) | (3)
AllWISE Designation | WISEA J153429.75$-$104303.3 | 2
CatWISE Designation | CWISE J153429.19$-$104318.9 | 3
Astrometric Fit
RA (ICRS) | 233.621873 deg $\pm$ 30.2 mas | 1
Dec (ICRS) |  $-$10.721775 deg $\pm$ 24.3 mas | 1
MJD | 57645.36 d | 1
Epoch | 2016.70 yr | 1
$\varpi_{\rm abs}$ | 61.4$\pm$4.7 mas | 1
$\mu_{\rm RA}$ |  $-$1253.1$\pm$8.9 mas yr-1 | 1
$\mu_{\rm Dec}$ |  $-$2377.0$\pm$7.0 mas yr-1 | 1
$\mu_{\rm total}$ | 2687.1$\pm$11.3 mas yr-1 | 1
Photometry
$J_{\rm MKO}$ |  $>$23.8 mag | 1
$H_{\rm MKO}$ |  $>$18.58 mag | 4,6
$K_{S}$ |  $>$17.85 mag | 4,6
F110W | 24.695$\pm$0.083 mag | 1
W1 | 18.182$\pm$0.189 magbbMagnitudes measured using the proper motion solution ($w1mpro\\_pm$ and $w2mpro\\_pm$), as these should be more accurate than those from the stationary solution. | 3
W2 | 16.145$\pm$0.084 magbbMagnitudes measured using the proper motion solution ($w1mpro\\_pm$ and $w2mpro\\_pm$), as these should be more accurate than those from the stationary solution. | 3
W3 |  $>$12.014 mag | 2
W4 |  $>$8.881 mag | 2
ch1 | 16.691$\pm$0.032 mag | 4,5
ch2 | 15.766$\pm$0.023 mag | 4,5
$J-$W2 |  $>$7.66 mag | 1
$J-$ch2 |  $>$8.03 mag | 1
$F110W-$W2 | 8.550$\pm$0.118 mag | 1
$F110W-$ch2 | 8.929$\pm$0.086 mag | 1
W1$-$W2 | 2.037$\pm$0.207 mag | 3
ch1$-$ch2 | 0.925$\pm$0.039 mag | 4,5
Derived Quantities
distance | 16.3${}^{+1.4}_{-1.2}$ pc | 1
$v_{\rm tan}$ | 207.4$\pm$15.9 km s-1 | 1
$M_{J}$ |  $>$22.7 mag | 1
$M_{F110W}$ | 23.636$\pm$0.19 mag | 1
$M_{W1}$ | 17.123$\pm$0.25 mag | 1
$M_{W2}$ | 15.086$\pm$0.19 mag | 1
$M_{ch1}$ | 15.632$\pm$0.17 mag | 1
$M_{ch2}$ | 14.707$\pm$0.17 mag | 1

Note. — Reference codes: (1) This Letter, (2) AllWISE Source Catalog - Cutri
et al. 2013; Kirkpatrick et al. 2014, (3) CatWISE2020 Catalog - Marocco et al.
2021, (4) Meisner et al. 2020, (5) Kirkpatrick et al. 2021, (6) VISTA
Hemisphere Survey, DR6.

Figure 2: Astrometric fit for WISE 1534$-$1043\. (Upper left) A square patch
of sky showing the measured positions and uncertainties at each epoch (black
points with error bars). The blue curve shows the best fit from the vantage
point of Spitzer, and the orange curve shows the best fit from the vantage
point of the Earth. Red lines connect each observation to its time on the
corresponding best-fit curve. (Upper right) A square patch of sky centered at
the mean equatorial position of the target. The green curve is the parallactic
fit, and the only observational data points plotted are those with the
smallest uncertainties from the previous panel, i.e., those from Spitzer
(filled blue squares) and HST (filled orange circle). Again, red lines connect
the time of observation with its prediction. Pale purple solid and dashed
lines mark lines of constant $\beta$ and $\lambda$, respectively, at
0$\farcs$1 spacing. (Lower left) RA and Dec as a function of time for the
Spitzer and HST points. The best fit parallax curve as seen from the vantage
points of Spitzer and HST is shown in blue and orange, respectively. The blue
curve ends in January 2020, the end of the Spitzer mission. (Lower right) The
RA and Dec residuals from the parallax fit as a function of time. Color coding
is the same as in previous panels.

##  3 Analysis

In Meisner et al. (2020), WISE 1534$-$1043 was labeled as a color outlier on
the $J-$ch2 vs. ch1$-$ch2 plot using a $J$-band magnitude limit of 20.56 mag
from follow-up observations using Palomar/WIRC. The new $\it HST$ detection
allows us to revisit this.

If we assume that WISE 1534$-$1043 is a typical cold brown dwarf, we can
convert between F110W and $J$ magnitudes. Using synthetic photometry on
thirteen high-S/N G102+G141 HST spectra of late-T and Y dwarfs from Schneider
et al. (2015), we find that F110W magnitudes are fainter than $J$ magnitudes
by 0.79 mag. This would imply that our F110W detection corresponds to
$J\approx 23.9$ mag. However, this is only 0.1 mag fainter than the measured
Keck $J$-band limit, meaning that the $Y$/$J$-band spectrum of WISE
1534$-$1043 likely differs from a typical late-T or Y dwarf. Because of this,
we choose to plot two points for WISE 1534$-$1043 in Figure 3. The first is
the $J$-band limit from Keck, and the second is the detection from HST, for
which we simply use the F110W magnitude as a proxy for $J$-band. As shown in
Figure 3 (left), WISE 1534$-$1043 is the sole occupant in its quadrant of the
color-color diagram: it is bluer by $\sim$1.5 mag in ch1$-$ch2 color than any
other object of similar $J-$ch2 color, and it is redder by at least 6 mag in
$J-$ch2 than the majority of objects with similar ch1$-$ch2 color.

Figure 3: Color-color and color-magnitude plots adapted from Figure 18 of
Kirkpatrick et al. (2021) and showing the 20-pc census of L, T, and Y dwarfs.
(Left) $J_{\rm MKO}-$ch2 color vs. ch1$-$ch2 color. (Right) Absolute $J_{\rm
MKO}$ magnitude vs. ch1$-$ch2 color. The locations of WISE 1534$-$1043 (orange
diamond), WISE 0855$-$0714 (green triangle), WISE 0830+2837 (recumbent pink
barbell), WISE 1828+2650 (standing yellow barbell), subdwarfs (blue squares),
and young objects (ruddy circles) are marked along with other L, T, and Y
dwarfs (black) within 20 pc of the Sun. There are two positions shown for WISE
1534$-$1043, as explained in the text. As a $J$-band limit for WISE 0830+2837,
we have used four Y1 dwarfs with G141 spectra from Schneider et al. (2015) to
measure an offset of F125W$-J\approx 0.87$ mag, and we apply this offset to
the F125W limit from Bardalez Gagliuffi et al. (2020). The young objects and
subdwarfs marked are those identified in Table 11 of Kirkpatrick et al.
(2021).

The unWISE- and Spitzer-based parallax in Kirkpatrick et al. (2021) had a
large (30%) relative uncertainty. The addition of the HST data point now
enables a parallax with a much smaller (7.7%) relative uncertainty, placing
the object securely within 20 pc. Comparing its absolute $J$ magnitude to
those of other L, T, and Y dwarfs within 20 pc (Figure 3, right), we find that
the object is the sole occupant in its quadrant of the color-magnitude
diagram. It is at least 8 mag fainter in $J$ than the bulk of objects of
similar ch1$-$ch2 color, and the $M_{J}$ value is comparable to that of Y1
dwarfs (see Figure 16a of Kirkpatrick et al. 2021).

##  4 Possible Interpretations

The unique position of WISE 1534$-$1043 on these diagrams makes deduction of
the object’s nature challenging. Given that the object is too faint at all
wavelengths for spectroscopy at any current facility, possible interpretations
must rely on diagrams like Figure 3, on which we try to recognize trends seen
among other known objects and to extend these into a terra incognita guided by
theoretical predictions. We consider four scenarios below.

###  4.1 Extremely low-metallicity (old) brown dwarf

As discussed in Meisner et al. (2020), the color abnormalities in this object,
even before a redder $J-$ch2 measurement was available, were reminiscent of
those of WISE J200520.38+542433.9 (hereafter, WISE 2005+5424; Mace et al.
2013). Shown as an outlying blue square on Figure 3, WISE 2005+5424 is an sdT8
companion to Wolf 1130, which has a metallicity of [Fe/H] = $-0.64{\pm}0.17$
as measured from the M subdwarf primary (Rojas-Ayala et al. 2012) and a much
bluer ch1$-$ch2 color (by $\sim$1 mag) than objects of comparable $J-$ch2 or
$M_{J}$ values.

Other WISE discoveries of unknown distance but occupying a similar locus of
color space as WISE 2005+5424 also show spectroscropic indications of low
metallicity (Schneider et al. 2020; Meisner et al. 2021). New atmospheric
models, called LOWZ, are presented in Meisner et al. (2021) and show that cold
brown dwarfs tend to drift bluer in ch1$-$ch2 and redder in $J-$ch2 at a fixed
temperature as metallicity decreases, which fits the trend expected if WISE
1534$-$1043 is a more metal-poor subdwarf than WISE 2005+5424. The bluer
ch1$-$ch2 color may indicate a low relative abundance of methane, as expected
in a low-Z atmosphere. Lower metallicity would also result in lower
atmospheric opacity, meaning that the visible photosphere would extend deeper
into the atmosphere where the pressure is higher. Because collision induced
absorption by H2 is predicted to be strongest in the near-infrared $JHK$ bands
(Borysow et al. 1997), this would suppress the $J$ flux and drive the $J-$ch2
color redward.

The LOWZ models that best match the position of WISE 1534$-$1043 in the $J-$W2
vs. W1$-$W2 diagram suggest [M/H] $\approx-2.0$ and $T_{\rm eff}<500$K,
pushing it possibly into the Y dwarf sequence. If correct, this would make
WISE 1534$-$1043 the first Y subdwarf. However, finding an object with a
metallicity of $-2.0$ is extremely unusual at this distance. Based on a
collection of the nearest, brightest K and M subdwarfs in the northern two-
thirds of the sky (Kesseli et al. 2019), we find that some objects in the
20-pc census (e.g., sdM3 LHS 272 and the aforementioned Wolf 1130) have
metallicities near $-1$, but more metal-poor objects are unlikely to be found
with 20 pc. At larger distances, for example, Kesseli et al. (2019) measure a
value of [Fe/H] = $-1.54{\pm}0.30$ for the usdM1 LHS 364, at 28 pc. It seems
implausible, then, that WISE 1534$-$1043 would have a value as low as [M/H] =
$-2$. Also, we note that the LOWZ models (Figure 7 of Meisner et al. 2021)
predict a value of $-1.5<$[M/H] $<-1.0$ for WISE 2005+5424, which is much
lower than the measured $-0.64{\pm}0.17$ value. Hence, the metallicity of WISE
1534$-$1043 is likely to be less extreme than the LOWZ models predict.

The low-metallicity hypothesis, which suggests old age, is also supported by
the tangential velocity, 207.4$\pm$15.9 km s-1 (Table 1), which exceeds by
$\sim$50 km s-1 the highest velocity known amongst the remainder of the 20-pc
L, T, and Y dwarf census (Figure 33 of Kirkpatrick et al. 2021). Figure 4 is a
Toomre diagram (Sandage & Fouts 1987) showing the space motions of Gaia DR2
stars within the 100-pc volume and having well measured parallaxes. Using the
kinematic criteria of Nissen (2004), objects are coded as thin disk ($V_{\rm
tot}\leq 85$ km s-1), thick disk ($85<V_{\rm tot}\leq 180$ km s-1), or halo
($V_{\rm tot}>180$ km s-1). For all values of its (unknown) radial velocity,
WISE 1534$-$1043 is suggested to be a halo member despite the fact that only
0.15% of objects in the solar neighborhood are halo members (Table A1 of
Bensby et al. 2014).

Figure 4: Toomre diagram of $UVW$ space motions corrected to the Local
Standard of Rest for 74,066 Gaia DR2 stars within 100 pc of the Sun and having
parallax errors $<10\%$. Thin disk (blue dots), thick disk (black crosses),
and halo (green circles) objects are marked, per the text. WISE 1534$-$1043
(orange diamonds) is shown at thirteen radial velocities ranging from $-300$
to 300 km s-1, in steps of 50 km s-1. All values suggest a rare member of the
halo.

###  4.2 Extremely low-mass, low-gravity (young) brown dwarf

Young objects within the 20-pc sample are highlighted by red circles on Figure
3. These show signs of low gravity in their spectra and/or are associated with
young moving groups (see Section 7.1 of Kirkpatrick et al. 2021), indicating
that they have not yet contracted to their final, equilibrium radii (Faherty
et al. 2016). A small population with ch1$-$ch2 $<$ 0.5 mag have much redder
$J-$ch2 colors than the others. It is believed that in such objects, grains
continue to amass in the photosphere as they cool because settling is unable
to clear the upper atmosphere of condensates (Marley et al. 2012) at the same
temperatures where this occurs in the older, field dwarf population
($\sim$1250K; Figure 22 of Kirkpatrick et al. 2021). As sufficiently cold
$T_{\rm eff}$ values are reached, the rainout will eventually commence,
pushing the L/T transition to a colder temperature.

Under this scenario, WISE 1534$-$1043 would be a young brown dwarf that still
has not crossed into the early-T dwarfs, although it would be $\sim$4.5 mag
redder in $J-$ch2 color than any other young L dwarf yet known. The ch1$-$ch2
color of $\sim$1 mag indicates that a considerable amount of methane opacity
is already present in the photosphere, which may be difficult to reconcile
with a young, low-gravity hypothesis since the conversion of CO to CH4 will
also be pushed to lower values of Teff (Zahnle & Marley 2014). Moreover, such
a young object would not yet be kinematically heated and thus would be very
unlikely to have a value of $v_{\tan}$ as large as that measured for WISE
1534$-$1043 (e.g., Figure 17 of Gonzales et al. 2019).

###  4.3 Ejected exoplanet

The two scenarios above assume WISE 1534$-$1043 is a brown dwarf formed via
the star formation process. Instead, what if it is an exoplanet formed out of
a protoplanetary disk and stripped later from its host system? Beichman et al.
(2013) posited this scenario as one possible explanation of the unusual
features of WISE 1828+2650. The giant planets of our solar system are composed
of material more metal rich than the Sun itself, showing that their formation
process involves elemental segregation. Carbon, as one example, is enhanced in
Jupiter by 3$\times$ the solar value (Owen et al. 1999), in Saturn by
7$\pm$2$\times$ (Flasar et al. 2005), and in Uranus and Neptune by 32$\times$
and 41$\times$, respectively (Lodders & Fegley 1994). Under this hypothesis,
WISE 1534$-$1043 is photometrically unusual because such elemental differences
would profoundly affect its atmospheric composition and emergent spectrum.
Unfortunately, forward modeling that incorporates a wide array of elemental
abundance differences does not yet exist, so our best method to test this
hypothesis is atmospheric retrieval, once a suitable spectrum for WISE
1534$-$1043 is obtained.

The high $v_{\rm tan}$ measured for this object is not at odds with the
ejected exoplanet hypothesis, but studies have found that the population of
short timescale, single events observed in microlensing searches can be
attributed to planets in wide orbits (Mróz et al. 2017, 2019) rather than to a
significant population of solivagant exoplanets. In addition, the "excess"
short-timescale microlensing events – those that cannot be attributed to the
distribution of stars and brown dwarfs – correspond to masses on the scale of
1 MEarth. Even if we assume that WISE 1534$-$1043 is an extremely young planet
($\sim$10 Myr) that has recently escaped and is still relatively bright, model
predictions (Burrows et al. 1997) suggest a mass of $\sim$0.3 MJup ($\sim$100
MEarth), far more massive than the bodies causing the short-timescale
microlensing events. Assuming an older age only drives this mass upward.
Having a rare, "high" mass, escaped planet falling within 20 pc of the Sun
appears to be unlikely.

###  4.4 Ultra-cold stellar remnant

Is it possible for WISE 1534$-$1043 to be a stellar remnant? Assuming a
blackbody with a typical white dwarf radius of $\sim$7000 km would require
$T_{\rm eff}\approx 1950$K to match the W2 flux and distance measured for WISE
1534$-$1043\. However, a 1950K blackbody would have its peak flux near
1.5$\mu$m, which does not match the photometry of WISE 1534$-$1043\. Although
white dwarfs below $\sim$4000K have strong collision-induced absorption,
reshaping their spectral energy distributions markedly away from a blackbody
(Lenzuni et al. 1991), this cannot explain the colors seen in WISE 1534$-$1043
(e.g., Blouin et al. 2017.) Besides, theoretical models predict that it is not
possible for a white dwarf to have cooled to this $T_{\rm eff}$ considering
the age of the Milky Way (e.g., Calcaferro et al. 2018).

WISE 1534$-$1043 could be a more exotic, cold stellar remnant – a bare core
stripped through ablation (Ray & Loeb 2017) and/or mass transfer (Hernández
Santisteban et al. 2016). For it to exist as an isolated object now, it would
had to have been ripped from its ablating source through a gravitational
encounter with a third object. Finding such a rare, ejected object within 20
pc of the Sun is highly improbable.

##  5 Conclusions

We have better constrained the $J-$ch2 color, measured a robust parallax, and
derived absolute magnitudes for WISE 1534$-$1043\. These results were obtained
through new $J$-band imaging from Keck that fails to detect the object and
even deeper F110W imaging from HST that reveals the source at S/N $\approx$
5\. On the resultant color-color and color-magnitude diagrams, it falls in a
region that contains no other known objects.

We conclude that the unique object WISE 1534$-$1043 is most likely a cold,
very metal poor brown dwarf – perhaps even the first Y-type subdwarf – given
its red $J-$ch2 color, its high tangential velocity, comparison to the
photometric signatures seen in other low-metallicity objects, and trends
suggested by theoretical predictions at low Z and low $T_{\rm eff}$.
Verification, refutation, or further befuddlement should be possible via
additional photometry and broad-wavelength spectroscopy from the James Webb
Space Telescope.

This research is based on observations made under program 16243 with the
NASA/ESA Hubble Space Telescope obtained from the Space Telescope Science
Institute, which is operated by the Association of Universities for Research
in Astronomy, Inc., under NASA contract NAS 5–26555. Some of the data
presented here were obtained at the W. M. Keck Observatory, which is operated
as a scientific partnership among the California Institute of Technology, the
University of California, and the National Aeronautics and Space
Administration. The Observatory was made possible by the generous financial
support of the W. M. Keck Foundation. The authors wish to recognize and
acknowledge the very significant cultural role and reverence that the summit
of Maunakea has always had within the indigenous Hawaiian community. We are
most fortunate to have the opportunity to conduct observations from this
mountain. This publication makes use of data products from WISE/NEOWISE, which
is a joint project of UCLA and JPL/Caltech, funding by NASA. Portions of this
research were carried out at JPL/Caltech under contract with NASA. JKF thanks
the Heising Simons Foundation for research support.

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*

Copyright for this paper by its authors. Use permitted under Creative Commons
License Attribution 4.0 International (CC BY 4.0).

dl4kg2023 @ ISWC: Workshop Deep Learning for Knowledge Graphs, November
6th-7th, 2023, Athen, Greece

<EMAIL_ADDRESS>[orcid=0000-0001-5260-5181<EMAIL_ADDRESS>[1]
[orcid=0000-0002-8130-8677] [orcid=0009-0008-5245-6655]
[orcid=0000-0002-1459-3754]

[1]Corresponding author. [1]These authors contributed equally.

# Benchmarking the Abilities of Large Language Models for RDF Knowledge Graph
Creation and Comprehension: How Well Do LLMs Speak Turtle?

Johannes Frey  Lars-Peter Meyer  Natanael Arndt  Felix Brei  Kirill Bulert
Institute for Applied Informatics, Goerdelerring 9, 04109 Leipzig, Germany,
https://infai.org Agile Knowledge Engineering and Semantic Web (AKSW),
https://aksw.org Leipzig University, Institute for Informatics, Germany,
https://www.uni-leipzig.de eccenca GmbH, Leipzig, Germany, https://eccenca.com

(2023)

###### Abstract

Large Language Models (LLMs) are advancing at a rapid pace, with significant
improvements at natural language processing and coding tasks. Yet, their
ability to work with formal languages representing data, specifically within
the realm of knowledge graph engineering, remains under-investigated. To
evaluate the proficiency of various LLMs, we created a set of five tasks that
probe their ability to parse, understand, analyze, and create knowledge graphs
serialized in Turtle syntax. These tasks, each embodying distinct degrees of
complexity and being able to scale with the size of the problem, have been
integrated into our automated evaluation system, the LLM-KG-Bench. The
evaluation encompassed four commercially available LLMs - GPT-3.5, GPT-4,
Claude 1.3, and Claude 2.0, as well as two freely accessible offline models,
GPT4All Vicuna and GPT4All Falcon 13B. This analysis offers an in-depth
understanding of the strengths and shortcomings of LLMs in relation to their
application within RDF knowledge graph engineering workflows utilizing Turtle
representation. While our findings show that the latest commercial models
outperform their forerunners in terms of proficiency with the Turtle language,
they also reveal an apparent weakness. These models fall short when it comes
to adhering strictly to the output formatting constraints, a crucial
requirement in this context.

###### keywords:

Large Language Model Knowledge Graph Engineering Large Language Model
Benchmark

##  1 Introduction

Large Language Models have gained significant attention in recent years, with
GPT-4 being among the most prominent [1]. However, other models also
demonstrate impressive performance in various tasks, as tracked in the LLMSYS
Chatbot Arena Leaderboard111https://huggingface.co/spaces/lmsys/chatbot-arena-
leaderboard/tree/a068e66fdd6f453812b307541e8c82f99472aabe .

In the field of Knowledge Graph Engineering (KGE) the overarching task is to
structure knowledge and encode it in a machine processable format. Using
machine learning techniques to create or process knowledge graphs is a well
researched topic that receives new momentum. The report of the Dagstuhl
Seminar 22372 [2] and the Knowledge Base Construction from Pre-trained
Language Models (LM-KBC) Challenge222Website: https://lm-
kbc.github.io/challenge2023/ emphasize the relevance of this topic. Pan et al.
[3] outline the potential of connecting LLMs and KGs and in particular with
“LLM-augmented KGs”.

RDF - the Resource Description Framework - serves as a standard for
representing Knowledge Graphs, while Turtle, a textual representation, is
widely used to store and exchange RDF data. We have opted for Turtle given its
strong resemblance to natural language, which aligns well with the primary
mode of interaction with LLMs.

In previous works, we have conducted manual experiments [4] and introduced the
framework _LLM-KG-Bench_ [5] for automated benchmarking of LLM performance on
KGE tasks.

In this paper, we expand upon that work by introducing two new tasks to the
_LLM-KG-Bench_ framework and evaluate the ability of various models to shed
light on the question “how well do LLMs speak Turtle”, i.e. parse, comprehend,
analyze, create and serialize RDF knowledge graphs using Turtle serialization.
Besides Claude 1.3-100k, GPT-3.5 Turbo, and GPT-4, we also include Claude 2 in
our evaluation and have extended the framework’s capabilities to allow for
benchmarking on a variety of freely available offline LLM models using
GPT4All. We have selected GPT4All Vicuna as a non-commercially usable model
and Falcon 13B as the top freely available commercially usable model (Apache
2.0 license) to be additionally included.

After describing related work in the next section, we introduce the benchmark
tasks in section 3. In section 4, we explain the study setup, present the
evaluation and discuss the strengths and weaknesses of the individual LLMs
regarding their utilization for RDF KGE workflows using Turtle. We conclude
with a discussion and outline future work in section 5.

##  2 Related Work

There are several evaluations and articles discussing the utilization of LLMs
for KG related tasks333Repository: https://github.com/zjukg/KG-LLM-Papers/,
e.g. [6, 7, 8, 9, 4]. Some of them include references to code for reproducing
the results. These works cover topics like, KG construction supported by LLMs,
KG reasoning supported by LLMs or question answering based on KGs and LLMs.

In the field of generic LLM benchmarking, the _BigBench_ framework [10] offers
a robust structure and collects already a large list of automated benchmark
tasks, but public code for integrating current LLMs like GPT or Claude is
missing. The Large Model Systems (LMSys) leaderboard [11] is built mainly on
manual testing and evaluation by the community. There is also the Language
Model Evaluation Harness [12] which tests Open Source LLMs on a variety of
reasoning and logic tasks, but none are related to knowledge graphs.

In the scope of this paper, we are focusing on the definition of automated
evaluation tasks testing KGE related capabilities using turtle syntax for RDF-
based KGs. We decided to use the KGE specific _LLM-KG-Bench_ framework[5],
which is compatible with _BigBench_ , but adds KGE specific helpers, includes
connectors for current LLMs like Claude and GPT, and supports tasks that can
scale in problem sizes.

##  3 Benchmark Tasks

To evaluate the capabilities of LLMs, we created five tasks with focus on the
ability to parse, understand, analyze, and create knowledge graphs using the
Turtle serialization format.

The tasks T2 _TurtleErrorsStatic_ (section 3.2), T3 _TurtleSampleGeneration_
(section 3.3), and T5 _FactExtractStatic_ (section 3.5) are extended versions
of the tasks described in [5], while the tasks T1
_TurtleConnectionExplainStatic_ (section 3.1) and T4 _TurtleFriendCount_
(section 3.4) are newly introduced in this paper.

The tasks are executed in two different manners, T1, T2, and T5 are executed
as static tasks, i.e. with a fixed prompt size and fixed expected responses,
while T3 and T4 are scalable in problem size (i.e. given prompt or expected
response length) using an estimated byte limit parameter. The byte limit can
be used by the scalable tasks to calculate a task specific problem size
(number of persons in the case of T3 and T4) to approximate that byte limit.

Task T1, T2 and T5 were executed 20 times per model. While the benchmark tasks
report a variety of metrics and info or debug scores, we report the F1
measures for these tasks for a unified comparison in the scope of this work
(shown in Figure 1). The scalable tasks T3 and T4 were executed 20 times per
combination of size and model for 8 different sizes. Byte limit and resulting
task problem sizes are depicted in table 1.

Table 1: Configured byte limit and resulting task problem sizes

Byte Limit | No. Persons Task T3 | No. Persons Task T4
---|---|---
1000 | 10 | 6
2000 | 20 | 16
⋮ | ⋮ | ⋮
8000 | 80 | 76

###  3.1 Task T1: Find Connection in Small Turtle File

To check basic support for knowledge graphs and Turtle syntax, we implemented
the _TurtleConnectionExplainStatic_ task similar to the first manual
experiment in our previous work [4].

Prompt 1: For the following turtle find the shortest non trivial connection
from Anne to Bob. Please summarize the connection with just a list of resource
IRIs, one per line, starting with https://abc.def/ghi/anne and ending with
https://abc.def/ghi/bob . Please leave out rdf:type infos, leave out
explanatory text and answer with just the IRI lines.

[… followed by the graph in section 3.1]

⬇

PREFIX : <https://abc.def/ghi/>

PREFIX rdfs: <http://www.w3.org/2000/01/rdf-schema#>

PREFIX owl: <http://www.w3.org/2002/07/owl#>

PREFIX foaf: <http://xmlns.com/foaf/0.1/>

PREFIX vcard: <http://www.w3.org/2006/vcard/ns#>

PREFIX org: <http://www.w3.org/ns/org#>

\par:anne a foaf:Person ; foaf:firstName "Anne" ; foaf:surname "Miller" ;

vcard:hasAddress [ a vcard:Home ; vcard:country-name "UK" ] .

:bob a foaf:Person ; foaf:firstName "Bob" ; foaf:surname "Tanner" ;

vcard:hasAddress [ a vcard:Home ; vcard:country-name "US" ] .

\par:wonderOrg a org:Organization .

:researchDep a org:OrganizationalUnit ; org:unitOf :wonderOrg ;

rdfs:label "Research Department" .

:marketingDep a org:OrganizationalUnit ; org:unitOf :wonderOrg ;

rdfs:label "Marketing Department" .

\par:chiefResearchOfficer a org:Role . :marketingManager a org:Role .

\par[ a org:Membership ; org:member :anne ; org:organization :researchDep ;

org:role :chiefResearchOfficer ] .

[ a org:Membership ; org:member :bob ; org:organization :marketingDep ;

org:role :marketingManager ] .

An organizational KG with two people working in different departments of the
same organization. Graph taken from [4].

In section 3.1, we provide a small organizational graph (see section 3.1) and
ask for the shortest non trivial connection (excluding the one via foaf:Person
type statement) between the two nodes _:Anne_ and _:Bob_. By finding the
connection
$anne\allowbreak\,\raisebox{-3.01385pt}{$\xleftarrow{org:member}$}\,\allowbreak
bnode1\allowbreak\,\raisebox{-3.01385pt}{$\xrightarrow{org:organization}$}\,\allowbreak
researchDep\allowbreak\,\raisebox{-3.01385pt}{$\xrightarrow{org:unitOf}$}\,\allowbreak
wonderOrg\allowbreak\,\raisebox{-3.01385pt}{$\xleftarrow{org:unitOf}$}\,\allowbreak
marketingDep\allowbreak\,\raisebox{-3.01385pt}{$\xleftarrow{org:organization}$}\,\allowbreak
bnode2\allowbreak\,\raisebox{-3.01385pt}{$\xrightarrow{org:member}$}\,\allowbreak
bob$ between them, the LLM demonstrates basic graph handling capabilities.
Note, that we ask the response output to be a list of resource/node IRIs
without any other text, to support the automated evaluation of the answer.
This also excludes both blank nodes and leads to the list
$anne,researchDep,wonderOrg,marketingDep,bob$. We use similar output
requirements in most tasks and argue that a strict adherence to task details
and output format instructions is a necessary capability for using LLMs as
part of a tool chain in KGE tasks or workflows. The task computes recall,
precision, and F1 measure for the list of IRIs mentioned in the model response
with regard to the list of IRIs representing the nodes of the shortest path.

###  3.2 Task T2: Find Errors in Small Turtle File

Prompt 2: Please check the following rdf turtle file for errors and answer
with no text but just the corrected turtle. Try to stick to the original
formatting. [… followed by the turtle document to check]

The _TurtleErrorsStatic_ task involves identifying and correcting syntax
errors in a small Turtle file, which is based on the same organizational graph
(section 3.1) with minor modifications. The turtle file has a period missing
at the end of line 9 and the first semicolon in line 16 was removed.
Correcting the errors demonstrates the LLM’s knowledge of Turtle grammar while
also showing its ability to transform it into a proper form without altering
existing facts and adhering strictly to the task requirements. One of the
scores calculated during evaluation is the F1 measure on parsable, normalized
triples, comparing the LLM’s answer with the perfect answer. In order to do
so, the response is directly consumed by `rdflib` in combination with an
iterative parsing failure heuristic. The heuristic removes invalid lines that
are reported as source of a syntax error until the document is fully parsable
or empty.

###  3.3 Task T3: Create Sample Graphs

We created the task _TurtleSampleGeneration_ to see if LLMs can understand and
honor the requirements that we postulate for creating a simple knowledge
graph, i.e. the number of resources in the graph as well as its structure. The
task makes use of the popular `FOAF` vocabulary because we assume that members
of it are very prevalent in the training data, since they are frequently used
in example snippets in online forums and datasets.

Prompt 3: Create a knowledge graph in turtle format that consists of $n$
different objects of type `foaf:person`. Each should have at least 2 and at
most $n-1$ connections to other persons via `foaf:knows`. Give no extra text.

In section 3.3 we instruct the LLM to generate a knowledge graph with a
certain number of persons who each have between two and $n-1$ friends (both
inclusive). The number of persons $n$ can be varied to get different answer
sizes. The task is motivated by the idea of using LLMs to generate test,
training, or example data of various sizes. Furthermore, it allows to study
the capacity of the models to generate content of increasing sizes while
maintaining the integrity of the graph and serialization. This task checks
first if the generated answer is parsable. If so, the structure of the graph
is evaluated. We count the number of resources in the graph and if they are
all correctly declared as `rdf:type` `foaf:Person`. The
_persons_relative_error_ scores measures the difference between the actual
number of person objects generated and the number asked for. This value is
normalized to be $=0$ if they match, $>0$ if there are more persons than asked
for and $<0$ if there are less persons, with the special case of $-1$ meaning
an empty graph.

###  3.4 Task T4: Count Links in Person Graph

Prompt 4: Please name the person which is known by the most persons according
to the following RDF graph in turtle syntax. Give just the IRI of this person
with most incoming links as answer, without abbreviation or explanation. […
followed by the graph serialization]

The task _TurtleFriendCount_ requires finding the resource with the most
incoming links in a simple generated KG. The structure of the KG is similar to
the previous tasks, consisting of a variable number of `foaf:Person` resources
connected by `foaf:knows` properties. Each person is known by two other
persons, but one designated `foaf:Person` is known by two additional persons
(one for small sizes), resulting in three to four incoming links instead of
two. This task tests basic RDF and turtle knowledge as well as graph
comprehension and processing skills by aggregating link counts for various KG
sizes. The number of `foaf:Person` resources in the prompt has a linear
correlation with the prompt length. The task computes recall, precision, and
f1 measure with respect to the expected person IRI.

###  3.5 Task T5: Create Knowledge Graph from Factsheet

Prompt 5: As a Linked Data expert and knowledge engineer please convert the 3d
printer specification given in the bottom into an RDF turtle formatted
Knowledge Graph. The main subject in this Knowledge Graph should be the
Printer itself using the https://data.semper-ki.org/resources/$encoded_label$
whereas $encoded_label$ label refers to the name of the printer in which
whitespaces have been replaced with underscores. Step by step attach the
following statements to that node.
1) Attach https://purl.org/tema/051993 as rdf:type
2) Attach the manufacturer via an object property using the schema.org
vocabulary. For the subject identifier of the manufacturer use the same
approach as for the printer subject and assign it the rdfs:label without a
language tag.
3) Attach all printable material as well all support materials the printer can
use via object properties. For the property identifiers use
https://data.semper-ki.org/properties/printMaterial respectively
https://data.semper-ki.org/properties/supportMaterial and for material
identifiers use the https://data.semper-ki.org/resources/materials/ namespace.
4) Attach the dimensions of the printer as 3 separate values using the
schema.org vocabulary for the property identifiers of the dimensions, but for
the values use qudt:QuantityValue objects from the QUDT ontologies family to
have the numerical value ( via qudt:numericValue property and typed as
xsd:double) and the appropriate QUDT unit of measurement identifier (via
qudt:hasUnit property) separately. Do not convert the dimensions into another
unit.
Follow best practices of Linked Data and common standard vocabularies as basis
for the modeling unless specified differently in the requirements above. Do
not extract any other properties or values from the specification than the
ones I mentioned but validate your output step by step, to check whether all
used prefixes are included and the given requirements as well as the grammar
of the RDF turtle serialization were respected. Only return the turtle format,
no additional text. [… followed by the fact sheet plaintext excerpt]

The task _FactExtractStatic_ assesses the LLM’s fact extraction and advanced
RDF modeling abilities, by utilizing a plaintext excerpt from one of our
previous experiments [4]. The excerpt (that is not shown for reasons of
brevity) describes various aspects of the specifications of a 3D printer in a
key-value format, including the formatting irregularities commonly found in
PDF extracts. We ask the model to generate a Turtle file that captures a
subset of this information to check for how well RDF facts can be extracted
from factsheet plaintexts and transformed into a knowledge graph. The prompt
is carefully designed with regard to the transformation aspect in order to be
very specific and unambiguous on how the data should be represented. The
prompt defines concrete namespace schemes and construction rules for IRI
identifiers for properties and subjects, but also challenges knowledge about
ontology members by requesting the use of concrete ontologies. Subsequently,
we can evaluate the quality of a single response using the F1 measure,
counting the set of parsable triples that (mis)match or are missing compared
to a manually curated reference document. Since we consider this as a quite
challenging task, we make use of multiple prompt engineering optimization
techniques, namely, asking for an expert answer in a domain context, providing
step by step instructions, and asking for critical self-evaluation.

##  4 Benchmark Study Results and Discussion

Using the _LLM-KG-Bench_ framework, we configured the 5 aforementioned
benchmark tasks (cf. section 3) to be evaluated for the 3 highest ranking LLMs
at the LLMSYS Chatbot Arena Leaderboard1 i.e. GPT-4 (gpt-4-0613), GPT-3.5
(gpt-3.5-turbo-0613), and Claude-1.3 (claude-1.3-100k), additionally we have
included Claude-2.0. These 4 systems were evaluated using the commercial APIs
of OpenAI and Anthropic. We also wanted to include freely available offline
LLMs. Based on the availability in GPT4All, we have selected GPT4All
Vicuna-13B (version ggml-vicuna-13b-1.1-q4_2.bin) as a non-commercially usable
representant and GPT4All Falcon (ggml-model-gpt4all-falcon-q4_0.bin) as the
top freely available commercially usable model (Apache 2.0 license).

(a) T1 - Connected Path

(b) T2 - Turtle Fixing

(c) T5 - Fact Extraction

Figure 1: Evaluation of Static Tasks: Distribution of F1 scores

(a) T3 - Person Graph Creation

(b) T4 - Friends Counts

Figure 2: Evaluation of Scalable Tasks: Mean of task metric with 95%
confidence interval

T1: As can be seen in Figure 1(a), Claude-2.0 answers perfectly. GPT-4 added
occasionally properties to the list, although only nodes were requested.
Claude-1.3 and GPT-3.5 seem to omit sometimes a resource from the requested
list. Falcon’s lists often contain only the first and last resource which was
already specified in the prompt, but a few attempts pose some basic
understanding of the task, but severely violate the output constraints.
Vicuna‘s responses are mostly in the form of "This is the shortest connection
from anne to bob".

T2: GPT-3.5 often claims that the given turtle file would be correct and
returns no turtle. This explains the high rate of zeros as F1 score (see
figure 1(b)) The answers given by Claude-1.3 and GPT-4 score better. Claude
2.0 fails in contrast to its predecessor in returning the plain turtle,
leading to unparseable documents although the errors seem fixed. Vicuna
replies with the empty string in all cases. While Falcon reports there would
be no errors in a few cases, it often does not follow the task and replies
with Turtle snippets or explanations of the content.

T3: Claude-1.3 clearly performed worst from the commercial models (Fig. 2(a)).
Common issues leading to missing persons for smaller sizes were omitting
rdf:type statements and missing prefix definitions (making the entire document
unparseable). For bigger sizes, Claude 1.3 additionally used ellipses with
comments like "continues with 70+ more persons" making it impractical for
sample generation. Claude 2.0 shows similar ellipses but sovereignly handles
types and prefixes, leading to the best results for small and medium sizes,
but bad ones for large sizes. GPT-4 has only a slightly better performance
compared to GPT-3.5 but in contrast uses ellipses much more frequently for
higher sizes (60-80 persons). Falcon and Vicuna scored worse. Vicuna creates
very few persons but then uses ellipses. However, it is missing the essential
rdf:type statement for size 10. Falcon just creates a list of prefix
declarations.

T4: All commercial models seem to be challenged by the size of 6 persons as
indicated in Figure 2(b), where the person with the most friends only has one
more friend compared to all others. GPT-3.5 consistently proposes the person
with the most outgoing friends as solution instead. For the other models this
also happens very frequently. Surprisingly, this potential misunderstanding
seems to occur significantly less frequent for all other sizes where the the
person with the most friends has 4 incoming relations. In fact, GPT-4 is
performing in an outstanding way, only doing one mistake for the largest size.
Claude-2.0 performs similar to GPT-3.5, both confuse one of the persons with
more outgoing links as correct solution, although interestingly GPT-3.5 also
answers often with a full sentence (instead the IRI only) when it has such a
confusion. Claude 1.3 also frequently confuses ingoing with outgoing links,
and similarly to GPT 3.5 this correlates with violating the output constraint,
but even stronger. Falcon and Vicuna seem to understand the task but have
incorrect reasoning. Falcon consistently answers with a wrong non-reasonable
person and textual explanations. Vicuna surprisingly identifies the correct
person for the smallest size, but fails for bigger sizes. Moreover the context
windows are exceeded for sizes greater than 26 persons for Vicuna and 36
persons for Falcon.

T5: Figure 1(c) shows that both GPT models outperform Claude 1.3 in this task.
While GPT4 has a better mean, due to one very good response (F1 score of
0.94), it however replied frequently with unparseable content, which in turn
did not happen for GPT3.5, leading to a slightly better median for that.
Claude 2.0 shows the highest values for F1 mean as well as the third quartile
and it returned less unparseable documents compared to Claude 1.3 and GPT4.
Vicuna did not return output that has any similarity with Turtle. Falcon
creates turtle preambles with varying prefix definitions but had problems to
continue the document and seemed very often stuck in repetitive namespace
gibberish and prefix patterns.

##  5 Conclusion and Future Work

The evaluation shows already promising results. Especially the newer versions
of both GPT and Claude speak turtle already at a level that it might be useful
for assistant tasks. A general problem is though, that the models, although
explicitly requested, do not consistently respond with plain Turtle but
include short explanations or markdown ticks. While our parsing failure
heuristic can mitigate some of these cases, this issue poses a challenge when
interfacing directly with RDF tools. It is noteworthy, that the newer versions
of both GPT and Claude tend to violate the output constraints more often.
While our failure tolerant parser heuristic allows to get more insights into
the quality of results, it can also reward solutions to some degree that might
not be useful without special post processing in practical scenarios.
Therefore, we see as a next step to define tests that are stricter, however
provide feedback to the models (e.g. parsing errors) to perform and evaluate
few-shot approaches. Moreover, it could be of value to assess the performance
using ntriples, which has less syntactical features but allows easier
retrieval of partially inconsistent responses. It also remains to be seen
whether finetuning LLMs on RDF syntax using large datasets like Wikidata and
DBpedia would be beneficial. Finally besides extending the framework with more
tests, we see integrating LangChain to study the combination of LLMs with KGE-
assistant plugins (e.g. prefix or ontology terminology lookup service) as an
interesting path to explore.

###### Acknowledgements.

This work was partially supported by grants from the German Federal Ministry
of Education and Research (BMBF) to the projects StahlDigital (13XP5116B) and
KupferDigital (F13XP5119F) as well as from the German Federal Ministry for
Economic Affairs and Climate Action (BMWK) to the CoyPu project (01MK21007A)
and KISS project (01MK22001A).

## References

  * Bubeck et al. [2023] S. Bubeck, V. Chandrasekaran, R. Eldan, J. Gehrke, E. Horvitz, E. Kamar, P. Lee, Y. T. Lee, Y. Li, S. Lundberg, H. Nori, H. Palangi, M. T. Ribeiro, Y. Zhang,  Sparks of artificial general intelligence: Early experiments with gpt-4 (2023). arXiv:2303.12712.
  * Groth et al. [2023] P. Groth, E. Simperl, M. van Erp, D. Vrandečić,  Knowledge graphs and their role in the knowledge engineering of the 21st century (dagstuhl seminar 22372) (2023). URL: https://www.dagstuhl.de/en/seminars/seminar-calendar/seminar-details/22372. doi:10.4230/DAGREP.12.9.60.
  * Pan et al. [2023] S. Pan, L. Luo, Y. Wang, C. Chen, J. Wang, X. Wu,  Unifying large language models and knowledge graphs: A roadmap (2023). arXiv:2306.08302.
  * Meyer et al. [2023a] L.-P. Meyer, C. Stadler, J. Frey, N. Radtke, K. Junghanns, R. Meissner, G. Dziwis, K. Bulert, M. Martin, Llm-assisted knowledge graph engineering: Experiments with chatgpt, 2023a. arXiv:2307.06917, to appear in conference proceedings of AI-Tomorrow-23, 29.+30. 6.2023 in Leipzig, Germany.
  * Meyer et al. [2023b] L.-P. Meyer, J. Frey, K. Junghanns, F. Brei, K. Bulert, S. Gründer-Fahrer, M. Martin, Developing a scalable benchmark for assessing large language models in knowledge graph engineering, 2023b. arXiv:2308.16622, to appear in poster proceedings of Semantics-23, 20-22. 9. 2023, Leipzig, Germany.
  * Trajanoska et al. [2023] M. Trajanoska, R. Stojanov, D. Trajanov,  Enhancing knowledge graph construction using large language models (2023). arXiv:2305.04676.
  * Carta et al. [2023] S. Carta, A. Giuliani, L. Piano, A. S. Podda, L. Pompianu, S. G. Tiddia,  Iterative zero-shot llm prompting for knowledge graph construction (2023). arXiv:2307.01128.
  * Zhu et al. [2023] Y. Zhu, X. Wang, J. Chen, S. Qiao, Y. Ou, Y. Yao, S. Deng, H. Chen, N. Zhang, Llms for knowledge graph construction and reasoning: Recent capabilities and future opportunities, 2023\. arXiv:2305.13168.
  * Guo et al. [2023] J. Guo, L. Du, H. Liu, M. Zhou, X. He, S. Han,  Gpt4graph: Can large language models understand graph structured data ? an empirical evaluation and benchmarking (2023). arXiv:2305.15066.
  * Srivastava et al. [2023] A. Srivastava, et al.,  Beyond the imitation game: Quantifying and extrapolating the capabilities of language models,  Transactions on Machine Learning Research (2023). arXiv:2206.04615.
  * Zheng et al. [2023] L. Zheng, W.-L. Chiang, Y. Sheng, S. Zhuang, Z. Wu, Y. Zhuang, Z. Lin, Z. Li, D. Li, E. P. Xing, H. Zhang, J. E. Gonzalez, I. Stoica, Judging llm-as-a-judge with mt-bench and chatbot arena, 2023\. arXiv:2306.05685.
  * Gao et al. [2021] L. Gao, J. Tow, S. Biderman, S. others Black, A. DiPofi, C. Foster, L. Golding, J. Hsu, K. McDonell, N. Muennighoff, J. Phang, L. Reynolds, E. Tang, A. Thite, B. Wang, K. Wang, A. Zou, A framework for few-shot language model evaluation, 2021. doi:10.5281/zenodo.5371628.

##  Appendix A Online Resources

  * •

Source Code Repository: https://github.com/AKSW/LLM-KG-Bench or
doi:10.5281/zenodo.8366061

  * •

Study Data: https://github.com/AKSW/LLM-KG-Bench-
Results/tree/main/2023-DL4KG_Turtle-KG-Eval or doi:10.5281/zenodo.8364535


# Gegenbauer kernel filtration on the unit hypersphere

Louis Omenyi1111Corresponding author.
1∗Email<EMAIL_ADDRESS>McSylvester Omaba2
2Email<EMAIL_ADDRESS>
(1Department of Mathematics/Computer Science/Statistics/Informatics, Alex
Ekwueme Federal University, Ndufu-Alike, Nigeria.
2Department of Mathematics, College of Science, University of Hafr Al Batin,
P. O Box 1803 Hafr Al Batin 31991, KSA.
)

###### Abstract

Filtration of quantifiable objects by smoothing kernels on Riemannian
manifolds for visualisation is an ongoing research. However, using common
filters created for linear domains on manifolds with non-Euclidean topologies
can yield misleading results. While there is a lot of ongoing research on
convolution of quantifiable functions with smoothing kernels on the lower
dimensional manifolds, higher-dimensional problems particularly pose a
challenge. One important generalization of lower dimensional compact
Riemannian manifolds is the unit hypersphere. In this paper, we derive
explicit forms of convolution formulae for Gegenbauer kernel filtration on the
surface of unit hypersphere. We prove that the Gegebauer filtration is the
limit of a sequence of finite linear combinations of the hyperspherical
Legendre harmonics, among other results.

_Keywords:_ Spherical Laplacian; hypersphere; Gegenbauer kernel; convolution;
rotations; special orthogonal group $SO(n).$
_2000 MSC Class:_ 11M06; 11M35; 11M36; 35P05.

##  1 Introduction

Spherical harmonic analysis is basically the spectral theory of a differential
operator such as the spherical Laplacian $\Delta_{S^{n}}$ on a compact
Riemannian manifold e.g. the hypersphere, $(S^{n},g).$ In this analysis,
spherical harmonics play salient roles. Spherical harmonics analysis is a
process of decomposing a function on a sphere into components of various
wavelengths using surface spherical harmonics as base functions, [5].
Spherical synthesis combines components of various wavelengths to generate
function values on a sphere and is the reverse process of harmonic analysis.
Spherical harmonic analysis and synthesis have many real-life and scientific
applications, see e.g. [5] and [6].

Harmonics analysis on Euclidean spaces and in particular the theory of
spherical harmonics provide an elegant and instructive application of group
theoretical concepts to various questions of classical function theory, see
e.g. [4]. Eigenspaces of spherical Laplacian with irreducible components with
respect to the natural action of the group $SO(n)$ within the $L^{2}$ space on
the sphere is an interesting occurrence. The role of classical orthogonal
polynomials such as the Gegenbauer polynomials as reproducing kernels for the
spaces of spherical harmonics of a given degree, or more generally, as
providing an explicit construction of symmetry adapted basis functions for
those spaces has been studied extensively, see e.g. [7] and [12]. Also [14]
studied the connection of the Fourier transform on the Euclidean space to the
Hankel transform obtained via restriction to $SO(n)$-finite functions and
various integral identities of the Hecke–Bochner type resulting there. The
generalized concept of convolution on groups is intimately related to the
concept of filtering on homogeneous spaces. Some insight into spherical
filtering with particular emphasis on wavelet transform can be found in [8],
[1] and [5].

The goal of the present paper is to present a novel form of the Gegenbauer
kernel filtration of harmonic functions on the hypersphere. This puts in
limelight signal processing methods on non-Euclidean spaces and in particular
on the hypersphere. The most basic is the notion of Fourier transform, that on
the sphere corresponds to the expansion of functions into series of familiar
spherical harmonics. Vast amount of literature is available on such
expansions, mostly from quantum mechanics and mathematical physics, see e.g.
[15].

It is known from group representation theory that Fourier transform can be
defined on any compact Lie group and consequently on homogeneous spaces of
these groups [2]. An example is the group of rotations of a three-dimensional
space about the origin, $SO(3),$ and the $2$-sphere, $S^{2},$ as a
corresponding homogeneous space. We derive some general formulae for the
convolution of $SO(n)$-finite functions, including recent generalizations of
Fourier Spherical Harmonic expansions and discuss their function theoretic
consequences. In particular, we derive the convolutions with Gegenbauer kernel
on the hypersphere, $S^{n-1}\subset\mathbb{R}^{n},$ regarded as a homogeneous
space of the group of rotations of the $n$-dimensional space, $SO(n).$ We
proceed with clarifying basic terms and concepts.

##  2 Notations and preliminaries

By a hypersphere in $\mathbb{R}^{n}$ we refer to a set of points whose
Euclidean distance from the origin is equal to unity, that is

$S^{n-1}=\\{x\in\mathbb{R}^{n}:x^{T}x=1\\}$ (2.1)

where $n>3.$ When $n=2,$ $S^{1}$ is the unit circle and for $n=3$ we have the
unit $2$-sphere, $S^{2}.$ The superscript $n-1$ in $S^{n-1}$ is due to the
fact that a sphere in $\mathbb{R}^{n}$ as defined by (2.1) is a submanifold of
dimension $n-1;$ see e.g. [9] and [10] for details. The hypersphere $S^{n-1}$
may be parameterized by a set of hyperspherical polar coordinates. If
$(x_{1},x_{2},\cdots,x_{n})$ are Cartesian coordinates in $\mathbb{R}^{n},$
then we define the angles $\theta_{1},\theta_{2},\cdots\theta_{n-1}$ with
$\theta_{1},\theta_{2},\cdots,\theta_{n-2}\in[0,\pi]$ and
$\theta_{n-1}\in[0,2\pi]$ such that

$\displaystyle\left.\begin{array}[]{rcl}x_{1}&=&\cos\theta_{1}\\\
x_{2}&=&\sin\theta_{1}\cos\theta_{2}\\\
x_{3}&=&\sin\theta_{1}\sin\theta_{2}\cos\theta_{3}\\\
x_{4}&=&\sin\theta_{1}\sin\theta_{2}\sin\theta_{3}\\\ \vdots\\\
x_{n-1}&=&\sin\theta_{1}\sin\theta_{2}\cdots\sin\theta_{n-2}\cos\theta_{n-1}\\\
x_{n}&=&\sin\theta_{1}\sin\theta_{2}\cdots\sin\theta_{n-2}\sin\theta_{n-1}.\end{array}\right\\}$
(2.9)

This choice of coordinates is not unique, but it is a natural generalization
of spherical polar coordinates in $\mathbb{R}^{3}.$ In the familiar case of
$S^{2}\subset\mathbb{R}^{3},$ $\theta_{1}$ corresponds to the elevation and
$\theta_{2}$ corresponds to the azimuth, often denoted
$(\theta,\psi)=(\theta_{2},\theta_{1}).$

Basic concepts related to function theory on the hypersphere used throughout
this paper include the Euler Gamma function, volume element, surface element
and some functional spaces over the hypersphere. We briefly explain them here.
As it is well known, for any $x\in\mathbb{R},$ we denote the smallest integer
that is greater or equal to $x$ by $[x].$ Also, the Euler Gamma function,
$\Gamma,$ in argument $s\in\mathbb{C}$ is

$\displaystyle{\Gamma(s):=\int_{0}^{\infty}e^{-t}t^{s-1}dt,~{}~{}\text{\rm
Re}\,(s)>0}.$

It can be shown that $\Gamma(\frac{1}{2})=\sqrt{\pi}$ and
$\displaystyle{\Gamma(n+\frac{1}{2})=\frac{(2n)}{2^{2n}n!}\sqrt{\pi}}.$ On the
hypersphere $S^{n},$ we use $dV_{n}$ for $n$-dimensional volume element and
$dS_{n}$ for $n$-dimensional surface element over the unit sphere $S^{n}.$ We
also use $d\xi$ for surface element over a general domain.

A point on the hypersphere is specified as follows. For $n\geq 3,$ let
$\xi=\xi_{(n)}\in S^{n}$ then $\xi_{(n)}=te_{n}+\sqrt{1-t^{2}}\xi_{(n-1)}$
with $t\in[-1,1],~{}\xi_{(n-1)}\in S^{n-1}$ and $e_{n}=(0,0,\cdots,0,1)^{T}$
is basis vector in $\mathbb{R}^{n}.$ When we write $\xi_{(n-1)}\in S^{n}$ we
mean that $\xi=(\xi_{1},\xi_{2},\cdots,\xi_{n-1},0)\in S^{n}.$

It can be shown that for $n\geq 3,$

$\Big{(}te_{n}+\sqrt{1-t^{2}}\xi_{(\xi_{(n-1)})}\Big{)}dS_{n-1}=(1-t^{2})^{\frac{n-3}{2}}dtdS_{n-2}(\xi_{(n-1)})=(1-t^{2})^{\frac{n-3}{2}}dtdS_{n-2}.$
(2.10)

The surface area of the hypersphere is by definition given by

$|S^{n-1}|:=\int_{S^{n-1}}dS_{n-1}$

and so from (2.10) we have

$|S^{n-1}|=\int_{-1}^{1}(1-t^{2})^{\frac{n-3}{2}}dt\int_{S^{n-2}}dS_{n-2}=|S^{n-2}|\int_{-1}^{1}(1-t^{2})^{\frac{n-3}{2}}dt.$

In the integral, $\displaystyle{\int_{-1}^{1}(1-t^{2})^{\frac{n-3}{2}}dt},$
let $s=t^{2}$ so that

$\int_{-1}^{1}(1-t^{2})^{\frac{n-3}{2}}dt=\int_{0}^{1}s^{-1/2}(1-s)^{\frac{n-3}{2}}ds=\beta(\frac{1}{2},\frac{n-1}{2})=\frac{\sqrt{\pi}\Gamma(\frac{n-1}{2})}{\Gamma(\frac{n}{2})}.$

Thus we have the recursive relation

$|S^{n-1}|=\frac{\sqrt{\pi}\Gamma(\frac{n-1}{2})}{\Gamma(\frac{n}{2})}|S^{n-2}|=\frac{2\pi^{n/2}}{\Gamma(n/2)};~{}~{}n\geq
3.$ (2.11)

If $A\in\mathbb{R}^{n\times n}$ is orthogonal, then there exists $\xi\in
S^{n-1}$ and $x\in\mathbb{R}^{n}$ such that

$dS_{n-1}(A\xi)=dS_{n-1}(\xi)~{}~{}\text{and}~{}~{}dV_{n}(Ax)=dV_{n}(x).$

In polar coordinates, we relate $x_{(n)}$ with $\xi_{(n)}$ using

$x_{(n)}=|x|\xi_{(n)},~{}~{}\xi_{(n)}=\xi\in S^{n-1}.$

Thus,

$dV_{n}(|x|\xi)=|x|^{n-1}d|x|dS_{n-1}(\xi).$ (2.12)

On the hypersphere, $(S^{n},g),$ where $g$ is a Riemannian metric, we define
the Riemannian (volume) measure $dV_{g}$ in terms of the metric $g$ in
standard hyperspherical coordinates (2.9) as

$dV_{g}=\sin^{n-1}d\theta
d\omega:=\sin^{n-1}d\theta\sin^{n-2}\sin^{n-3}\cdots\sin\theta_{2}d\theta_{2}\cdots
d\theta_{n-1}.$ (2.13)

The space of real or complex-valued continuous functions $C(S^{n-1})$ over the
unit hypersphere $S^{n-1}$ is a Banach space with the canonical norm

$||f||_{\infty}=\sup_{\xi\in S^{n-1}}|f(\xi)|.$

Similarly, the space of real or complex-valued square integrable functions
$L^{2}(S^{n-1})$ over $S^{n-1}$ is an Hilbert space with canonical inner
product

$\langle f,g\rangle:=\int_{S^{n-1}}f\bar{g}dS_{n-1}$ (2.14)

with the induced norm

$||f||_{2}:=\sqrt{\langle f,f\rangle}.$ (2.15)

The space $L^{2}(S^{n-1})$ is the closure of $C(S^{n-1})$ with respect to the
norm (2.15). We study a function $f$ on $S^{n-1}$ through its extension

$f^{\ast}(x):=f(\frac{x}{|x|})=f(\xi)~{}~{}\text{for}~{}~{}x\in\Omega_{\epsilon}:=\\{x\in\mathbb{R}^{n}:|x|\in[1-\epsilon,1+\epsilon]\\}.$

##  3 Gegenbauer Polynomial

Gegenbauer polynomials enables one express Gegenbauer filtration kernels.
Thus, following [3, 2, 16, 12] and [11], we briefly explore them in this
section.

The system of Gegenbauer polynomials, which we denote by $C_{l}^{\alpha}$ for
degree $l$ and index $\alpha,$ are ultraspherical polynomials expressible
through hyperspherical Legendre polynomials $\\{P_{l,n}(x):l=0,1,2,\cdots\\}.$
Hyperspherical Legendre polynomials are a classical family of polynomials
orthogonal with respect to the weight function
$w_{\alpha}=(1-x^{2})^{\alpha-\frac{1}{2}}$ on the support interval $[-1,1].$
We make these definitions formal in what follows.

######  Definition 3.1.

The function $P_{l,n}$ is hyperspherical Legendre polynomial of degree $l$ in
$n$ dimension and it is given by

$P_{l,n}(t):=l!\Gamma(\frac{n-1}{2})\sum_{j=0}^{[\frac{l}{2}]}(-1)^{j}\frac{(1-t^{2})^{j}t^{l-2j}}{4^{j}j!(l-2j)!\Gamma(j+\frac{n-1}{2})}$
(3.1)

for $t\in[-1,1].$

Specifically, $P_{l,n}(1)=1,$ for example.

There are other forms of definition of $P_{l,n}.$ Popular among those
definitions is the Rodrigues formulation.

######  Theorem 3.2.

(Rodrigues representation). The hyperspherical Legendre polynomial of degree
$l$ in $n$ dimension (3.1) in Rodrigues form is given by

$P_{l,n}(t)=(-1)^{l}R_{l,n}(1-t^{2})^{\frac{3-n}{2}}\frac{d^{l}}{dt^{l}}(1-t^{2})^{l+\frac{n-3}{2}}~{}~{}\mathrm{with}~{}~{}n\geq
2,$ (3.2)

where the Rodrigues constant $R_{l,n}$ is given by

$R_{l,n}=\frac{\Gamma(\frac{n-1}{2})}{2^{l}\Gamma(l+\frac{n-1}{2})}.$ (3.3)

For proof, see e.g. [11, 2] and [13].

We remark that for $n=2,$ we have the relation

$\Gamma(l+\frac{1}{2})=\frac{(2l)!}{2^{2l}l!}\Gamma(\frac{1}{2})$

and so

$P_{l,2}(t)=(-1)^{l}\frac{2^{l}l!}{(2l)!}(1-t^{2})^{1/2}\Big{(}\frac{d}{dt}\Big{)}^{l}(1-t^{2})^{l-1/2},~{}~{}~{}l\in\mathbb{N}_{0}$

which can be simplified to

$P_{l,2}(t)=\cos(l\arccos t);~{}~{}t\in[-1,1].$

For $n=3,$ one recovers from (3.2) the standard Rodrigues representation
formula for the standard Legendre polynomials as

$P_{l,3}(t)=\frac{1}{2^{l}l!}\Big{(}\frac{d}{dt}\Big{)}^{l}(t^{2}-1)^{l},~{}~{}~{}l\in\mathbb{N}_{0}.$
(3.4)

Moreover, [11] proved an integral representation of the hyperspherical
harmonics as the theorem (3.3) that follows.

######  Theorem 3.3.

For $l\in\mathbb{N}_{0},$ $n\geq 3$ and for any $t\in[-1,1],$

$P_{l,n}(t)=\frac{|S^{n-3}|}{|S^{n-2}|}\int_{-1}^{1}\Big{[}t+i(1-t^{2})^{\frac{1}{2}})s\Big{]}^{l}(1-s^{2})^{\frac{n-4}{2}}ds.$
(3.5)

Now, let $\mathcal{H}_{l,n}$ denote the space homogeneous Legendre polynomials
of degree $l$ in dimension $n.$ We call function $f\in\mathcal{H}_{l,n}$ such
that $\Delta f=0$ hyperspherical harmonic. We denote the space of
hyperspherical harmonic polynomials restricted to the unit hypersphere,
$S^{n},$ by $\mathcal{Y}_{l,n}.$ So any $Y_{l}\in\mathcal{Y}_{l,n}$ is related
to a homogeneous harmonic $h_{l}\in\mathcal{H}_{l,n}$ as follows
$h_{l}(r\xi)=r^{l}Y_{l}(\xi)$ where $r=|h_{l}|.$ So, they have the same
dimension.

The hyperspherical harmonic polynomials have the following properties.

######  Theorem 3.4.

(Addition theorem).

Let $\\{\psi_{l,j}:1\leq j\leq d_{l}(n)\\}$ be an orthonormal basis of the
space of $n$-dimensional spherical harmonics $\mathcal{H}_{l}(S^{n}),$ i.e:

$\int_{S^{n}}\psi_{l,j}(x)\bar{\psi}_{l,m}(x)dV_{g}(x)=\delta_{jm};~{}~{}1\leq
j,m\leq d_{l}(n).$ (3.6)

Then

$\sum_{j=1}^{d_{l}(n)}\psi_{l,j}(x)\bar{\psi}_{l,m}(y)=\frac{d_{l}(n)}{|S^{n}|}P_{l,\frac{(n-1)}{2}}(x\cdot
y)$ (3.7)

where $P_{l,n}(t)$ are the hyperspherical Legendre polynomials of degree $l$
in $n$ dimensions. The dimension of $\mathcal{H}_{l}(S^{n})$ is $d_{l}(n)$
given by the formula

$d_{l}(n)=\begin{pmatrix}l+n\\\ n\end{pmatrix}-\begin{pmatrix}l+n-2\\\
n\end{pmatrix}=\frac{(2l+n-1)(l+n-2)!}{l!(n-1)!},~{}~{}\text{for}~{}~{}l\in\mathbb{N}.$

For proof, one may see [12] and [11]. This means in particular that
$P_{l,(n-1)/2}(x\cdot y)$ is a harmonic function on $S^{n}$ with eigenvalue
$\lambda_{l}=l(l+n-1)$ for the eigenvalue problem

$\Delta_{S^{n}}P_{l,(n-1)/2}(\theta)=\lambda_{l}P_{l,(n-1)/2}(\theta)$

where $\Delta_{S^{n}}$ is the usual Laplacian on $S^{n}$ defined by

$\Delta_{S^{n}}:=\frac{1}{\sin^{n-1}\theta}\frac{\partial}{\partial\theta}\big{(}\sin^{n-1}\frac{\partial}{\partial\theta}\big{)}+\frac{1}{\sin^{2}\theta}\Delta_{S^{n-1}}.$
(3.8)

We note that (3.7) can also be expressed in terms of Jacobian polynomials
$P_{l}^{\alpha,\alpha}$ as follows:

$\sum_{j=1}^{d_{l}(n)}\psi_{l,j}(x)\bar{\psi}_{l,m}(y)=\frac{1}{P_{l}^{\big{(}\frac{(n-2)}{2},\frac{(n-2)}{2}\big{)}}(1)}P_{l}^{(\frac{(n-2)}{2},\frac{(n-2)}{2})}(\cos\theta)~{}~{}\text{for}~{}~{}x\cdot
y=\cos\theta.$ (3.9)

A special case of the hyperspherical Legendre polynomials are the familiar
spherical harmonics on $S^{2}$ usually given by

$Y_{l,m}(\theta,\psi)=\tilde{N}_{lm}P_{l,m}(\cos\theta)\exp(jm\psi)$ (3.10)

with normalisation constant

$\tilde{N}_{lm}=\sqrt{(2l+1)\frac{(l-m)!}{(l+m)!}}$ (3.11)

chosen so that the norm of each harmonic would be unity.

We have the following lemma.

######  Lemma 3.5.

The hyperspherical harmonic polynomials are orthogonal:

$\displaystyle\int_{S^{n}}P_{l,n}(x\cdot y)P_{j,n}(x\cdot y)dV_{n}(x)$
$\displaystyle=$
$\displaystyle\begin{cases}\frac{|S^{n}|}{d_{l}(n)}&\text{if}~{}~{}l=j.\\\
0&\text{if}~{}~{}l\neq j.\end{cases}$ (3.12)

###### Proof.

That

$\int_{S^{n}}P_{l,n}(x\cdot y)P_{j,n}(x\cdot
y)dV_{n}(x)=0~{}~{}\text{for}~{}~{}l\neq j$

follows directly from (3.6). When $l=j$ we observe from (3.7) that

$\displaystyle\int_{S^{n}}|P_{l,n}(x\cdot y)|^{2}dV_{g}(y)$ $\displaystyle=$
$\displaystyle\big{(}\frac{|S^{n}|}{d_{l}(n)}\big{)}^{2}\int_{S^{n}}\big{|}\sum_{j=1}^{d_{l}(n)}\psi_{l,j}(x)\bar{\psi}_{l,m}(y)\big{|}^{2}dV_{g}(y)$
$\displaystyle=$
$\displaystyle\big{(}\frac{|S^{n}|}{d_{l}(n)}\big{)}^{2}\sum_{j=1}^{d_{l}(n)}\big{|}\psi_{l,j}(x)\big{|}^{2}=\big{(}\frac{|S^{n}|}{d_{l}(n)}\big{)}^{2}\frac{d_{l}(n)}{|S^{n}|}.$

∎

We also have

$\int_{-1}^{1}P_{l,n}(t)P_{j,n}(t)(1-t^{2})^{\frac{n-3}{2}}dt=\frac{|S^{n-1}|}{d_{l}(n)|S^{n-2}|}\delta_{lj}.$
(3.13)

Besides, that

$\int_{S^{n-1}}P_{l,n}(\xi\cdot\eta)P_{j,n}(\xi\cdot\eta)dS_{n-1}(\xi)=\left\\{\begin{array}[]{rcl}0{}{}{}&=&\text{if}~{}~{}l\neq
j\\\ \frac{|S^{n-1}|}{d_{l}(n)}&=&\text{if}~{}~{}l=j\end{array}\right.$

follows from the following. Take $t=\xi\cdot\eta$ and observe that

$\int_{-1}^{1}|P_{l,n}(t)|^{2}(1-t^{2})^{\frac{n-3}{2}}dt=\frac{|S^{n-1}|}{d_{l}(n)|S^{n-2}|}.$

So,

$\int_{S^{n-1}}\Big{(}\int_{-1}^{1}P_{l,n}(t)P_{j,n}(t)(1-t^{2})^{\frac{n-3}{2}}dt\Big{)}dS_{n-2}=|S^{n-2}|\int_{-1}^{1}P_{l,n}(t)P_{j,n}(t)(1-t^{2})^{\frac{n-3}{2}}dt$

where

$\int_{-1}^{1}P_{l,n}(t)P_{j,n}(t)(1-t^{2})^{\frac{n-3}{2}}dt=0~{}~{}\text{for}~{}~{}l\neq
j.$

Another interesting property is the projection of integrable function onto the
space of spherical harmonics on the hypersphere. We make the following
definition.

######  Definition 3.6.

The projection of $f\in L^{1}(S^{n-1})$ into $\mathcal{Y}_{l,n}$ is

$(F_{l,n}f)(\xi):=\frac{d_{l}(n)}{|S^{n-1}|}\int_{S^{n-1}}P_{l,n}(\xi\cdot\eta)f(\eta)dS_{n-1}(\eta),~{}~{}\eta\in
S^{n-1}.$ (3.14)

Let us now consider the problem of finding the best approximation for $f\in
L^{2}(S^{n-1}).$ That is,

$\inf_{Y_{n}\in\mathcal{Y}_{l,n}}||f-Y_{l}||_{L^{2}(S^{n-1})}.$ (3.15)

In terms of the orthonormal $\\{Y_{l,j}:1\leq j\leq d_{l}(n)\\}$ of
$\mathcal{Y}_{l,n},$ the solution to (3.15) is

$(F_{l,n}f)(\xi)=\sum_{j=1}^{d_{l}(n)}\langle f,Y_{l,j}\rangle Y_{l,j}(\xi).$
(3.16)

The operator $F_{l,n}$ is the projection of any $f$ into $\mathcal{Y}_{l,n}$
and it is defined for $f\in L^{1}(S^{n-1}).$ One can apply this formula
without explicit knowledge of the orthonormal basis by applying the addition
lemma (3.5).

######  Lemma 3.7.

If $f\in L^{2}(S^{n-1})$ then for any $\xi\in S^{n-1}$ we have

$||(F_{l,n}f)||_{L^{2}(S^{n-1})}\leq||f||_{L^{2}(S^{n-1})}.$

###### Proof.

Let $f\in L^{2}(S^{n-1})$ and $\xi\in S^{n-1}$ given, we have

$\displaystyle|(F_{l,n}f)(\xi)|^{2}$ $\displaystyle\leq$
$\displaystyle\frac{d_{l}(n)}{|S^{n-1}|}\int_{S^{n-1}}|P_{l,n}(\xi\cdot\eta)|^{2}dS_{n-1}(\eta)\cdot\int_{S^{n-1}}|f(\eta)|^{2}dS_{n-1}(\eta)$
$\displaystyle\implies||(F_{l,n}f)||_{L^{2}(S^{n-1})}$ $\displaystyle\leq$
$\displaystyle\sqrt{d_{l}(n)}||f||_{L^{2}(S^{n-1})}.$

Similarly,

$||(F_{l,n}f)||_{L^{2}(S^{n-1})}\leq\sqrt{\frac{d_{l}(n)}{|S^{n-1}|}}||f||_{L^{2}(S^{n-1})}.$

These imply that

$||(F_{l,n}f)||_{L^{2}(S^{n-1})}\leq||f||_{L^{2}(S^{n-1})}.$

∎

All these give rise to the following theorem.

######  Theorem 3.8.

We have the orthogonal decomposition

$L^{2}(S^{n-1})=\bigoplus_{l=0}^{\infty}\mathcal{Y}_{l,n}.$

This means that any $f\in L^{2}(S^{n-1})$ can be uniquely represented as

$f(\xi)=\sum_{l=0}^{\infty}f_{l}(\xi);~{}~{}\text{with}~{}~{}f_{l}\in\mathcal{Y}_{l,n};~{}l\geq
0.$ (3.17)

We call $f_{l}\in\mathcal{Y}_{l,n}$ hyperspherical component of $f$ given by

$f_{l}(\xi)=\frac{d_{l}(n)}{|S^{n-1}|}\int_{S^{n-1}}f(\eta)P_{l,n}(\xi\cdot\eta)dS_{n-1}(\eta);~{}~{}~{}\eta\geq
0.$ (3.18)

We now define another generalisation of the hyperspherical harmonics.

######  Definition 3.9.

For $\alpha>0,$ $l\in\mathbb{N}_{0}$, the Gegenbauer polynomial of degree $l$
with index $\alpha$ is defined by

$C_{l}^{\alpha}(t):=\binom{l+2\alpha-1}{l}\frac{\Gamma(\alpha+1/2)}{\sqrt{\pi}\Gamma(\alpha)}\int_{-1}^{1}(t+i(1-t^{2})^{l/2}s)^{l}(1-s^{2})^{\alpha-1}ds.$
(3.19)

######  Theorem 3.10.

The Gegenbauer function $C_{l}^{\alpha}$ is indeed a polynomial and has a
representation in terms of the hyperspherical harmonics as

$C_{l}^{\frac{n-2}{2}}(t):=\binom{l+n-3}{l}P_{l,n}(t);~{}~{}~{}\text{for}~{}~{}~{}n\geq
3.$ (3.20)

###### Proof.

From the definition of $C_{l}^{\alpha}$ in (3.19), we observe that by Binomial
expansion

$[t+i(1-t^{2})^{\frac{1}{2}}s]^{l}=\sum_{j=0}^{l}\binom{l}{j}t^{l-j}(1-t^{2})^{\frac{j}{2}}(is)^{j}.$

So for $j=2k+1$ odd, the integral corresponding term is

$\int_{-1}^{1}s^{2k+1}(1-s^{2})^{\alpha-1}ds=0.$

Thus, $C_{l}^{\alpha}$ is real valued and has the form

$C_{l}^{\alpha}(t)=\binom{l+2\alpha-1}{l}\frac{\Gamma(\alpha+1/2)}{\sqrt{\pi}\Gamma(\alpha)}\sum_{k=0}^{[\frac{l}{2}]}\binom{l}{2k}t^{l-2k}(-1)^{k}(1-t^{2})^{k}\int_{-1}^{1}s^{2k}(1-s^{2})^{\alpha-1}ds$

which is a polynomial of degree less or equal to $l.$

Besides, the coefficient of $t^{l}$ in $C_{l}^{\alpha}(t)$ is

$\binom{l+2\alpha-1}{l}\frac{\Gamma(\alpha+1/2)}{\sqrt{\pi}\Gamma(\alpha)}\sum_{k=0}^{[\frac{l}{2}]}\binom{l}{2k}\int_{-1}^{1}s^{2k}(1-s^{2})^{\alpha-1}ds.$

Therefore, $C_{l}^{\alpha}(t)$ is a polynomial.

Moreover, from (3.5), it follows that $C_{l}^{\alpha}(t)$ has the form (3.19).
∎

We remark the following standard result.

######  Lemma 3.11.

The Gegenbauer polynomials $C_{l}^{(n-1)/2}(\xi)$ are eigenfunctions of the
Laplace-Beltrami operator on $S^{n-1}$ which satisfy the following eigenvalue
problem:

$(1-\xi^{2})(C_{l}^{(n-1)/2})^{\prime\prime}(\xi)-n\xi(C_{l}^{(n-1)/2})^{\prime}(\xi)+l(l+n-1)C_{l}^{(n-1)/2}(\xi)=0.$
(3.21)

###### Proof.

Let $\psi\in C^{\infty}(S^{n}).$ Then,

$\Delta_{S^{n}}\psi=\frac{1}{\sin^{n-1}\theta}\frac{\partial}{\partial\theta}\big{(}\sin^{n-1}\frac{\partial\psi}{\partial\theta}\big{)}+\frac{1}{\sin^{2}\theta}\Delta_{S^{n-1}}\psi.$

For a fixed $\xi\in S^{n},$ the function $\psi$ given by
$C_{l}^{n}(\xi\cdot\eta)=C_{l}^{{(n-1)/2}}(\cos\theta)$ belongs to
$\mathcal{H}_{l}(S^{n})$ where $\theta$ is the geodesic angle between $\xi$
and $\eta\in S^{n}.$ A direct computation gives

$\Delta_{S^{n}}\psi=l(l+n-1)\psi.$

Since $\psi$ only depends on $\xi,$ so, $\Delta_{S^{n-1}}\psi=0$ and thus

$l(l+n-1)C_{l}^{n}(\xi)=\Delta_{S^{n}}C_{l}^{n}(\xi)=(1-\xi^{2})\frac{\partial^{2}C_{l}^{n}(\xi)}{\partial
t^{2}}-n\xi\frac{\partial C_{l}^{n}(\xi)}{\partial\xi}$ (3.22)

proving the theorem. ∎

##  4 Spherical harmonic analysis and filtration

Consider the following generating function of the Gegenbauer harmonic (3.19)
for a fixed index $\alpha$:

$\Phi(r)=\sum_{l=0}^{\infty}C_{l}^{\alpha}(t)r^{l}=\frac{1}{(1+r^{2}-2rt)^{(\frac{n-2}{2})}}~{}~{}~{}~{}\text{for}~{}~{}|r|<1,~{}~{}|t|\leq
1.$ (4.1)

Consequently from (3.20),

$\Phi(r)=\sum_{l=0}^{\infty}\binom{l+n-3}{n-3}P_{l,n}(t)r^{l}=\frac{1}{(1+r^{2}-2rt)^{(\frac{n-2}{2})}}~{}~{}~{}~{}\text{for}~{}~{}|r|<1,~{}~{}|t|\leq
1.$ (4.2)

######  Definition 4.1.

A Gegenbauer filtration kernel is

$G_{n}(r,t):=\frac{|S^{n-2}|}{|S^{n-1}|}\frac{1-r^{2}}{(1+r^{2}-2rt)^{(\frac{n}{2})}}~{}~{}~{}~{}\text{for}~{}~{}|r|<1,~{}~{}|t|\leq
1~{}~{}~{}\text{and}~{}~{}~{}n\geq 3.$ (4.3)

######  Lemma 4.2.

The function (4.3) satisfies the following properties

$\int_{-1}^{1}G_{n}(r,t)(1-t^{2})^{\frac{n-3}{2}}dt=1$ (4.4)

and

$\lim_{r\rightarrow 1^{-}}G_{n}(r,t)=0.$ (4.5)

###### Proof.

A direct computation shows that

$\displaystyle\int_{-1}^{1}G_{n}(r,t)(1-t^{2})^{\frac{n-3}{2}}dt$
$\displaystyle=$
$\displaystyle\frac{|S^{n-2}|}{|S^{n-1}|}\int_{-1}^{1}\sum_{l=0}^{\infty}d_{l}(n)r^{l}P_{l,n}(t)(1-t^{2})^{\frac{n-3}{2}}dt$
$\displaystyle=$
$\displaystyle\frac{|S^{n-2}|}{|S^{n-1}|}\int_{-1}^{1}(1-t^{2})^{\frac{n-3}{2}}dt=1$

showing (4.4).

Besides for $n\geq 2$ we have

$\displaystyle\lim_{r\rightarrow
1^{-}}\Big{(}\frac{1-r^{2}}{(1+r^{2}-2rt)^{\tfrac{n}{2}}}\Big{)}$
$\displaystyle=$ $\displaystyle\lim_{r\rightarrow
1^{-}}\Big{(}\frac{1-r^{2}}{[(1-r)^{2}+2r(1-t)]^{\tfrac{n}{2}}}\Big{)}$
$\displaystyle\leq$ $\displaystyle\lim_{r\rightarrow
1^{-}}\Big{(}\frac{1-r^{2}}{[2r(1-t_{0})]^{\tfrac{n}{2}}}\Big{)}=0$

for $t\in[-1,t_{0}]$ arbitrary, showing (4.5). ∎

Now we define the Gegenbauer filtration operator $\mathcal{G}_{n}(r)$ to be

$(\mathcal{G}_{n}(r)f)(\xi):=\frac{1}{|S^{n-2}|}\int_{S^{n-1}}G_{n}(r,\xi\cdot\eta)f(\eta)dS_{n-1}.$
(4.6)

We immediately have the following result.

######  Theorem 4.3.

The Gegebauer filtration $\mathcal{G}_{n}(r)$ is the limit of a sequence of
finite linear combinations of the hyperspherical Legendre harmonics.

###### Proof.

Let $f\in L^{2}(S^{n-1}).$ From (4.6), we have

$\displaystyle(\mathcal{G}_{n}(r)f)(\xi)$ $\displaystyle=$
$\displaystyle\frac{1}{|S^{n-2}|}\int_{S^{n-1}}G_{n}(r,\xi\cdot\eta)f(\eta)dS_{n-1}$
$\displaystyle=$
$\displaystyle\frac{1}{|S^{n-2}|}\int_{S^{n-1}}\Big{(}\int_{-1}^{1}G_{n}(r,t)P_{l,n}(t)(1-t^{2})^{\tfrac{n-3}{2}}dt\Big{)}f(\eta)dS_{n-1}\Big{|}_{t=\xi\cdot\eta}$
$\displaystyle\implies(\mathcal{G}_{n}(r)f)(\xi)$ $\displaystyle=$
$\displaystyle\frac{1}{|S^{n-2}|}\sum_{l=0}^{\infty}r^{l}d_{l}(n)\int_{S^{n-2}}C_{l}^{\alpha}(\xi\cdot\eta)f(\eta)dS_{n-1}(\eta)$
$\displaystyle=$ $\displaystyle\sum_{l=0}^{\infty}r^{l}(F_{l,n}f)(\xi)$

and for $|r|<1,$

$(\mathcal{G}_{n}(r)f)(\xi)=\sum_{l=0}^{\infty}d_{l}(r)P_{l,n}(\xi\cdot\eta)r^{l}(F_{l,n}f)(\xi)=(F_{l,n}f)(\xi)$
(4.7)

which completes the proof. ∎

The inner product of two complex valued functions on the surface of the
hypersphere is

$\langle
f,h\rangle_{S^{n-1}}=\int_{S^{n-1}}f(\xi,\eta)h(\xi,\eta)^{\ast}dS_{n-1}(\eta)$
(4.8)

where $\ast$ denotes complex conjugation. Using the fact that the Gegenbauer
filtration is rotation invariant over $SO(n),$ one can move $h$ to any point
$(\xi_{0},\eta_{0})\in S^{n-1}.$ Then we define a generalised convolution as a
function on the rotation group $SO(n)$ to be

$(h\ast f)(R)=\int_{S^{n-1}}f(\xi,\eta)h_{R}(\xi,\eta)^{\ast}dS_{n-1}(\eta)$
(4.9)

for $R\in SO(n)$ and where $h_{R}$ is $h$ rotated by $R$ defined as

$h_{R}(A)=h(R^{-1}A).$ (4.10)

Thus, every well-behaved function $f\in L^{2}(S^{n-1})$ admits the expansion

$f(\xi,\eta)=\sum_{\alpha=0}^{\infty}\sum_{l,m}\hat{f}_{l,m}(\xi,\eta)C_{l}^{\alpha}(\xi,\eta)$
(4.11)

where the generalised Fourier transform on the surface of the hypersphere
$\hat{f}_{l,m}$ is defined to be

$\hat{f}_{l,m}(\xi)=\int_{S^{n-1}\ni\eta\in
SO(n)}f(R)\rho_{l,m}(R)dS_{n-1}(\eta).$ (4.12)

Here, $\rho(R)$ is a function on $SO(n)$ containing fixed matrix valued
functions called the irreducible representations of $SO(n).$

As a consequence of (3.16), we observe that for a suitable $\psi$, we have

$\displaystyle(F_{l,n}\psi)(\xi)$ $\displaystyle=$
$\displaystyle\frac{d_{l}(n)}{|S^{n-1}|}\int_{S^{n-1}}P_{l,n}(\xi\cdot\eta)\psi(\eta)dS_{n-1}(\eta)$
$\displaystyle=$
$\displaystyle\frac{d_{l}(n)}{|S^{n-1}|}\frac{|S^{n-1}|}{d_{l}(n)}\psi_{l}(\xi)=\sum_{j=1}^{d_{l}(n)}\langle\psi_{l},\psi_{l,j}\rangle_{S^{n-1}}\psi_{l,j}(\xi)$
$\displaystyle\implies(F_{l,n}\psi)(\xi)$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{d_{l}(n)}\hat{\psi}_{l}\psi_{l,j}(\xi).$ (4.13)

Another interesting result of this work is that Gegenbauer filtration on
$S^{n-1}$ coincides with convolution on $S^{n-1}.$ Particularly, convolution
of two functions on the hypersphere equals the multiplication of their Fourier
coefficients. This is the next theorem (4.4) below.

######  Theorem 4.4.

Let $f,h\in L^{2}(S^{n-1})$ and $\mathcal{G}(r)$ the Gegenbauer filtration
operator. Then the spectrum of the Gegenbauer kernel convolution is given by

$\widehat{f\ast
h}(l,\alpha)=\sqrt{\frac{\Gamma(l+n-2)(2l+n-2)}{l!\Gamma(n-1)}}\hat{f}(l,\alpha)\hat{h}(l,0).$
(4.14)

A special case of this property has been proved for the special case of
convolutions on $S^{2}$, see e.g. [8, 6] and [4]. We get a proof for this
generalisation on $S^{n-1}.$

###### Proof.

Since $f,h\in L^{2}(S^{n-1})$ and by definition of the Gegenbauer filtration,

$\displaystyle\widehat{f\ast h}(l,\alpha)$ $\displaystyle=$
$\displaystyle\int_{S^{n-1}}f\ast
h(\xi)\overline{\mathcal{G}_{n}(r)(\xi)}dS_{n-1}(\xi)$ $\displaystyle=$
$\displaystyle\int_{S^{n-1}}\Big{(}\int_{SO(n)}f(R\eta)h(R^{-1}\xi)dR\Big{)}\overline{\mathcal{G}_{n}(r)(\xi)}dS_{n-1}(\xi).$

After rearranging the integrals and using the (3.16) we get

$\widehat{f\ast
h}(l,\alpha)=\int_{SO(n)}\Big{(}\int_{S^{n-1}}h(R^{-1}\xi)\overline{(F_{l,n})(F_{j,n})C_{l}^{\alpha}C_{j}^{\alpha}(\xi)}dS_{n-1}(\xi)\Big{)}f(RN)dR$

where $N$ is the north pole. This implies that from (4) and the addition
theorem (3.4),

$\widehat{f\ast
h}(l,\alpha)=\int_{SO(n)}f(RN)\Big{(}\int_{S^{n-1}}h(R^{-1}\xi)\sum_{|j|\leq
d_{l}(n)}d_{l}(n)C_{j}^{\alpha}(R\xi)F_{j,n}(R^{-1}\xi)dS_{n-1}(\xi)\Big{)}C_{l}^{\alpha}(R^{-1}\xi)dR$

which is $0$ unless $\alpha=0.$ This follows since the measure on $SO(n)$ is
rotation-invariant.

Therefore,

$\widehat{f\ast
h}(l,\alpha)=\int_{SO(n)}f(RN)C_{l}^{\alpha}(R^{-1}\xi)dR\hat{h}(l,0).$

Finally using the relationship between $C_{l}^{\alpha}$ and the hyperspherical
harmonics $P_{l,n}$ (3.20), we conclude that (4.14) holds. ∎

##  5 Conclusion

We have derived formulae for Gegebauer kernel filtration of different
functions on the unit hypersphere. Particularly, we derived a Gegebauer
filtration operator via hyperspherical Legendre harmonics. We related the
filtration to convolution on the hypersphere and proved that Gegenbauer
filtration on $S^{n-1}$ coincides with convolution on $S^{n-1}.$ We showed
that convolution of two functions on the hypersphere equals the multiplication
of their Fourier coefficients.

The results obtained suggest that filtrations and convolutions on the
Euclidean space can be successfully extended to spaces with non-Euclidean
topologies such as the hypersphere using the example of Gegenbauer filters.
The formulae and approach are readily extendable to other non-Euclidean
geometries such as the flat torus and cylinder in higher dimensions. These can
be done as further extension of this study.

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