A Note on the Derivatives of Isotropic Positive Definite Functions on the Hilbert Sphere
In this note, we give a recursive formula for the derivatives of isotropic positive definite functions on the Hilbert sphere. We then use it to prove a conjecture stated by Trübner and Ziegel, which says that for a positive definite function on the Hilbert sphere to be in C²ˡ([0,π]), it is necessary...
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| citation_txt | A Note on the Derivatives of Isotropic Positive Definite Functions on the Hilbert Sphere / J. Jäger // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 23 назв. — англ. |
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| description | In this note, we give a recursive formula for the derivatives of isotropic positive definite functions on the Hilbert sphere. We then use it to prove a conjecture stated by Trübner and Ziegel, which says that for a positive definite function on the Hilbert sphere to be in C²ˡ([0,π]), it is necessary and sufficient for its ∞ Schoenberg sequence to satisfy ∑ₘ₌₀ ∞ aₘmˡ < ∞.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 081, 7 pages
A Note on the Derivatives of Isotropic Positive
Definite Functions on the Hilbert Sphere
Janin JÄGER
Lehrstuhl Numerische Mathematik, Justus-Liebig University,
Heinrich-Buff Ring 44, 35392 Giessen, Germany
E-mail: janin.jaeger@math.uni-giessen.de
Received May 22, 2019, in final form October 16, 2019; Published online October 23, 2019
https://doi.org/10.3842/SIGMA.2019.081
Abstract. In this note we give a recursive formula for the derivatives of isotropic positive
definite functions on the Hilbert sphere. We then use it to prove a conjecture stated by
Trübner and Ziegel, which says that for a positive definite function on the Hilbert sphere
to be in C2`([0, π]), it is necessary and sufficient for its ∞-Schoenberg sequence to satisfy
∞∑
m=0
amm
` <∞.
Key words: positive definite; isotropic; Hilbert sphere; Schoenberg sequences
2010 Mathematics Subject Classification: 33B10; 33C45; 42A16; 42A82; 42C10
1 Introduction and main results
In the last five years there has been a tremendous number of publications stating new results
on positive definite functions on spheres, see for example [1, 5, 6, 8, 11, 14, 18, 20, 21]. Isotropic
positive definite functions are used in approximation theory, where they are often referred to
as spherical radial basis functions [2, 3, 4, 23] and are for example applied in geostatistics and
physiology [10, 15]. They are also of importance in statistics where they occur as correlation
functions of homogeneous random fields on spheres [16].
A function g : Sd × Sd → R is called positive definite on the d-dimensional sphere
Sd =
{
ξ ∈ Rd+1 : ‖ξ‖2 = 1
}
if it satisfies∑
ξ∈Ξ
∑
ζ∈Ξ
λξλζg(ξ, ζ) ≥ 0,
for any finite subset Ξ ⊂ Sd of distinct points on Sd and all λξ ∈ R. It is called strictly positive
definite if the above inequality is strict unless λξ = 0 for all ξ ∈ Ξ.
Further, a function g : Sd × Sd → R is called isotropic if there exists a univariate function
φ : [0, π]→ R for which
g(ξ, ζ) = φ(ρ(ξ, ζ)), ∀ ξ, ζ ∈ Sd,
where ρ(ξ, ζ) = arccos
(
ξT ζ
)
is the geodesic distance between ξ and ζ.
The class of isotropic positive definite function on spheres has received more attention during
the last years, even though the theory was in fact started by Schoenberg in 1942. He showed
in [19] that:
mailto:janin.jaeger@math.uni-giessen.de
https://doi.org/10.3842/SIGMA.2019.081
2 J. Jäger
Theorem 1 (Schoenberg, [19]). Every φ : [0, π] → R that is positive definite on Sd can be
represented as
φ(θ) =
∞∑
k=0
ak,d
Cλk (cos(θ))
Cλk (1)
, θ ∈ [0, π],
where ak,d ≥ 0, for all k, and
∞∑
k=0
ak,d < ∞, λ := (d − 1)/2, and the Cλk are the Gegenbauer
polynomials as defined in [12, formula (8.930)].
The sequence (ak,d)k∈N0 is referred to as a d-Schoenberg sequence. A criterion for the strict
positive definiteness of such functions was given by Chen et al. in [9].
A variety of 18 open problems on strictly and non-strictly positive definite spherical functions
has been posed in the supplement material of Gneiting’s article [11]. Some of the results on these
problems are described in [1, 8, 18, 20]. This note will provide some additional information to
the known solution of Problem 6.
Problem 6 was concerned with the smoothness properties of the members of the class of
positive definite functions on Sd. Trübner and Ziegel gave a solution to the problem in [20, 22]
and in the course of their proof stated an interesting connection between the existence of the
derivative of such a function at zero and the decay of its d-Schoenberg sequence. The result was
also described in the paper [13, Theorem 1] by Guinness and Fuentes.
Lemma 1 (Trübner and Ziegel, [20, Lemma 2.1a]). Let ` ≥ 1. Suppose φ is positive definite
on Sd with d-Schoenberg sequence (ak,d)k∈N0. Then, φ(2`)(0) exists if and only if
∞∑
k=0
ak,dk
2`
converges.
We show that a different connection holds for functions which are positive definite on all
spheres. This function class is equivalent to the class of functions positive definite on the
Hilbert sphere S∞. For these functions Schoenberg derived a simple representation in [19]. The
characterisation of strictly positive definite functions on S∞ was later completed by Menegatto
in [17].
Theorem 2 (Schoenberg, [19]). A function φ is positive definite on Sd for all d ≥ 1 if and only
if it has the form
φ(θ) =
∞∑
m=0
am(cos(θ))m,
where am ≥ 0, for all m ∈ N0, and
∞∑
m=0
am <∞.
The series (am)m∈N0 is referred to as an ∞-Schoenberg sequence. In this note, we will prove
the following theorem, which was shown to be true for ` ∈ {1, 2} in [20], and in the process
prove an interesting recursion formula for the derivatives of these positive definite functions.
Theorem 3. Let φ be positive definite on S∞ with ∞-Schoenberg sequence (am)m∈N0. Then
φ(2`)(0) exists if and only if
∞∑
m=0
amm
` converges.
The important difference between Lemma 1 and Theorem 3 is the decay property of the
Schoenberg sequence which is connected to the smoothness of the kernel. Theorem 3 is not
the limit of Lemma 1 when d → ∞. An explanation of the discrepancy between the cases Sd
and S∞ is possible using a probabilistic viewpoint. The positive definite functions are used there
A Note on the Derivatives of Isotropic Positive Definite Functions on the Hilbert Sphere 3
as covariance functions of stationary isotropic Gaussian processes on spheres. The behaviour
of these processes is governed by the decay of the Schoenberg sequences. Faster decay of the
sequence induces higher smoothness of the paths of the process.
As described in [7], processes on the Hilbert sphere have very different properties from those
on Euclidean spheres because the Hilbert sphere is not locally compact. Gaussian processes
on S∞ can be discontinuous and locally deterministic, for exact definitions and explanations
see [7] and the references therein. One might expect local determinism to be a sign of greater
smoothness, as for holomorphic functions in complex analysis, but it is the other way around.
The processes are extremely wild. Therefore even with similarly smooth covariance functions,
the process in S∞ is expected to have Schoenberg sequences with slower decay.
In [20] it was proven that Theorem 3 is equivalent to an interesting series relation introduced
as Conjecture 2.2. The conjecture contained a small typographical error in the sign of the
exponent of 22j , in the second part of the formula. We can now prove the corrected conjecture
which we state in the next lemma, and thereby prove Theorem 3.
Lemma 2. For ` > 1, there is a constant c(`) > 0 such that, as j →∞,
2−2j+1
j∑
n=1
(2n)2`
(
2j
j + n
)
∼ c(`)j`, 2−2j
j∑
n=1
(2n− 1)2`
(
2j − 1
j + n− 1
)
∼ c(`)j`.
In Section 2 we establish necessary preliminary results which allow us to prove Lemma 2.
2 Preliminaries
First we will introduce the following lemma which might prove helpful in other areas of the
discussion of positive definite functions on the Hilbert sphere.
Lemma 3. For φ(x) = cosj(x) and j > `,
φ(`)(x) =
∑
n1+n2=`,
0≤n2≤n1
(−1)n1bjn1,n2
cosj−n1+n2(x) sinn1−n2(x),
where the coefficients can be computed recursively by bj0,0 = 1,
bjn1,n2
= bjn1−1,n2
(j − (n1 − 1) + n2) + bjn1,n2−1(n1 − (n2 − 1)), 0 < n2 < n1,
bjn1,0
= bj`,0 = (j − (n1 − 1))bj`−1,0 =
j!
(j − `)!
,
and in the case of ` even bjn2,n2 = bj`/2,`/2 = bj`/2,`/2−1.
Proof. We prove the result by induction starting with ` = 1.
Let ` = 1 and j ∈ N>1. Then
φ′(x) = −j cosj−1(x) sin(x)
thereby bj1,0 = j. The step of the induction will be proven for odd values of ` and even values
of ` separately.
Let ` be even, show `→ `+ 1. We assume
φ(`)(x) =
∑
n1+n2=`,
n2<n1
(−1)n1bjn1,n2
cosj−n1+n2(x) sinn1−n2(x) + (−1)
`
2 bj`/2,`/2 cosj(x).
4 J. Jäger
Therefore for j > `+ 1
φ(`+1)(x) =
∑
n1+n2=`,
n2<n1−1
(−1)n1+1bjn1,n2
(j − n1 + n2) cosj−(n1+1)+n2(x) sin(n1+1)−n2(x)
+
∑
n1+n2=`,
0≤n2<n1−1
(−1)n1bjn1,n2
cosj−n1+n2+1(x)(n1 − n2) sinn1−(n2+1)(x)
+ (−1)`/2+1j bj`/2,`/2 cosj−1(x) sin(x)
=
∑
ñ1+ñ2=`+1,
ñ2<ñ1−2
(−1)ñ1bjñ1−1,ñ2
(j − (ñ1 − 1) + ñ2) cosj−ñ1+ñ2(x) sinñ1−ñ2(x)
+
∑
ñ1+ñ2=`+1,
0<ñ2<ñ1
(−1)ñ1bjñ1,ñ2−1 cosj−ñ1+ñ2(x)(ñ1 − (ñ2 − 1)) sinñ1−ñ2(x)
+ (−1)`/2+1j bj`/2,`/2 cosj−1(x) sin(x)
= (−1)`+1bj`,0 cosj−`−1(x) sin`+1(x)(j − `)
+
∑
ñ1+ñ2=`+1,
0<ñ2≤ñ1
(−1)ñ1bjñ1,ñ2
cosj−ñ1+ñ2(x) sinñ1−ñ2(x),
with bñ1,ñ2 as defined above.
Let ` be odd, show `→ `+ 1. We assume
φ(`)(x) =
∑
n1+n2=`,
n2<n1
(−1)n1bjn1,n2
cosj−n1+n2(x) sinn1−n2(x).
Therefore for j > `+ 1
φ(`+1)(x) =
∑
n1+n2=`,
0≤n2<n1
(−1)n1+1bjn1,n2
(j − n1 + n2) cosj−(n1+1)+n2(x) sin(n1+1)−n2(x)
+
∑
n1+n2=`,
0≤n2<n1
(−1)n1bjn1,n2
cosj−n1+(n2+1)(x)(n1 − n2) sinn1−(n2+1)(x)
=
∑
ñ1+ñ2=`+1,
0≤ñ2<ñ1−1
(−1)ñ1bjñ1−1,ñ2
(j − (ñ1 − 1) + ñ2) cosj−ñ1+ñ2(x) sinñ1−ñ2(x)
+
∑
ñ1+ñ2=`+1,
1≤ñ2<ñ1+1
(−1)ñ1bjñ1,ñ2−1 cosj−ñ1+ñ2(x)(ñ1 − (ñ2 − 1)) sinñ1−ñ2(x)
= bj`,0(j − `)(−1)`+1 cosj−`−1(x) sin`+1(x)
+
∑
ñ1+ñ2=`+1,
0<ñ2≤ñ1−1
(−1)ñ1bjñ1,ñ2
cosj−ñ1+ñ2(x) sinñ1−ñ2(x)
+ (−1)(`+1)/2bj(`+1)/2,(`+1)/2 cosj(x),
with bjñ1,ñ2
as defined above. �
Now the behaviour of the coefficients bjn1,n2 for j →∞ is described.
A Note on the Derivatives of Isotropic Positive Definite Functions on the Hilbert Sphere 5
Lemma 4. The coefficients bjn1,n2 satisfy
bjn1,n2
∼ cn1,n2j
n1 , for j →∞, for fixed n1, n2, (1)
where ∼ means the sequences are asymptotically equivalent. Here cn1,n2 are defined recursively
by c1,1 = 1, cn1,0 = 1 and for n1 > 1, 1 ≤ n2 < n1
cn1,n2 = cn1−1,n2 + (n1 − n2 + 1)cn1,n2−1
and cn1,n1 = cn1,n1−1.
Proof. We show this property by induction over the pairs (n1, n2). For all pairs of coefficients
(n1, 0) we have
bjn1,0
=
j!
(j − n1)!
∼ jn1 ,
further bj1,1 = bj1,0 = j ∼ j1.
We assume (1) holds for all combinations (n1, n2) with n1 ≤ n′ and n2 ≤ n1 and for all pairs
(n′ + 1, n2) up to a certain n2 ≤ n′′ < n′. Then
bjn′+1,n′′+1 = bjn′,n′′+1(j − n′ + n′′ + 1) + bjn′+1,n′′(n
′ + 1− n′′)
∼ cn′,n′′+1j
n′
(j − n′ + n′′ + 1) + cn′+1,n′′jn
′+1(n′ + 1− n′′)
∼
(
cn′,n′′+1 + cn′+1,n′′(n′ + 1− n′′)
)
jn
′+1
∼ cn′+1,n′′+1j
n′+1.
For the last choice of n′′ = n′ we find
bjn′+1,n′+1 = bjn′+1,n′ ∼ cn′+1,n′+1j
n′+1. �
3 Proof of Theorem 3
Proof of Lemma 2. We use the identities of the powers of cos from Gradshteyn and Ryzhik
[12, formulas (1.320.5) and (1.320.7)]
cos2j(x) =
1
22j
{
j−1∑
k=0
2
(
2j
k
)
cos(2(j − k)x) +
(
2j
j
)}
(2)
and
cos2j−1(x) =
1
22j−2
j−1∑
k=0
(
2j − 1
k
)
cos((2j − 2k − 1)x). (3)
Differentiating each side of the above equations 2`-times, using Lemma 3 for the left-hand side,
and evaluating the derivative at zero we find
b2j`,` =
1
22j
j−1∑
k=0
2
(
2j
k
)
(2(j − k))2`
and
b2j−1
`,` =
1
22j−2
j−1∑
k=0
(
2j − 1
k
)
(2j − 2k − 1)2`, for j > `.
The result now follows by applying (1) and rearranging of the coefficients. �
6 J. Jäger
Proof of Theorem 3. Let φ be positive definite on Sd for all d,
φ(θ) =
∞∑
j=0
aj,1 cos(jθ) =
∞∑
m=0
am cosm(θ), aj,1 ≥ 0.
Employing the Ziegel–Trübner result (Lemma 1), we know that φ2`(0) exists if and only if
∞∑
k=0
ak,1k
2` converges. The following relation between the Schoenberg sequences was proven,
also by Ziegel and Trübner (see [20, Proposition 5.1]):
a2n,1 =
∞∑
j=n
2−2j+1a2j
(
2j
j + n
)
, a2n−1,1 =
∞∑
j=n
2−2ja2j−1
(
2j − 1
j + n− 1
)
.
This yields
∞∑
n=0
an,1n
2` =
∞∑
n=0
a2n,1(2n)2` +
∞∑
n=1
a2n−1,1(2n− 1)2`
=
∞∑
n=0
∞∑
j=n
2−2j+1a2j
(
2j
j + n
)
(2n)2` +
∞∑
n=1
∞∑
j=n
2−2ja2j−1
(
2j − 1
j + n− 1
)
(2n− 1)2`.
This with an,1, am ≥ 0 for all m,n ∈ N0 and after application of Lemma 2, proves the theo-
rem. �
Acknowledgments
The author was a post-doctoral fellow funded by the Justus Liebig University during the de-
velopment of this research. I would like to express my gratitude to Professor M. Buhmann for
his helpful comments on the paper. Thanks are also due to the anonymous referees for their
thorough advice on how to improve this note.
References
[1] Arafat A., Gregori P., Porcu E., Schoenberg coefficients and curvature at the origin of continuous isotropic
positive definite kernels on spheres, Statist. Probab. Lett. 156 (2020), 108618, 6 pages, arXiv:1807.02363.
[2] Baxter B.J.C., Hubbert S., Radial basis functions for the sphere, in Recent Progress in Multivariate Ap-
proximation (Witten-Bommerholz, 2000), Internat. Ser. Numer. Math., Vol. 137, Birkhäuser, Basel, 2001,
33–47.
[3] Beatson R.K., zu Castell W., Dimension hopping and families of strictly positive definite zonal basis functions
on spheres, J. Approx. Theory 221 (2017), 22–37, arXiv:1510.08658.
[4] Beatson R.K., zu Castell W., Thinplate splines on the sphere, SIGMA 14 (2018), 083, 22 pages,
arXiv:1801.01313.
[5] Beatson R.K., zu Castell W., Xu Y., A Pólya criterion for (strict) positive-definiteness on the sphere, IMA J.
Numer. Anal. 34 (2014), 550–568, arXiv:1110.2437.
[6] Berg C., Peron A.P., Porcu E., Schoenberg’s theorem for real and complex Hilbert spheres revisited, J. Ap-
prox. Theory 228 (2018), 58–78, arXiv:1701.07214.
[7] Bingham N.H., Symons T.L., Gaussian random fields on the sphere and sphere cross line, Stochastic Process.
Appl., to appear, arXiv:1812.02103.
[8] Bissiri P.G., Menegatto V.A., Porcu E., Relations between Schoenberg coefficients on real and complex
spheres of different dimensions, SIGMA 15 (2019), 004, 12 pages, arXiv:1807.08184.
[9] Chen D., Menegatto V.A., Sun X., A necessary and sufficient condition for strictly positive definite functions
on spheres, Proc. Amer. Math. Soc. 131 (2003), 2733–2740.
https://doi.org/10.1016/j.spl.2019.108618
https://arxiv.org/abs/1807.02363
https://doi.org/10.1007/978-3-0348-8272-9_4
https://doi.org/10.1016/j.jat.2017.04.001
https://arxiv.org/abs/1510.08658
https://doi.org/10.3842/SIGMA.2018.083
https://arxiv.org/abs/1801.01313
https://doi.org/10.1093/imanum/drt008
https://doi.org/10.1093/imanum/drt008
https://arxiv.org/abs/1110.2437
https://doi.org/10.1016/j.jat.2018.02.003
https://doi.org/10.1016/j.jat.2018.02.003
https://arxiv.org/abs/1701.07214
https://doi.org/10.1016/j.spa.2019.08.007
https://doi.org/10.1016/j.spa.2019.08.007
https://arxiv.org/abs/1812.02103
https://doi.org/10.3842/SIGMA.2019.004
https://arxiv.org/abs/1807.08184
https://doi.org/10.1090/S0002-9939-03-06730-3
A Note on the Derivatives of Isotropic Positive Definite Functions on the Hilbert Sphere 7
[10] Fornberg B., Flyer N., A primer on radial basis functions with applications to the geosciences, CBMS-NSF
Regional Conference Series in Applied Mathematics, Vol. 87, Society for Industrial and Applied Mathematics
(SIAM), Philadelphia, PA, 2015.
[11] Gneiting T., Strictly and non-strictly positive definite functions on spheres, Bernoulli 19 (2013), 1327–1349,
arXiv:1111.7077.
[12] Gradshteyn I.S., Ryzhik I.M., Table of integrals, series, and products, 8th ed., Academic Press Inc., San
Diego, CA, 2014.
[13] Guinness J., Fuentes M., Isotropic covariance functions on spheres: some properties and modeling consider-
ations, J. Multivariate Anal. 143 (2016), 143–152.
[14] Hubbert S., Lê Gia Q.T., Morton T.M., Spherical radial basis functions, theory and applications, Springer-
Briefs in Mathematics, Springer, Cham, 2015.
[15] Jäger J., Klein A., Buhmann M., Skrandies W., Reconstruction of electroencephalographic data using radial
basis functions, Clinical Neurophysiology 127 (2016), 1978–1983.
[16] Lang A., Schwab C., Isotropic Gaussian random fields on the sphere: regularity, fast simulation and stochas-
tic partial differential equations, Ann. Appl. Probab. 25 (2015), 3047–3094, arXiv:1305.1170.
[17] Menegatto V.A., Strictly positive definite kernels on the Hilbert sphere, Appl. Anal. 55 (1994), 91–101.
[18] Nie Z., Ma C., Isotropic positive definite functions on spheres generated from those in Euclidean spaces,
Proc. Amer. Math. Soc. 147 (2019), 3047–3056.
[19] Schoenberg I.J., Positive definite functions on spheres, Duke Math. J. 9 (1942), 96–108.
[20] Trübner M., Ziegel J.F., Derivatives of isotropic positive definite functions on spheres, Proc. Amer. Math.
Soc. 145 (2017), 3017–3031, arXiv:1603.06727.
[21] Xu Y., Positive definite functions on the unit sphere and integrals of Jacobi polynomials, Proc. Amer. Math.
Soc. 146 (2018), 2039–2048, arXiv:1701.00787.
[22] Ziegel J., Convolution roots and differentiability of isotropic positive definite functions on spheres, Proc.
Amer. Math. Soc. 142 (2014), 2063–2077, arXiv:1201.5833.
[23] zu Castell W., Filbir F., Radial basis functions and corresponding zonal series expansions on the sphere,
J. Approx. Theory 134 (2005), 65–79.
https://doi.org/10.1137/1.9781611974041.ch1
https://doi.org/10.1137/1.9781611974041.ch1
https://doi.org/10.3150/12-BEJSP06
https://arxiv.org/abs/1111.7077
https://doi.org/10.1016/j.jmva.2015.08.018
https://doi.org/10.1007/978-3-319-17939-1
https://doi.org/10.1016/j.clinph.2016.01.003
https://doi.org/10.1214/14-AAP1067
https://arxiv.org/abs/1305.1170
https://doi.org/10.1080/00036819408840292
https://doi.org/10.1090/proc/14454
https://doi.org/10.1215/S0012-7094-42-00908-6
https://doi.org/10.1090/proc/13561
https://doi.org/10.1090/proc/13561
https://arxiv.org/abs/1603.06727
https://doi.org/10.1090/proc/13913
https://doi.org/10.1090/proc/13913
https://arxiv.org/abs/1701.00787
https://doi.org/10.1090/S0002-9939-2014-11989-7
https://doi.org/10.1090/S0002-9939-2014-11989-7
https://arxiv.org/abs/1201.5833
https://doi.org/10.1016/j.jat.2004.12.018
1 Introduction and main results
2 Preliminaries
3 Proof of Theorem 3
References
|
| id | nasplib_isofts_kiev_ua-123456789-210307 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:25:06Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Jäger, J. 2025-12-05T09:30:55Z 2019 A Note on the Derivatives of Isotropic Positive Definite Functions on the Hilbert Sphere / J. Jäger // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 23 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33B10; 33C45; 42A16; 42A82; 42C10 arXiv: 1905.08655 https://nasplib.isofts.kiev.ua/handle/123456789/210307 https://doi.org/10.3842/SIGMA.2019.081 In this note, we give a recursive formula for the derivatives of isotropic positive definite functions on the Hilbert sphere. We then use it to prove a conjecture stated by Trübner and Ziegel, which says that for a positive definite function on the Hilbert sphere to be in C²ˡ([0,π]), it is necessary and sufficient for its ∞ Schoenberg sequence to satisfy ∑ₘ₌₀ ∞ aₘmˡ < ∞. The author was a post-doctoral fellow funded by Justus Liebig University during the development of this research. I would like to express my gratitude to Professor M. Buhmann for his helpful comments on the paper. Thanks are also due to the anonymous referees for their thorough advice on how to improve this note. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Note on the Derivatives of Isotropic Positive Definite Functions on the Hilbert Sphere Article published earlier |
| spellingShingle | A Note on the Derivatives of Isotropic Positive Definite Functions on the Hilbert Sphere Jäger, J. |
| title | A Note on the Derivatives of Isotropic Positive Definite Functions on the Hilbert Sphere |
| title_full | A Note on the Derivatives of Isotropic Positive Definite Functions on the Hilbert Sphere |
| title_fullStr | A Note on the Derivatives of Isotropic Positive Definite Functions on the Hilbert Sphere |
| title_full_unstemmed | A Note on the Derivatives of Isotropic Positive Definite Functions on the Hilbert Sphere |
| title_short | A Note on the Derivatives of Isotropic Positive Definite Functions on the Hilbert Sphere |
| title_sort | note on the derivatives of isotropic positive definite functions on the hilbert sphere |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210307 |
| work_keys_str_mv | AT jagerj anoteonthederivativesofisotropicpositivedefinitefunctionsonthehilbertsphere AT jagerj noteonthederivativesofisotropicpositivedefinitefunctionsonthehilbertsphere |