Estimates for logarithmic and Riesz energies for spherical $t$-designs

TL;DR

This paper derives asymptotic formulas for the discrete logarithmic energy and order estimates for Riesz s-energy of well-separated spherical t-designs on high-dimensional spheres, advancing understanding of their energy properties.

Contribution

It provides the first asymptotic equalities and order estimates for energies of spherical t-designs, linking design properties with energy asymptotics.

Findings

Asymptotic equalities for logarithmic energy of spherical t-designs

Order estimates for Riesz s-energy for s ≥ d

Results applicable to high-dimensional spheres

Abstract

In this paper we find asymptotic equalities for the discrete logarithmic energy of sequences of well separated spherical $t$-designs on the unit sphere ${\mathbb{S}^{d}\subset\mathbb{R}^{d+1}}$, $d\geq2$. Also we establish exact order estimates for discrete Riesz $s$-energy, $s\geq d$, of sequences of well separated spherical $t$-designs.

Equations

$$\frac{1}{N}\sum _{\mathbf{x}\in X_N}p(\mathbf{x})={\int }_{{\mathbb{S}}^d}p(\mathbf{x})d{\sigma }_d(\mathbf{x}),\mathrm{deg}(p)\le t.$$

$$E_{log}^{(d)}(X_N):=\sum _{\genfrac{}{}{0ex}{}{i,j=1,}{i\ne j}}^N\log \frac{1}{|{\mathbf{x}}_i-{\mathbf{x}}_j|}=2\sum _{1\le j<i\le N}\log \frac{1}{|{\mathbf{x}}_i-{\mathbf{x}}_j|}.$$

$$\underset{i\ne j}{\min }|{\mathbf{x}}_i-{\mathbf{x}}_j|>\frac{c_1}{N^{\frac{1}{d}}}.$$

$${\mathcal{E}}_{log}^{(d)}(N):=\underset{X_N}{\inf }{E}_{log}^{(d)}(X_N),$$

$${\mathcal{E}}_{log}^{(d)}(N)=N^2\underset{{\mathbb{S}}^d}{\int }\underset{{\mathbb{S}}^d}{\int }\log \frac{1}{|\mathbf{x}-\mathbf{y}|}d{\sigma }_d(\mathbf{x})d{\sigma }_d(\mathbf{y})-\frac{1}{d}N\log N+\mathcal{O}(N).$$

$$E_{log}^{(d)}(X_N)=N^2\underset{{\mathbb{S}}^d}{\int }\underset{{\mathbb{S}}^d}{\int }\log \frac{1}{|\mathbf{x}-\mathbf{y}|}d{\sigma }_d(\mathbf{x})d{\sigma }_d(\mathbf{y})-\frac{1}{d}N\log N+\mathcal{O}(N).$$

$$E_s^{(d)}(X_N):=\frac{1}{2}\sum _{\genfrac{}{}{0ex}{}{i,j=1,}{i\ne j}}^N|{\mathbf{x}}_i-{\mathbf{x}}_j{|}^{-s},$$

$$E_s^{(2)}(X_N)\le c_s{N}^{1+\frac{s}{2}},$$

$$E_s^{(2)}(X_N)\le \frac{\sum _{k=0}^t\frac{1}{k+1}}{4}{N}^2+c_2{N}^2,$$

$$\underset{N\to \infty }{\lim }\frac{E_s^{(2)}(X_N)}{N^2\log N}=\frac{1}{8}.$$

$${\mathcal{E}}_s^{(d)}(N):=\underset{X_N}{\inf }{E}_s^{(d)}(X_N),$$

$$C_{d,s}^{(1)}{N}^{1+\frac{s}{d}}\le {\mathcal{E}}_s^{(d)}(N)\le C_{d,s}^{(2)}{N}^{1+\frac{s}{d}}.$$

$$\underset{N\to \infty }{\lim }(N^2\log N{)}^{-1}{\mathcal{E}}_s^{(d)}(N)=\frac{1}{2d}\frac{\Gamma (\frac{d+1}{2})}{\Gamma (\frac{d}{2})\Gamma (\frac{1}{2})}.$$

$$E_s^{(d)}(X_N)\ll N^{1+\frac{s}{d}},s>d$$

$$\underset{N\to \infty }{\lim }\frac{E_s^{(d)}(X_N)}{N^2\log N}=\frac{1}{2d\sqrt{\pi }}\frac{\Gamma (\frac{d}{2}+\frac{1}{2})}{\Gamma (\frac{d}{2})},s=d.$$

$$(a{)}_0:=1,(a{)}_n:=a(a+1)\dots (a+n-1)\mathrm{for}n\in \mathbb{N},$$

$$(a{)}_{\ell }=\frac{\Gamma (\ell +a)}{\Gamma (a)}.$$

$$\frac{\Gamma(n+a)}{\Gamma(n+b)}=n^{a-b}\Big{(}1+\mathcal{O}\Big{(}\frac{1}{n}\Big{)}\Big{)}\ \ \mathrm{as}\ \ n\rightarrow\infty.$$

$$\underset{{\mathbb{S}}^d}{\int }f(⟨\mathbf{x},\mathbf{y}⟩)d{\sigma }_d(\mathbf{x})=\frac{\Gamma (\frac{d+1}{2})}{\sqrt{\pi }\Gamma (\frac{d}{2})}\underset{-1}{\overset{1}{\int }}f(t)(1-t^2{)}^{\frac{d}{2}-1}dt\forall \mathbf{y}\in {\mathbb{S}}^d.$$

$$P_{\ell}^{(\alpha,\beta)}(1)=\binom{\ell+\alpha}{\ell}=\frac{(1+\alpha)_{\ell}}{\ell!}=\frac{1}{\Gamma(1+\alpha)}\ell^{\alpha}\left(1+\mathcal{O}\Big{(}\frac{1}{\ell}\Big{)}\right),\quad\alpha,\beta>-1.$$

$$P_{\ell }^{(\alpha ,\beta )}(-x)=(-1{)}^l{P}_{\ell }^{(\alpha ,\beta )}(x)$$

$$P_m^{(\gamma ,\gamma )}(x)=\frac{(\gamma +1{)}_m}{(2\gamma +1{)}_m}\sum _{k=0}^{[\frac{m}{2}]}\frac{(2\alpha +1{)}_{m-2k}}{(\alpha +1{)}_{m-2k}}\frac{(\gamma +\frac{1}{2}{)}_{m-k}(\alpha +\frac{3}{2}{)}_{m-2k}(\gamma -\alpha {)}_k}{(\alpha +\frac{3}{2}{)}_{m-k}(\alpha +\frac{1}{2}{)}_{m-2k}k!}{P}_{m-2k}^{\alpha ,\alpha }(x).$$

$$P_{\ell}^{(\alpha,\beta)}(\cos\theta)=\frac{1}{\sqrt{\pi}}\ell^{-1/2}\Big{(}\sin\frac{\theta}{2}\Big{)}^{-\alpha-1/2}\Big{(}\cos\frac{\theta}{2}\Big{)}^{-\beta-1/2}\\ \times\Big{\{}\cos\Big{(}\Big{(}\ell+\frac{\alpha+\beta+1}{2}\Big{)}\theta-\frac{2\alpha+1}{4}\pi\Big{)}+\mathcal{O}(\ell\sin\theta)^{-1}\Big{\}}.$$

$$|P_{\ell }^{(\alpha ,\beta )}(\cos \theta )|\le {\tilde{c}}_{\alpha ,\beta }{\ell }^{-1/2}(\sin \theta {)}^{-\alpha -1/2}+{\tilde{c}}_{\alpha ,\beta }{\ell }^{-3/2}(\sin \theta {)}^{-\alpha -3/2},\alpha \ge \beta .$$

$$\frac{d}{dx}{P}_n^{(\alpha ,\beta )}(x)=\frac{\alpha +\beta +n+1}{2}{P}_{n-1}^{(\alpha +1,\beta +1)}(x).$$

$$(1-x)^{-\frac{s}{2}}=2^{2\lambda-\frac{s}{2}}\pi^{-\frac{1}{2}}\Gamma(\lambda)\Gamma\Big{(}\lambda-\frac{s}{2}+\frac{1}{2}\Big{)}\\ \times\sum\limits_{n=0}^{\infty}\frac{(n+\lambda)(\frac{s}{2})_{n}}{\Gamma(n+2\lambda-\frac{s}{2}+1)}\frac{(2\lambda)_{n}}{(\lambda+\frac{1}{2})_{n}}P_{n}^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}(x).$$

$$S(\mathbf{x};\varphi):=\big{\{}\mathbf{y}\in\mathbb{S}^{d}\big{|}\langle\mathbf{x},\mathbf{y}\rangle\geq\cos\varphi\big{\}}.$$

$$|S(\mathbf{x};\phi )|=\frac{\Gamma ((d+1)/2)}{\sqrt{\pi }\Gamma (d/2)}\underset{\cos \phi }{\overset{1}{\int }}(1-t^2{)}^{\frac{d}{2}-1}dt\asymp (1-\cos \phi {)}^{\frac{d}{2}}\text{as }\phi \to 0.$$

$$\alpha_{N}:=\arccos\Big{(}1-\frac{c^{2}_{1}}{8N^{\frac{2}{d}}}\Big{)},$$

$$\sin \theta \le \theta \le \frac{\pi }{2}\sin \theta ,0\le \theta \le \frac{\pi }{2},$$

Full text

1 Introduction

Let $\mathbb{S}^{d}=\{\mathbf{x}\in\mathbb{R}^{d+1}:\ |\mathbf{x}|=1\}$, where $d\geq 2$, be the unit sphere in the Euclidean space $\mathbb{R}^{d+1}$, equipped with the Lebesgue measure $\sigma_{d}$ normalized by $\sigma_{d}(\mathbb{S}^{d})=1$.

Definition 1

A spherical $t$-design is a finite subset $X_{N}\subset\mathbb{S}^{d}$ with a characterising property that an equal weight integration rule with nodes from $X_{N}$ integrates all spherical polynomials $p$ of total degree at most $t$ exactly; that is,

[TABLE]

Here $N$ is the cardinality of $X_{N}$ or the number of points of spherical design.

The concept of spherical $t$-design was introduced by Delsarte, Goethals and Seidel in the groundbreaking paper Delsarte-Goethals-Seidel1977:spherical_designs , since then they attracted a lot of interest from scientific community (see e.g., BrauchartGrabner ).

The logarithmic energy of a set of $N$ distinct points (or an $N$-point set) $X_{N}$ on $\mathbb{S}^{d}$ is defined as

[TABLE]

This paper investigates the logarithmic energy for spherical $t$-designs. Spherical $t$-designs of a fixed strength $t$ can have points arbitrary close together (see, e.g. HesseLeopardiTheCoulombEnergy ), hence the logarithmic energy of $N$-point spherical $t$-designs can have no asymptotic bounds in terms of $t$ and $N$. That’s why we will have additional condition and consider the sequences of well–separated spherical $t$-designs.

Definition 2

A sequence of $N$-point sets $X_{N}$, X_{N}=\big{\{}\mathbf{x}_{1},\ldots,\mathbf{x}_{N}\big{\}}, is called well-separated if there exists a positive constant $c_{1}$ such that

[TABLE]

The existence of $N$-point spherical $t$-designs with $N(t)\asymp t^{d}$ was proven by Bondarenko, Radchenko and Viazovska Bondarenko-Radchenko-Viazovska2013:optimal_designs . They showed that for ${d\geq 2}$, there exists a constant $c_{d}$, which depends only of $d$, such that for every $N\geq c_{d}t^{d}$ there exists a spherical $t$-design on $\mathbb{S}^{d}$ with $N$ points. Two years later by these authors in Bondarenko-Radchenko-Viazovska2015:Well_separated the existence of $N$-point well–separated spherical $t$-designs with $N(t)\asymp t^{d}$ was proven. Namely, they showed that for each ${d\geq 2}$, $t\in\mathbb{N}$, there exist positive constants $c_{d}$ and $\lambda_{d}$, depending only on $d$, such that for every $N\geq c_{d}t^{d}$, there exists a spherical $t$-design on $\mathbb{S}^{d}$, consisting of $N$ points $\{\mathbf{x}_{i}\}_{i=1}^{N}$ with $|\mathbf{x}_{i}-\mathbf{x}_{j}|\geq\lambda_{d}N^{-\frac{1}{d}}$ for $i\neq j$.

On the basis of these results we always assume that $N=N(t)\asymp t^{d}$.

We write $a_{n}\asymp b_{n}$ to mean that there exist positive constants $C_{1}$ and $C_{2}$ independent of $n$ such that $C_{1}a_{n}\leq b_{n}\leq C_{2}a_{n}$ for all $n$.

Denote by $\mathcal{E}_{log}^{(d)}(N)$ the minimal discrete logarithmic energy for $N$-points on the sphere

[TABLE]

where the infimum is taken over all $N$-points subsets of $\mathbb{S}^{d}$.

From the papers of Wagner Wagner , Kuijlaars and Saff KuijlaarsSaff:1998Asymptotics and Brauchart Brauchart2008 it follows that for $d\geq 2$ and as $N\rightarrow\infty$ the following asymptotic equality holds

[TABLE]

Also in BoyvalenkovDragnevHardinSaffStoyanova some general upper and lower bounds for the energy of spherical designs were found.

We show that for every well-separated sequence of $N$-point spherical $t$-designs on $\mathbb{S}^{d}$, $d\geq 2$, with $N\asymp t^{d}$ the following asymptotic equality holds

[TABLE]

Comparing two last formulas, we have that the leading and second terms are exactly the same, and third terms are of the same order. So, we can summarize, that for logarithmic energy well-separated spherical $t$-designs are as good as point sets which minimize the logarithmic energy.

For given $s>0$ the discrete Riesz $s$-energy of a set of $N$ distinct points (or an $N$-point set) $X_{N}$ on $\mathbb{S}^{d}$ is defined as

[TABLE]

where $|\mathbf{x}|$ denotes the Euclidian norm in $\mathbb{R}^{d+1}$ of the vector $\mathbf{x}$. In the case $s=d-1$ the energy (5) is called as Coulomb energy.

Hesse Hesse:2009s-energy showed, that if spherical $t$-designs with ${N=\mathcal{O}(t^{2})}$ exist, then they have asymptotically minimal Riesz energy $E_{s}^{(2)}(X_{N})$ for $s\geq 2$. In particular, under the assumption that $N\leq\kappa t^{2}$, it was shown that for $s>2$, there exists a positive constant $c_{s}$ such that for every well separated sequence $N$ -point spherical $t$-designs the following estimate holds

[TABLE]

and for $s=2$, there exists a positive constant $c_{2}$, such that

[TABLE]

and

[TABLE]

Denote by $\mathcal{E}_{s}^{(d)}(X_{N})$ the minimal discrete $s$-energy for $N$-points on the sphere

[TABLE]

where the infimum is taken over all $N$-points subsets of $\mathbb{S}^{d}$.

Kuijlaars and Saff KuijlaarsSaff:1998Asymptotics proved that for $d\geq 2$ and $s>d$, there exist constants $C_{d,s}^{(1)},C_{d,s}^{(2)}>0$, such that

[TABLE]

Also in KuijlaarsSaff:1998Asymptotics it was showed that for $s=d$ the following formula holds

[TABLE]

We show that for every well-separated sequence of $N$-point spherical $t$-designs on $\mathbb{S}^{d}$, $d\geq 2$, with $N\asymp t^{d}$ the following relations are true:

[TABLE]

and

[TABLE]

Here and further we use the Vinogradov notation $a_{n}\ll b_{n}$ to mean that there exists positive constant $C$ independent of $n$ such that $a_{n}\leq Cb_{n}$ for all $n$.

First, we observe, that since $\mathcal{E}_{s}^{(d)}(N)\leq{E}_{s}^{(d)}(X_{N})$ for any $N$-point set, the lower bound in (10) provides the lower bound for the $s$-energy of any $N$-point set. So, asymptotically for Riesz $s$-energy, $s\geq d$, well-separated spherical $t$-designs are as good as point sets which minimize the $s$-energy.

This paper is organised as follows: Section 2 provides basic notations and necessary background for Jacobi polynomials, Section 3 contains formulation of main results and proofs of theorems.

2 Preliminaries

In this paper we use the Pochhammer symbol $(a)_{n}$, where $n\in\mathbb{N}_{0}$ and $a\in\mathbb{R}$, defined by

[TABLE]

which can be written in the terms of the gamma function $\Gamma(z)$ by means of

[TABLE]

For fixed $a,b$ the following asymptotic equality is true

[TABLE]

For any integrable function $f:[-1,1]\rightarrow\mathbb{R}$ (see, e.g., Mueller1966:spherical_harmonics ) we have

[TABLE]

The Jacobi polynomials $P_{\ell}^{(\alpha,\beta)}(x)$ are the polynomials orthogonal over the interval $[-1,1]$ with the weight function $w_{\alpha,\beta}(x)=(1-x)^{\alpha}(1+x)^{\beta}$ and normalised by the relation

[TABLE]

(see, e.g., (Magnus-Oberhettinger-Soni1966:formulas_theorems, , (5.2.1))).

We will also use formula

[TABLE]

and the connection coefficient formula (see, e.g., Theorem 7.1.4 from SpecialFunctions )

[TABLE]

For fixed ${\alpha,\beta>-1}$ and ${0<\theta<\pi}$, the following relation gives an asymptotic approximation for $\ell\rightarrow\infty$ (see, e.g., (Szegoe1975:orthogonal_polynomials, , Theorem 8.21.13))

[TABLE]

Thus, for $c_{\alpha,\beta}\ell^{-1}\leq\theta\leq\pi-c_{\alpha,\beta}\ell^{-1}$ the last asymptotic equality yields

[TABLE]

The following differentiation formula holds

[TABLE]

If $\lambda>s-1$, $s\geq d$, then taking into account formula (Magnus-Oberhettinger-Soni1966:formulas_theorems, , (5.3.4))) and the fact that the Gegenbauer polynomials are a special case of the Jacobi polynomials $P_{n}^{(\alpha,\beta)}(x)$ (see, e.g., (Magnus-Oberhettinger-Soni1966:formulas_theorems, , (5.3.1))), we have that for ${-1<x<1}$ the following expansion holds

[TABLE]

3 Main results

By a spherical cap $S(\mathbf{x};\varphi)$ of centre $\mathbf{x}$ and angular radius $\varphi$ we mean

[TABLE]

The normalised surface area of a spherical cap is given by

[TABLE]

If for sequence $(X_{N})_{N}$ condition (2) holds, then any spherical cap $S(\mathbf{x};\alpha_{N})$, $\mathbf{x}\in\mathbb{S}^{d}$, where

[TABLE]

contains at most one point of the set $(X_{N})_{N}$.

From the elementary estimates

[TABLE]

we obtain

[TABLE]

The following two theorems are the main result of this paper.

Theorem 3.1

Let $d\geq 2$ be fixed, $(X_{N(t)})_{t}$ be a sequence of well-separated spherical $t$-designs on $\mathbb{S}^{d}$ and $N(t)\asymp t^{d}$. Then for the logarithmic energy $E_{log}^{(d)}(X_{N})$ the following estimate holds

[TABLE]

Theorem 3.2

Let $d\geq 2$ be fixed, and $(X_{N(t)})_{t}$ be a sequence of well-separated spherical $t$-designs on $\mathbb{S}^{d}$ and $N(t)\asymp t^{d}$. Then for $s>d$ the $s$-energy $E_{s}^{(d)}(X_{N})$ satisfies the estimate

[TABLE]

and for $s=d$, the $s$-energy $E_{s}^{(d)}(X_{N})$ satisfies following estimates

[TABLE]

and

[TABLE]

3.1 Proof of Theorem 3.1

For each $i\in\{1,\ldots,N\}$ we divide the sphere $\mathbb{S}^{d}$ into an upper hemisphere $H_{i}^{+}$ with ’north pole’ $\mathbf{x}_{i}$ and a lower hemisphere $H_{i}^{-}$:

[TABLE]

[TABLE]

Noting that

[TABLE]

the logarithmic energy can be written in the form

[TABLE]

Let $\lambda>d+1$. The, putting $s=2$ in (20), we get

[TABLE]

Formula (19) implies, that

[TABLE]

Integrating from [math] to $x$, we have

[TABLE]

We split the $\log$-energy into two parts

[TABLE]

From (2) and the fact the spherical cap $S(-\mathbf{x}_{j};\alpha_{N})$ contains at most one point of $X_{N}$, the second term in (34), where the scalar product is close to $-1$, can be bounded from above by

[TABLE]

Taking into account (30), (33)–(35), we deduce

[TABLE]

where

[TABLE]

[TABLE]

and

[TABLE]

Let us show that

[TABLE]

Applying (12), (13) and (18) to (38), we have

[TABLE]

From (Brauchart-Hesse2007:numerical_integration, , (3.30) and (3.33)), it follows that

[TABLE]

Estimates (24) and (42) imply

[TABLE]

This proves (40).

Now let us find the estimate for $E_{H_{\log,t}}(X_{N})$. The polynomial $H_{log,t}$ is a spherical polynomial of degree $t$ and $X_{N}$ is a spherical $t$-design. That is why, an equal weight integration rule with nodes from $X_{N}$ integrates $H_{log,t}$ exactly, and

[TABLE]

Let $b_{0}\in\mathbb{R}_{+}$ is such, that for $\beta_{N}:=\arccos(1-b_{0}N^{-\frac{2}{d}})$ the following relation holds

[TABLE]

It is clear, that

[TABLE]

Then

[TABLE]

where

[TABLE]

Now we shall prove that

[TABLE]

Using (14), (41) and (46), we get

[TABLE]

From the definition of $\beta_{n}$ it is easy to see, that

[TABLE]

According to the definition of $\beta_{N}$ (45) we deduce

[TABLE]

Formulas (15), (19) and (37) imply

[TABLE]

From (16), (18) and (37) it follows that

[TABLE]

Relations (13) and (15) allow us to write

[TABLE]

Hence, (54) and (55) enable us to obtain

[TABLE]

Using (14), (53) and (56), we deduce

[TABLE]

Applying (56), we have

[TABLE]

where $\xi\in[-1,-1+b_{0}N^{-\frac{2}{d}}]$.

Relations (50)-(53), (57) and (58) prove (49).

Integrating by parts, we obtain

[TABLE]

So, combining (36), (40), (47), (49) and (59), we get

[TABLE]

This implies (25). Theorem 3.1 is proved.

∎

3.2 Proof of Theorem 3.2

In the same way as in the case for logarithmic energy, we split the $s$-energy into two parts

[TABLE]

Taking into account that the Jacobi series (20) converges uniformly in {\Big{[}-1+\frac{c^{2}_{1}}{8N^{\frac{2}{d}}},1-\frac{c^{2}_{1}}{8N^{\frac{2}{d}}}\Big{]}}, for $\lambda>s-1$ we get that

[TABLE]

where

[TABLE]

[TABLE]

Formula (65) from GrabnerStepanyukJAT implies

[TABLE]

Hence,

[TABLE]

where we have used formulas (3.2), (3.2) and (65).

The polynom $H_{s,t}$ is a spherical polynomial of degree $t$ and $X_{N}$ is a spherical $t$-design. So, an equal weight integration rule with nodes from $X_{N}$ integrates $H_{s,t}$ exactly, and

[TABLE]

From relations (12), (13), (15) and (3.2) we obtain

[TABLE]

Let now estimate the integral from (3.2). Substituting $\gamma=\lambda-\frac{1}{2}$, $\alpha=\frac{d}{2}-1$ in formula (17), we have

[TABLE]

Since

[TABLE]

then (3.2) yields

[TABLE]

So,

[TABLE]

where we have used (37) and (71).

Thus, if $s>d$, then

[TABLE]

and the relations (3.2), (3.2) and (73) imply

[TABLE]

This implies (26).

If $s=d$, then using (12) and (13) from (72) we have

[TABLE]

Formulas (3.2), (3.2) and (3.2) imply (27) and (28). Theorem 3.2 is proved. ∎

Acknowledgements.

The author is supported by the Austrian Science Fund FWF project F5503 (part of the Special Research Program (SFB) “Quasi-Monte Carlo Methods: Theory and Applications”