Isotropic Positive Definite Functions on Spheres Generated from Those in Euclidean Spaces

ISOTROPIC POSITIVE DEFINITE FUNCTIONS ON SPHERES GENERATED FROM THOSE IN EUCLIDEAN SPACES

A Dissertation by

Zhihui Nie

Master of Science, Tianjin University of Technology and Education, 2014

Bachelor of Science, Tianjin University of Technology and Education, 2011

Submitted to the Department of Mathematics, Statistics, and Physics and the faculty of the Graduate School of Wichita State University in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy

The following faculty members have examined the ﬁnal copy of this dissertation for form and content, and recommend that it be accepted in partial fulﬁllment of the requirement for the degree of Doctor of Philosophy with a major in Applied Mathematics.

Chunsheng Ma, Committee Chair

Tianshi Lu, Committee Member

Xiaomin Hu, Committee Member

Adam Jaeger, Committee Member

Hyuck M. Kwon, Committee Member

Accepted for the College of Liberal Arts and Sciences

Andrew Hippisley, Dean

Accepted for the Graduate School

Kerry Wilks, Interim Dean

iii DEDICATION

To my parents

iv ACKNOWLEDGEMENTS

This dissertation would not have been possible without the guidance and help of several individuals who in one way or another contributed and extended their valuable as- sistance in the preparation and completion of this study. My ﬁrst debt of gratitude must go to my advisor, Professor Chunsheng Ma, who was truly an inspiration. Without his invalu- able guidance, this thesis would not have been possible. I would also like to to express my gratitude to Professor Daowei Ma, Professor Tianshi Lu, Professor Xiaomin Hu, Professor Adam Jaeger, Professor Hyuck M. Kwon and Professor Hua Liu for the contribution of their valuable time. In addition, I am grateful to acknowledge my friends in Wichita State Uni- versity, J. Chen, X. Wei, L. Liang, Y. Wang, N. Nguyen, S. Rahmati, S. Zhang and F. Yan, for their friendship and warmth. Last but not the least, huge thank you to my parents, for their full supports.

v ABSTRACT

For a continuous function g(x) on [0, π] with g(π) = 0, if it satisﬁes the inequality

Z π α+ 1 u 2 g(u)J 1 (xu)du ≥ 0, x ≥ 0, α− 2 0 it is shown in this thesis that

Z π (α) 2α g(u)Pn (cos u) sin udu ≥ 0, n ∈ N0, 0

(ν) where α is a nonnegative integer, and Jν(x) and Pn (x) denote the Bessel function and the ultraspherical polynomial, respectively. As a consequence, an isotropic and continuous positive deﬁnite function in the Euclidean space, if it is compactly supported, can be adopted as an isotropic positive deﬁnite function on a sphere.

vi TABLE OF CONTENTS

Chapter Page

1 INTRODUCTION ...... 1

2 BACKGROUND INFORMATION ...... 5

2.1 Deﬁnitions ...... 5 2.2 The Gamma Function ...... 7 2.3 Orthogonal Polynomial ...... 8 2.4 Bessel Functions ...... 9 2.5 Abel-Summability ...... 10

3 ISOTROPIC POSITIVE DEFINITE FUNCTIONS IN EUCLIDEAN SPACE . . 12

3.1 Integral Transforms ...... 12 3.2 Characterizations of Positive Deﬁnite Functions ...... 13 3.3 Schoenberg’s Characterization ...... 14 3.4 Properties of Positive Deﬁnite Functions ...... 15

4 ISOTROPIC POSITIVE DEFINITE FUNCTIONS ON SPHERES ...... 20

4.1 Some Classical Results Concerning Ultraspherical Polynomials ...... 20 4.2 Schoenberg’s Characterization ...... 23 4.3 Recent Results ...... 24

5 ISOTROPIC POSITIVE DEFINITE FUNCTIONS ON SPHERES GENERATED FROM THOSE IN EUCLIDEAN SPACES ...... 26

5.1 Main Results ...... 26 5.2 Some Lemmas and Their Proofs ...... 30 5.3 Proof of Theorems 5.1.1 and 5.1.2 ...... 38

6 PLANS FOR FUTURE WORK ...... 41

REFERENCES ...... 42

vii CHAPTER 1

INTRODUCTION

There has run on renewed interest in isotropic positive deﬁnite functions in the Euclidean

space Rd or on the sphere Sd, with applications in approximation theory [4], [8], [10], [28],

machine learning [20], probability, and spatial statistics [6], [15], [16], where Sd is the spherical

shell of radius 1 and center 0 in Rd+1, i.e., Sd = {x ∈ Rd+1, kxk = 1}, kxk is the Euclidean norm of x, and d is a positive integer.

An isotropic (radial, or spherically symmetric) function g(kxk): Rd → R, is said to be positive deﬁnite, if the inequality

n n X X aiajg(kxi − xjk) ≥ 0 i=1 j=1

d holds for every n ∈ N, any ai ∈ R and xi ∈ R (i = 1, . . . , n), where N is the set of positive integers.

When g(kxk) is continuous in Rd or g(·) is continuous on R, it is well-known [21] that

g(kxk) is positive deﬁnite in Rd if and only if g(x) admits a representation of the form Z ∞ − d−2 g(x) = (xu) 2 J d−2 (xu)dF (u), x ≥ 0, (1.0.1) 0 2 where F (x) is a bounded and nondecreasing function on [0, ∞), and Jν(x) denotes a Bessel function of order ν.

As an example [2],  λ λ  (a − x) , 0 ≤ x ≤ a, g(x) = (a − x)+ = (1.0.2)  0, x > a,

d d makes g(kxk) an isotropic positive deﬁnite function in R whenever λ ≥ 2 , where a is a positive constant and [x] denotes the greatest integer less than or equal to x.

For two points x and y on Sd, their spherical (great circle, angular, or geodesic) distance,

denotes by ϑ(x, y), is the distance between x and y on the largest circle on Sd that passes

1 through them. More precisely,

0 d ϑ(x, y) = arccos(x y), x, y ∈ S , (1.0.3) where x0y denotes the inner product between x and y. Since

kx − yk2 = kxk2 + kyk2 − 2x0y = 2 − 2x0y,

another expression for ϑ(x, y) in terms of the Euclidean distance kx − yk is

1 ϑ(x, y) = arccos 1 − kx − yk2 , x, y ∈ d. 2 S

Evidently, 0 ≤ ϑ(x, y) ≤ π.

A function g(ϑ(x1, x2)), or g(ϑ) for simplicity, is called an isotropic positive deﬁnite

function on Sd if n n X X aiajg(ϑ(xi, xj)) ≥ 0 i=1 j=1 d holds for every n ∈ N, any ai ∈ R and xi ∈ S , i = 1, . . . , n. Thus, an isotropic positive function on a sphere depends on its arguments via the spher-

0 ical distance ϑ(x1, x2),or equivalently, via the inner product x1x2.

For a continuous function g(ϑ) on [0, π], it is positive deﬁnite on Sd if and only if it can be expressed [22] as

∞ d−1 X ( 2 ) g(ϑ) = bnPn (cos ϑ), ϑ ∈ [0, π], (1.0.4) n=0

∞ d−1 P ( 2 ) (ν) where {bn, n ∈ N0} is a sequence of nonnegative constants, bnPn (1) converges, Pn (x) n=0 denotes the Gagenbauer or ultraspherical polynomial [25] of degree n, and N0 is the set of nonnegative integers. In other words, g(ϑ) is positive deﬁnite on Sd if and only if

Z π d−1 ( 2 ) d−1 bn = g(ϑ)Pn (cos ϑ) sin ϑdϑ ≥ 0, n ∈ N0, (1.0.5) 0

∞ d−1 ∞ d−1 P ( 2 ) P ( 2 ) and bnPn (1) converges. For d ≥ 2, the convergence of bnPn (1) is equivalent n=0 n=0 ∞ P d−2 to that of n bn [6]. n=0

2 A question of interest is: if a function g(x) on [0, ∞) makes g(kxk) an isotropic positive

deﬁnite function in Rd, would it also make g(ϑ) an isotropic positive deﬁnite function on

Sd? Such a question may be answered on a case-by-case basis. A counterexample is g(x) = exp(−x2), x ≥ 0, it is known that g(kxk) is positive deﬁnite in all dimensions, but g(ϑ) is

not positive deﬁnite even on S1. On the other hand, a stimulating example is (1.0.2), for which it is shown in [4] that g(ϑ) is an isotropic positive deﬁnite function on Sd if d ≤ 7,

but, actually, the restriction d ≤ 7 can be removed [28]. More interestingly, for α, β ∈ N0, and max(α, β) > 0, Xu [28] proves that

Z a 1 1 2α 2β (α− 2 , β− 2 ) ϑ ϑ g(ϑ)Pn (cos ϑ) sin cos dϑ > 0, n ∈ N, 0 2 2

(α,β) for every a ∈ (0, π] and λ ≥ α + 1, where Pn (x) is the Jacobi polynomial [25] of degree n.

(α,β) When α = β, Pn (x) reduces to the ultraspherical polynomial, with

1 Γ α + Γ(n + 2α) (α− 1 ,α− 1 ) 1 P (α)(x) = 2 P 2 2 (x), n ∈ , α > − ; (1.0.6) n 1 n N0 2 Γ(2α)Γ n + α + 2 see, e.g., (4.7.1) of [25].

A simple but important feature of the function (1.0.2) is that it is of compact support

and takes nonnegative values. This motivates us to consider the following question: for a

compactly supported function g(x) on [0, ∞), if it makes g(kxk) an isotropic positive deﬁnite

function in Rd, will it also make g(ϑ) an isotropic positive deﬁnite function on Sd? The answer is yes, in a particualr case d = 1 or 3, but, as is conjectured in [15], it would be of interest to

see whether such a result holds for a general d, for which a diﬃculty arises when one deals

with the connection between the two bases, the Bessel functions for Rd and ultraspherical

polynomials for Sd. A useful bridge for the connection employed in [28] is Theorem 8.1 of [25], which is generalized in Theorem 5.1.3 below for us to answer the above question.

In this thesis we develop isotropic positive deﬁnite functions on odd dimensional spheres from those in Euclidean spaces. The rest of the thesis is organized as follows. We state the background information in chapter 2. The characterizations of isotropic positive deﬁnite functions on Rd were reviewed in chapter 3. We discuss the characterizations of positive

3 deﬁnite functions on Sd in chapter 4. We introduce and prove our main result in chapter 5. There is an open question conjectured in chapter 6 for further research.

4 CHAPTER 2

BACKGROUND INFORMATION

2.1 Deﬁnitions

In this expository section, we list the deﬁnitions of isotropic functions in Rd and on

Sd, and those of positive deﬁnite functions in Rd and on Sd, as well as their connections to positive deﬁnite matrices.

Deﬁnition 2.1.1. (i) A function G : Rd × Rd → R is said to be isotropic, radial or spherical

symmetric if there exists a function g : [0, ∞) → R such that

G(x, y) = g(kx − yk)

for all x, y ∈ Rd.

(ii) A function G : Rd → R is said to be isotropic, radial or spherical symmetric if there

exists a function g : [0, ∞) → R such that

G(x) = g(kxk)

for all x ∈ Rd.

Deﬁnition 2.1.2. A complex-valued function g(x, y) of two arbitrary points of Rd is called positive deﬁnite, if it enjoys the following properties: (i) Hermitian symmetry

g(x − y) = g(x − y).

d (ii) For any ﬁnite set of xj ∈ R and complex numbers zj (with conjugate complexz ¯j), 1 ≤ j ≤ n, we have n n X X g(xj − xk)zjz¯k ≥ 0. (2.1.1) j=1 k=1 If the quadratic form (2.1.1) is zero only for z ≡ 0 , then g is called strictly positive deﬁnite.

5 Here, we introduce a general deﬁnition for complex-valued positive deﬁnite functions.

The reason for this is that it allows us to use techniques such as Fourier transforms more naturally in Chapter 3. However, we will see that for even, real-valued functions it suﬃces

to investigate the quadratic form only for real numbers z ∈ R.

Deﬁnition 2.1.3. A d×d symmetric real matrix M is said to be positive deﬁnite if x0Mx ≥ 0 for all x ∈ Rd.

Strictly positive deﬁnite matrices are deﬁned similarly, except that the above scalar x0Mx is positive for every x 6= 0.

Now, we define positive definite functions by using positive definite matrices.

Deﬁnition 2.1.4. A function G : Rd × Rd → R is said to be positive deﬁnite if for any

d N ∈ N and any ﬁnite set of points {x1, ··· , xN } in R , the N × N matrix

N G[XN ] = [G(xi, xj)]i,j=1

is positive definite. It is said to be strictly positive definite if the matrix G[XN ] is strictly positive definite.

Deﬁnition 2.1.4 is equivalent to deﬁnition 2.1.5 (i).

Deﬁnition 2.1.5. (i) A function G : Rd × Rd → R is said to be positive deﬁnite, if the inequality n n X X aiajG(xi, xj) ≥ 0 i=1 j=1 d holds for every n ∈ N, any ai ∈ R, and any xi ∈ R (i = 1, ··· , n).

(ii) An isotropic function g(x): Rd → R, is said to be positive deﬁnite, if the inequality

n n X X aiajg(kxi − xjk) ≥ 0 i=1 j=1

d holds for every n ∈ N, any ai ∈ R and xi ∈ R (i = 1, . . . , n).

6 Deﬁnition 2.1.6. A function G : Sd × Sd → R is said to be isotropic, if there exists a

function g : [0, π] → R such that G(x, y) = g(ϑ(x, y)) for all x, y ∈ Sd, where ϑ(x, y) is the spherical distance between x and y deﬁned in (1.0.3).

Deﬁnition 2.1.7. (i) A function G : Sd × Sd → R is said to be positive deﬁnite on Sd, if the inequality n n X X aiajG(xi, xj) ≥ 0 i=1 j=1 d holds for every n ∈ N, any ai ∈ R and xi ∈ S (i = 1, ··· , n).

(ii) A function g(ϑ(x1, x2)), or g(ϑ) for simplicity, is called an isotropic positive deﬁnite

function on Sd if n n X X aiajg(ϑ(xi, xj)) ≥ 0 i=1 j=1 d holds for every n ∈ N, any ai ∈ R and xi ∈ S , i = 1, . . . , n.

2.2 The Gamma Function

The Euler integral of the second kind Z ∞ Γ(z) = xz−1e−xdx 0 deﬁnes the Gamma function Γ(z) for all complex numbers which have the positive real part.

In particular, Γ(1) = 1. Using integration by parts, we get the recurrence formula Z ∞ z−1 −x ∞ z−2 −x Γ(z) = −[x e ]0 + (z − 1) x e dx = (z − 1)Γ(z − 1). (2.2.1) 0 If we restrict the domain on the set of positive integers N, then we get

Γ(n) = (n − 1)!

Now, we list the following lemma [19] as a reference.

Lemma 2.2.1. If n → ∞, then Γ(n + k) ∼ nk, k ∈ , (2.2.2) Γ(n) R where the symbol ∼ means that the ratio of the two sides converges to 1 or asymptotically converges.

7 2.3 Orthogonal Polynomial

(α) Gegenbauer polynomials or ultraspherical polynomials, denoted by Pn (x), are orthog-

2 α− 1 onal polynomials on the interval [−1, 1] with respect to the weight function (1 − x ) 2 .

They are special cases of Jacobi polynomials, occasionally called hypergeometric polynomi-

(α,β) als, denoted by Pn (x), which are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight (1 − x)α(1 − x)β on the interval [−1, 1].

(α) The Gegenbauer polynomials Pn (x) of degree n can be deﬁned in terms of their generating function [9]:

∞ 1 X = P (α)(x)tn, t ∈ (−1, 1). (2.3.1) (1 − 2xt + t2)α n n=0

(α) The function Pn (x) is even if n is even and odd if n is odd. For α = 0, we follow [22] and set

(0) Pn (cos ϑ) = cos nϑ, n ∈ N0, ϑ ∈ [0, π].

The Gegenbauer polynomials satisfy the recurrence relation [24]:

α P(0)(x) = 1,

α P(1)(x) = 2αx, 1 P α (x) = [2x(n + α − 1)P α (x) − (n + 2α − 2)P α (x)], n ∈ . (n) n n−1 n−2 N0

The Jacobi polynomials are deﬁned as follows:

(α + 1) P (α,β)(x) = n F −n, 1 + α + β + n; α + 1; 1 (1 − x) , n n! 2 1 2

where ∞ n X (a)n(b)n z F (a, b; c; z) = 2 1 (c) n! n=0 n

is the hypergeometric function which is deﬁned for |z| < 1 by the power series and (α + 1)n denotes the Pochhammer’s symbol (for the rising factorial) ,i.e.,

(α + 1)n = (α + 1)(α + 2), ··· , (α + n + 1).

8 In this case, the series for the hypergeometric function is ﬁnite, and an explicit expression

(α,β) [25] of Pn (x) is

n m X Γ(α + n + 1) n Γ(α + β + n + m + 1) x − 1 P (α,β)(x) = , n ∈ N . n n! Γ(α + β + n + 1) m Γ(α + m + 1) 2 0 m=0 For reference later on, we note the identities of Jacobi polynomials

n + α P (α,β)(1) = , n n

(α,β) P0 (x) = 1.

(α,β) If α = β, the Jacobi polynomial Pn (x) reduces to the Gegenbauer polynomial, with

1 (α− 1 ,α− 1 ) Γ(2α)Γ n + α + 1 P 2 2 (x) = 2 P (α)(x), n ∈ , α > − . n 1 n N0 2 Γ α + 2 Γ(n + 2α) 2.4 Bessel Functions

The Bessel function of the ﬁrst kind of order α, denoted as Jα(z), is deﬁned by its series expansion around z = 0, ∞ X (−1)ν(z/2)α+2ν J (z) = . α ν!Γ(ν + α + 1) ν where Γ(x) is the Gamma function, α is arbitrary real.

Bessel functions are solutions of Bessel’s diﬀerential equation

d2y dy x2 + x + x2 − α2 y = 0. dx2 dx

1 If α = − 2 , 2 1 2 J− 1 (z) = ( ) cos z. 2 πz 1 If α = 2 , 2 1 J 1 (z) = ( ) 2 sin z. 2 πz Bessel functions have the following asymptotic forms, when α is ﬁxed and z → 0,

Jα(z) → 1,

z 1 ( ) 2 J (z) ∼ 2 . α Γ(α + 1)

9 The integral representation

z 1 1 ( ) 2 Z 1 2 2 (α− 2 ) izx Jα(z) = 1 1 (1 − x ) e dx Γ(α + 2 )Γ( 2 ) −1

1 holds for α > − 2 .

2.5 Abel-Summability

In order to use the Abel-summability in the proof of the convergence of the series

d−1 P∞ 2 n=0 bnPn (1) in Chapter 4 and Chapter 5, we state Abel-summability and Abel theorem here.

Abel-summability is a generalized convergence criterion for power series. It extends the usual definition of the sum of a series, and gives a way of summing up certain divergent P∞ series. Let us start with a series n=0 cn, convergent or not, and use that series to define P∞ n a power series f(x) = n=0 cnx . Note that for |x| < 1 the summability of f(x) is easier to achieve than the summability of the original series. Starting with this observation we P∞ say that the series n=0 cn is Abel-summable if the defining series f(x) is convergent for all |x| < 1, and if f(x) converges to some limit s as x → 1−. If this is so, we shall say that P∞ n=0 cn Abel converges to s. In other words, If

∞ X n lim cnx = s, x→1− n=0 then ∞ X cn = s. n=0 .

Of course it is important to ask whether an ordinary convergent series is also Abe- summable, and whether it converges to the same limit? This is true, and the result is known as Abel’s limit theorem, or simply as Abel’s theorem [3].

Theorem 2.5.1 (Abel’s Theorem). Let

∞ X n f(x) = cnx n=0

10 be a power series with real coeﬃcients cn and radius of convergence 1. Suppose that the series ∞ X cn n=0 is convergent. Then f(x) is continuous from the left at x = 1, i.e.

∞ ∞ X n X lim cnx = cn. x→1− n=0 n=0

11 CHAPTER 3

ISOTROPIC POSITIVE DEFINITE FUNCTIONS IN EUCLIDEAN SPACE

This chapter reviews some properties of isotropic deﬁnite functions in Euclidean space

Rd. The main result is the ﬁrst inequality in the abstract which is the key to prove the positive deﬁniteness of an isotropic function on spheres Sd. 3.1 Integral Transforms

Before getting into the discussion of the characterizations of positive deﬁnite functions we summarize some formulas for various integral transforms to be used later.

We start with the conventions of the Fourier transform, for which a reference book is

Bochner and Chandrasekhara [5].

Deﬁnition 3.1.1. The Fourier transform of f ∈ L1(Rd) is given by Z iu0x d gˆ(u) = g(x)e dx, u ∈ R , Rd and its inverse Fourier transform is given by Z 1 −ix0u d gˇ(x) = d g(u)e du, x ∈ R (2π) Rd where x0u is the inner product between x and u.

In this thesis, we will focus on isotropic positive deﬁnite functions. Note that the Fourier transform of an isotropic function is again isotropic [26]. Indeed, the Fourier transform of an isotropic function is its own inverse.

d Theorem 3.1.2. Let G ∈ L1(R ) be continuous and isotropic, i.e, G(x) = g(kxk). Then its ˆ ˆ Fourier transform is also isotropic, i.e, G is also isotropic, i.e, G(x) = Fdg(kxk) with

Z ∞ d − d−2 d Fdg(r) = (2π) 2 x 2 g(u)u 2 J d−2 (xu)du, (3.1.1) 0 2

d−2 where r = kxk and J d−2 (x) is the classical Bessel function of the ﬁrst kind of order . 2 2

12 Remark 1: The integral transform in (3.1.1) is also referred to as a Fourier-Bessel transform or Hankel transform.

Remark 2: The Hankel inversion theorem [23] ensures that the Fourier transform for an isotropic function is its own inverse, i.e., for an isotropic function g,

Fd[Fdg] = g. (3.1.2)

3.2 Characterizations of Positive Deﬁnite Functions

One of the most celebrated results on positive deﬁnite function in Euclidean space Rd is Bochner’s integral characterization, which is established by Bochner in 1932 for d = 1 and

1933 for d > 1.

The following theorem established the identity of positive deﬁnite functions in Rd with characteristic functions of probability measures in Rd.

Theorem 3.2.1 (Bochner’s Theorem). A complex-valued function g ∈ C(Rd) is positive

deﬁnite in Rd if and only if it is the Fourier transform of a ﬁnite nonnegative Borel measure

ν on Rd, i.e., Z ix0y d g(x) = e ν(dy), x ∈ R . Rd Remark: This is the Bochner’s integral characterization for the positive deﬁnite func-

tions in Rd. One of the proof is given by Wendland [27]. The proof for d = 1 can be found in the probability textbook [7].

From Bochner’s Theorem, an isotropic positive deﬁnite function g in Rd is the Fourier

transform of a ﬁnite nonnegative Borel measure ν on Rd. Then the inverse Fourier transform

of an isotropic positive definite function g is a finite nonnegative Borel measure ν on Rd, so it is nonnegative. From (3.1.2), the inverse Fourier transform for an isotropic function is its Fourier transform. Therefore, the inverse Fourier transform or Fourier transform of an isotropic positive definite function is nonnegative. The Fourier transform of an isotopic

13 positive deﬁnite function is given by (3.1.1). Now we obtain,

Z ∞ d − d−2 d (2π) 2 x 2 g(u)u 2 J d−2 (xu)du ≥ 0, x ≥ 0, 0 2

or, simplify, ∞ Z d g(u)u 2 J d−2 (xu)du ≥ 0, x ≥ 0. (3.2.1) 0 2 3.3 Schoenberg’s Characterization

A well-known result on isotropic positive deﬁnite functions in Euclidean space Rd is Schoenberg’s characterization [21] established in 1938. It is stated as the following theorem

[10].

Theorem 3.3.1. A continuous function g(kxk): Rd → R is isotropic positive deﬁnite if and only if g(x) is the Bessel transform of a ﬁnite nonnegative Borel measure ν on [0, ∞), i.e.,

Z ∞ g(x) = Ωd(xt)dν(t) 0

where ( cos x d = 1 Ωd(x) = d 2 d−2 Γ( )( ) 2 J d−2 (x) d ≥ 2 2 r 2 is an integral transcendental function and J d−2 (x) is the Bessel function of the ﬁrst kind of 2 d−2 order 2 .

If a function is isotropic positive deﬁnite in Rd, then it is also isotropic positive deﬁnite

in Rd−1. Therefore, we are interested in those functions which are isotropic positive deﬁnite

in Rd for all d ∈ N. Those functions are identiﬁed by the following theorem.

Theorem 3.3.2. A continuous function g : [0, ∞) → R makes g(kxk) an isotropic positive

deﬁnite function in all Rd if and only if it is of the form

Z ∞ g(x) = e−x2t2 dν(t), 0

where ν is a ﬁnite nonnegative Borel measure on [0, ∞).

14 3.4 Properties of Positive Deﬁnite Functions

In order to derive the properties of positive deﬁnite functions easily, we have expanded

the range of positive deﬁnite functions on the set of complex numbers in Deﬁnition 2.1.2.

The reader should note that we call a function positive deﬁnite if the left side of (2.1.1) is

nonnegative and strictly positive definite if the left side of (2.1.1) is positive for z 6= 0 . This seems to be natural. However, there is an alternative terminology around in the literature: some authors call a function positive semi-definite if the left side of (2.1.1) is nonnegative and positive definite if if the left side of (2.1.1) is positive for z 6= 0. We do not follow this approach here, but the reader should always keep this in mind when looking at other texts.

From Definition 2.1.2, we can read off the elementary properties [27] of such a positive definite function.

Theorem 3.4.1. Suppose g is a complex-valued positive deﬁnite function in Rd. Then the following properties are satisﬁed.

(a) g(0) ≥ 0.

(b) g(x) = g(−x) for all x ∈ Rd.

(e) For d = 1, g is continuous at x = 0 implies that it is uniformly continuous on R.

(f) For d = 1, g is uniformly continuous in R, we have for every continuous complex- valued function ζ on R and every T > 0:

Z T Z T g(s − t)ζ(s)ζ(t)dsdt ≥ 0. (3.4.1) 0 0

Pm (g) If c1, ··· , cm are nonnegative, and g1, ··· , gm are positive definite, then G := l=1 clgl is also positive definite. If one of gl is strictly positive definite and the corresponding cl is positive then G also is strictly positive definite.

(h) For any a ∈ R, f(x) = g(x)eiax is also positive deﬁnite. Generally, the product of two positive deﬁnite functions is positive.

15 d (i) If {gn(x)} be a sequence of positive deﬁnite in R , gn(x) → g(x) as n → ∞, and g(x)

is continuous, then g(x) is also a positive deﬁnite function in Rd.

Proof. (a) Assume that n = 1 and z1 = 1. By (2.1.1), we obtain

g(0) ≥ 0.

(b) Assume that n = 2. By (2.1.1), we have

2 2 g(0)|z1| + g(x1 − x2)z1z2 + g(x2 − x1)z2z1 + g(0)|z2| ≥ 0 (3.4.2)

Taking x1 = 0, x2 = x, z1 = z2 = i in (3.4.2), we have

2g(0) + g(−x) + g(x) ≥ 0 which implies g(−x) + g(x) is real.

Taking x1 = 0, x2 = x, z1 = i, z2 = 1 in (3.4.2), we have

2g(0) + ig(−x) − ig(x) ≥ 0 which implies g(−x) − g(x) is pure imaginary. Hence, we have

g(x) = g(−x)

g(0)|g(x)|2 + g(−x)g(x)(−|g(x)|) + g(x)(−|g(x)|)g(x) + g(0)|g(x)|2 ≥ 0

By (b), we have

g(0)|g(x)|2 + g(x)g(x)(−|g(x)|) + g(x)(−|g(x)|)g(x) + g(0)|g(x)|2

= 2g(0)|g(x)|2 + 2g(−x)g(x)(−|g(x)|)

= 2|g(x)|2(g(0) − |g(x)|) ≥ 0

Therefore,

g(0) ≥ |g(x)|.

16 (d) For the “ only if ” part , g(0) = 0 implies |g(x)| = 0 for all x ∈ Rd by (c). The converse is obvious.

n (e) From (2.1.1) and (b), we have that [g(xj − xk)]j,k=1 is a Hermitian positive deﬁnite

matrix. In particular, for n = 3, x1 = t, x2 = s and t3 = 0, the matrix   g(0) g(t − s) g(t)     g(t − s) g(0) g(s)   g(t) g(s) g(0) must have a nonnegative determinant, which gives

0 ≤ g(0)3 + g(s)g(t − s)g(t) + g(t)g(s)g(t − s) − g(0)(|g(t)|2 + |g(s)|2 + |g(t − s)|2)

= g(0)3 + 2Re[g(s)g(t − s)g(t)] − g(0)(|g(t) − g(s)|2 + 2Re[g(s)g(t)] + |g(t − s)|2)

= g(0)3 + 2Re[g(s)g(t)(g(t − s) − g(0)] − g(0)(|g(t) − g(s)|2 + |g(t − s)|2)

≤ g(0)3 + 2g(0)2|g(0) − g(t − s)| − g(0)(|g(t) − g(s)|2 + |g(t − s)|2) where in the last inequality we used (c). After simplifying and adding |g(t) − g(s)|2 to both

sides of the inequality we have

|g(t) − g(s)|2 ≤ g(0)2 − |g(t − s)|2 + 2g(0)|g(0) − g(t − s)|

= (g(0) − |g(t − s)|)(g(0) + |g(t − s)|) + 2g(0)|g(0) − g(t − s)|

≤ (g(0) − |g(t − s)|)(g(0) + g(0)) + 2g(0)|g(0) − g(t − s)|

≤ 4g(0)|g(0) − |g(t − s)|

Thus the “ modulus of continuity” at each t is bounded by twice the square root of the that at 0, and so the continuity at 0 implies uniform continuity everywhere on R. (f) The integrand of the double integral in (3.4.1) is continuous and the integral is the limit of the Riemann sums, hence it is nonnegative by (2.1.1), i.e.

Z T Z T g(s − t)ζ(s)ζ(t) ≥ 0 0 0

17 (g) Since G := clgl, we have

n n X X G(xj − xk)zjz¯k j=1 k=1 n n m X X X = ( clgl(xj − xk))zjz¯k j=1 k=1 l=1 m n n X X X = clgl(xj − xk)zjz¯k l=1 j=1 k=1 m n n X X X = cl( gl(xj − xk)zjz¯k) l=1 j=1 k=1 ≥ 0

where in the last line we used cl ≥ 0 and the positive deﬁniteness of gl for 1 = 1, 2, ··· , m.

If one of gl is strictly positive deﬁnite and the corresponding cl is positive, without loss of generality, let l = p (0 ≤ p ≤ m), then we have

n n X X cp gp(xj − xk)zjz¯k) > 0, j=1 k=1 hence, m n n X X X cl( gl(xj − xk)zjz¯k) > 0. l=1 j=1 k=1 By Deﬁnition 2.1.2, G is strictly positive deﬁnite.

(h) Changing g to f in (2.1.1), we have

n n X X f(xj − xk)zjzk j=1 k=1 n n X X ia(xj −xk) = g(xj − xk)e zjzk j=1 k=1 m n X X iaxj −iaxj = g(xj − xk)e e zjzk l=1 j=1 m n X X iaxj iax = g(xj − xk)e e j zjzk l=1 j=1 m n X X iaxj iax = g(xj − xk)e zje j zk l=1 j=1 ≥ 0

18 where the inequality is because of the positive deﬁniteness of g. By the deﬁnition, f(x) =

g(x)eiax is also positive deﬁnite. The proof for the general case can be found in [27].

d (i) Since {gn(x)} be a sequence of positive deﬁnite in R , gn(x) → g(x) as n → ∞, and g(x) is continuous, we have

n n X X g(xj − xk)zjzk j=1 k=1 n n X X = lim gi(xj − xk)zjzk i→∞ j=1 k=1 n n X X = lim gi(xj − xk)zjzk i→∞ j=1 k=1 ≥ 0.

By Deﬁnition 2.1.2, g(x) is also positive deﬁnite in Rd.

19 CHAPTER 4

ISOTROPIC POSITIVE DEFINITE FUNCTIONS ON SPHERES

In this chapter, we proceed to review Schogenberg’s characterizations and constructions

of isotropic positive deﬁnite functions on Sd in terms of Gegenbauer polynomials, for which we require some classical results concerning Gegenbauer polynomials.

4.1 Some Classical Results Concerning Ultraspherical Polynomials

We introduce a representation of a real continuous function by Gegenbauer polynomials

[22].

Lemma 4.1.1. A real continuous function G(x) deﬁned on Sd can be approximated by ∞ X n + α Z G(x) ∼ G(y)P (α)(cos θ)dω . (4.1.1) αω n y n=0 d Sd

d+1 d−1 2π 2 d where α = 2 , ωd = d+1 is the surface area of S and θ is the spherical distance between Γ( 2 ) x and y.

d Proof. Suppose that G(x) is a real continuous function deﬁned on S . Let xr be a point on the radius ox such that oxr = r, 0 ≤ r < 1. Deﬁne G(xr) as a Poisson integral

1 Z 1 − r2 G(xr) = d+1 G(y)dωy. (4.1.2) ω 2 2 d Sd (1 − 2r cos θ + r ) By Proposition 2.3.2 in [14], we have that

lim G(xr) = G(x). (4.1.3) r→1

Let t = r and x = cos θ in (2.3.1), we obtain

∞ 1 X = rnP (α)(cos θ). (4.1.4) (1 − 2r cos θ + r2)α n n=0 Diﬀerentiating both sides of (4.1.5) with respect to r, we get

∞ −2 cos θ + 2r X = nrn−1P (α)(cos θ). (4.1.5) (1 − 2r cos θ + r2)α+1 n n=0

20 r Multiplying (4.1.5) by α , we have ∞ −2r cos θ + 2r2 1 X = nrnP (α)(cos θ). (4.1.6) α(1 − 2r cos θ + r2)α+1 α n n=0 Adding (4.1.4) and (4.1.6), we now have ∞ 1 − r2 X n = ( + 1)rnP (α)(cos θ). (4.1.7) (1 − 2r cos θ + r2)α+1 α n n=0 Multiplying (4.1.7) by G(y) dωy and integrating both sides over d, we have ωd S Z 2 Z ∞ 1 1 − r X n + α n (α) d+1 G(y)dωy = r G(y)Pn (cos θ)dωy. (4.1.8) ω 2 2 αω d Sd (1 − 2r cos θ + r ) Sd n=0 d As r → 1 in (4.1.8), we develop an approximation of G(x) by ∞ X n + α Z G(x) ∼ G(y)P (α)(cos θ)dω . (4.1.9) αω n y n=0 d Sd

The relation (4.1.3) implies that the approximation (4.1.9) is Abel-summable at any

point x on Sd to the sum G(x).

d+1 We may convert the Cartesian coordinates of x = (x, x1, ··· , xd) ∈ R to spherical coordinates with:

x = cos ϑ,

x1 = sin ϑ cos ϑ1,

x2 = sin ϑ sin ϑ1 cos ϑ2, . .

xd−2 = sin ϑ sin ϑ1 ··· cos ϑd−2,

xd−1 = sin ϑ sin ϑ1 ··· sin ϑd−2 cos φ,

xd = sin ϑ sin ϑ1 ··· sin ϑd−2 sin φ, where the angles ϑ, ϑ1, ··· , ϑd−2 ∈ [0, π] , and φ ∈ [0, 2π).

d−1 Let G(x) = g(ϑ) and α = 2 . The integral in (4.1.9) can be written Z π Z π Z 2π 0 (α) d−1 0 d−2 0 0 0 0 0 An = ··· g(ϑ )Pn (cos θ) sin ϑ sin ϑ1 ··· sin ϑd−2dϑ ··· dϑd−2dφ . (4.1.10) 0 0 0

21 0 0 0 0 0 0 0 where y = (x , x1, ··· , xd) in terms of Euclidean coordinate and y = (ϑ , ϑ1, ··· , ϑd−2, φ ) in

0 (α) d−1 0 terms of spherical coordinates. Since the integrand g(ϑ )Pn (cos θ) sin ϑ only involves in

0 0 0 0 ϑ , we may integrate ﬁrst with the variable ϑ1, ··· , ϑd−2, φ as follows: Z π 0 d−1 0 0 An = g(ϑ )Cn sin ϑ dϑ . (4.1.11) 0 where

Z π Z π Z 2π (α) d−2 0 0 0 0 0 Cn = ··· Pn (cos θ) sin ϑ1 ··· sin ϑd−2dϑ ··· dϑd−2dφ . (4.1.12) 0 0 0

This integral can be reduced as follows. For two points p, q both lie in the equator Sd−1 π deﬁned by ϑ = 2 with the polar coordinate: π π p = ( , ϑ , ··· , ϑ , φ), q = ( , ϑ0 , ··· , ϑ0 , φ0). 2 1 d−2 2 1 d−2

Then we get

0 0 0 0 0 0 cos θ = x y = xx + x1x1 + ··· + xdxd = cos ϑ cos ϑ + sin ϑ sin ϑ cos ψ, (4.1.13) where ψ is the spherical distance between p and q.

(α) d d Now, Cn is an integration of Pn (cos θ) over S . If we take in S a diﬀerent polar coordinates (ψ, ψ1, ··· , ψd−3, ϕ) of pole p, we obtain

Z π Z π Z 2π (α) d−2 d−3 Cn = ··· Pn (cos θ) sin ψ sin ψ1 ··· sin ψd−3dψdψ1 ··· dψd−3dϕ 0 0 0 Since cos θ depends only on ψ by (4.1.13), we have

Z π (α) d−2 Cn = ωd−2 Pn (cos θ) sin ψdψ 0 which can be explicitly computed by

Z π 1 (α) 2α−1 Γ(α)Γ( 2 )Γ(n + 1)Γ(2α) α α 0 Pn (cos θ) sin ψdψ = 1 Pn (cos ϑ)Pn (cos ϑ ) 0 Γ(α + 2 )Γ(n + 2α)

Substituting Cn into (4.1.11), the expression (4.1.9) reduces to

∞ X (n + α)Γ(α)Γ(n + 1)Γ(2α) Z π g(ϑ) ∼ P α(cos ϑ) P α(cos ϑ0)g(ϑ0) sind−1 ϑ0dϑ0 (4.1.14) Γ(α + 1 )Γ( 1 )Γ(n + 2α) n n n=o 2 2 0

22 We note that this expansion is also Abel-summable.

We need the following two lemmas due to Schoenberg to prove the Schoenberg’s charac-

terization.

Lemma 4.1.2. If g(x) is a real continuous function on [−1, 1] such that g(θ) is positive

deﬁnite on Sd, then we have Z g(θ)dωy ≥ 0, (4.1.15) Sd where θ is the spherical distance between x and y.

Lemma 4.1.3. The Gegenbauer polynomials

d−1 2 Pn (cos ϑ), n ∈ N0, ϑ ∈ [0, π]

are all positive deﬁnite on Sd.

4.2 Schoenberg’s Characterization

Theorem 4.2.1 (Schoenberg’s Characterization). A real continuous function g(x) on [0, π]

is positive deﬁnite on Sd if and only if g(ϑ) can be expressed as the series in (4.1.14) with d−1 2 the series converges for ϑ = 0 and the coeﬃcients of Pn (cos ϑ) are nonnegative, i.e.

∞ d−1 d−1 π (n + )Γ( )Γ(n + 1)Γ(d − 1) Z d−1 d−1 X 2 2 2 d−1 2 Pn (cos ϑ)g(ϑ) sin ϑdϑPn (1) Γ( d )Γ( 1 )Γ(n + d − 1) n=0 2 2 0 converges, and

d−1 d−1 π (n + )Γ( )Γ(n + 1)Γ(d − 1) Z d−1 2 2 2 d−1 d 1 Pn (cos ϑ)g(ϑ) sin ϑdϑ ≥ 0, Γ( 2 )Γ( 2 )Γ(n + d − 1) 0 which is equivalent to

Z π d−1 ( 2 ) d−1 bn = g(ϑ)Pn (cos ϑ) sin ϑdϑ, n ∈ N0, (4.2.1) 0

are nonnegative. In other words, g(ϑ) is positive deﬁnite on Sd if and only if ∞ X α g(ϑ) = bnPn (cos ϑ) (4.2.2) n=0 with bn ≥ 0 and the series converges for ϑ = 0.

23 α Proof. We can rewrite the integral coeﬃcient of Pn (cos ϑ) in (4.1.14) as Z π 1 Z D = P α(cos ϑ0)g(ϑ0) sind−1 ϑ0dϑ0 = P α(cos θ0)g(θ0)dω n n ω n y 0 d−1 Sd where θ0 is the spherical distance between a = (1, 0, ··· , 0) and y. Note that g(θ0) is positive

d α 0 d deﬁnite in S , and Pn (cos θ ) is also positive deﬁnite on S , so is their product. By lemma

(4.1.2), Dn is nonnegative, so are all coeﬃcients of (4.1.14). Thus, (4.1.14) may be rewritten as ∞ X α g(ϑ) ∼ bnPn (cos ϑ), (4.2.3) n=0 where bn is the same as in (4.2.1). On the other hand, we know this series is Abel-summable for all ϑ, hence, in particular, for ϑ = 0. Thus k k ∞ X α X (α) X n (α) bn |Pn (cos ϑ)| ≤ bnPn (1) ≤ lim bnr Pn (1) = g(1). r→1− n=0 n=0 n=0 This shows that the series (4.2.3) is absolutely and uniformly convergent for all ϑ, hence convergent to its Abel-sum which is g(ϑ).

The other side of prove is by the property i in theorem 3.4.1. Note that g(ϑ) is continuous because the series (4.2.2) must be convergent uniformly. As g(ϑ) is the continuous limit of a sequence of positive deﬁnite functions, it is also positive deﬁnite.

4.3 Recent Results

In [15], it has been proved a compactly supported continuous function is positive deﬁnite

on Sd, if the function is positive deﬁnite in Rd, where d = 1, 3. In [28], Xu showed that the integral of Jacobi polynomials deﬁned as

Z a 1 1 2α 2β (α,β),δ δ (α− 2 ,β− 2 ) ϑ ϑ Fn = (a − ϑ) Pn (cos ϑ) sin cos dϑ 0 2 2 is nonnegative. It can be used to prove the truncated power function  δ δ  (a − ϑ) , 0 ≤ ϑ ≤ a, g(x) = (a − ϑ)+ = (4.3.1)  0, ϑ > a,

24 d−1 d 1 is positive deﬁnite on S for δ > 2 + 2 .

25 CHAPTER 5

ISOTROPIC POSITIVE DEFINITE FUNCTIONS ON SPHERES GENERATED FROM THOSE IN EUCLIDEAN SPACES

5.1 Main Results

One of the main results in this thesis is the following theorem, which provides an approach

for generating isotropic positive deﬁnite functions on the sphere from those on the Euclidean

space.

Theorem 5.1.1. Suppose that g(x) is a continuous function on [0, ∞), and g(x) = 0, x ≥ π.

For an odd integer d, if g(kxk) is a positive deﬁnite function in Rd, then g(ϑ), ϑ ∈ [0, π], is

a positive deﬁnite function on Sd.

d Note: The variables x in 5.1.1 can be any point in R , but we should have kx1 − x2k ∈

[0, π] in order to keep g(kxk) is positive deﬁnite in Rd. For the function g(x) = cos x, which

is isotropic positive deﬁnite in R and on S1, it is not satisﬁes the conditions of 5.1.1, since g(π) = −1.

It is not clear whether a similar result holds for an even integer d. Nevertheless, the next

corollary is a consequence of Theorem 5.1.1, since an isotropic positive deﬁnite function in

Rd is also an isotropic positive deﬁnite function in Rd−1 (d ≥ 2), and d − 1 is odd for an even d.

Corollary 1. Let g(x) be as in Theorem 5.1.1. For an even integer d, if g(kxk) is a positive

deﬁnite function in Rd, then g(ϑ), ϑ ∈ [0, π], is a positive deﬁnite function on Sd−1. Although the requirement g(x) = 0, x ≥ π, is crucial in Theorem 5.1.1, it is always

possible to change the scale for a compactly supported function. This result is in the following corollary. Corollary 2. Let g(x) be a continuous function on [0, ∞), and g(x) = 0, x ≥ l. For an odd

d l integer d, if g(kxk) is a positive deﬁnite function in R , then g π ϑ , ϑ ∈ [0, π], is a positive deﬁnite function on Sd.

26 Since g(x) is continuous with compact support on [0, ∞), the positive deﬁniteness of g(kxk) in Rd implies that

π Z d 2 u J d −1(xu)g(u)du ≥ 0, x ≥ 0, 0 2

Hence, Theorem 5.1.1 can be rephrased as the two inequality implication statement as follows, in term of the Bessel function and the ultraspherical polynomial.

Theorem 5.1.2. For a nonnegative integer α and a continuous function g(x) on [0, π] with

g(π) = 0, if Z π α+ 1 u 2 J 1 (xu)g(u)du ≥ 0, x ≥ 0, (5.1.1) α− 2 0 then Z π (α) 2α bn = g(ϑ)Pn (cos ϑ) sin ϑdϑ ≥ 0, n ∈ N0, (5.1.2) 0 ∞ P (α) and bnPn (1) converges. n=0 Moreover, if inequality (5.1.1) is strict for every x ≥ 0, then so is inequality (5.1.2) for

every n ∈ N0.

The proof of Theorems 5.1.1 and 5.1.2 occupies Section 4, for which we need some lemmas in Section 3 and Theorem 5.1.3 that builds a useful bridge between the Jacobi polynomial and the Bessel function, as an extension of Theorem 8.1.1 of [25].

The base for an isotropic positive deﬁnite function in Rd is the Bessel function, as is released by the integral representation (1.0.1), and the base for that on Sd is the ultraspherical polynomial, by (1.0.4). The latter is a special case of the Jacobi polynomial. To prove Theorem 5.1.1 or 5.1.2 we need a bridge to connect these two types of special functions.

According to Theorem 8.1.1 of [25], the Bessel function is related to the Jacobi polynomial by 1 1 − α− 1 1 (α− ,β− ) z z ( 2 ) −(α− 2 ) 2 2 lim m Pm cos = Jα− 1 (z), z ∈ C, (5.1.3) m→∞ m 2 2

27 where the limit holds uniformly in every bounded region of the complex z-plane C. This relation is empolyed in [28] to deduce the positivity of the integrals of the Jacobi polynomial from that of the Bessel function. A generalized version of (5.1.3) is given in the next theorem,

where γm is assumed to be a rational number for ensuring that the degree, mγm + γ0, of a Jacobi polynomial is a positive integer.

1 1 Theorem 5.1.3. For α > − 2 , β > − 2 , and γ0 ∈ N0, if {γm, m ∈ N} is a sequence of

bounded positive rational numbers such that mγm ∈ and lim γm = γ > 0, then N m→∞

1 1 1 1 −(α− ) − α− (α− 2 ,β− 2 ) z z 2 ( 2 ) 1 lim m Pmγm+γ0 cos = Jα− (zγ), z ∈ C. (5.1.4) m→∞ m 2 2

1 1 (α− 2 ,β− 2 ) Proof. For mγm +γ0 ∈ N, an explicit expression of Pmγm+γ0 (x) is given by formula (4.21.2) of [25], which reads,

1 1 (α− 2 ,β− 2 ) Pmγm+γ0 (x) mγm+γ0 1 ν X 1 Γ(mγm + γ0 + α + β + ν)Γ mγm + γ0 + α + x − 1 = 2 . ν!Γ ν + α + 1 Γ(mγ + γ + α + β)Γ(mγ + γ − ν + 1) 2 ν=0 2 m 0 m 0

z after substituting x by cos m ,

(α− 1 ,β− 1 ) z P 2 2 (cos ) mγm+γ0 m mγm+γ0 1 z ν X 1 Γ(mγm + γ0 + α + β + ν)Γ mγm + γ0 + α + cos − 1 = 2 m ν!Γ ν + α + 1 Γ(mγ + γ + α + β)Γ(mγ + γ − ν + 1) 2 ν=0 2 m 0 m 0 mγm+γ0 1 X 1 Γ(mγm + γ0 + α + β + ν)Γ mγm + γ0 + α + z ν = 2 − sin2 ν!Γ ν + α + 1 Γ(mγ + γ + α + β)Γ(mγ + γ − ν + 1) 2m ν=0 2 m 0 m 0

Γ(n+k) k Keeping ν and z ﬁxed, and letting m → ∞, by the asymptotic formula (2.2.2), Γ(n) ∼ n z z and sin 2m ∼ 2m , we obtain the following asymptotic expression for the (ν + 1)st term in the

28 above series,

1 Γ(mγ + γ + α + β + ν)Γ mγ + γ + α + 1 z ν m 0 m 0 2 − sin2 1 Γ(mγ + γ + α + β)Γ(mγ + γ − ν + 1) 2m ν!Γ ν + α + 2 m 0 m 0 1 Γ(mγ + γ + α + β + ν)Γ mγ + γ + α + 1 z 2ν ∼ 0 0 2 1 Γ(mγ + γ + α + β)Γ(mγ + γ − ν + 1) 2m ν!Γ ν + α + 2 0 0 ν ν+α− 1 (mγ) (mγ) 2 z 2ν ∼ 1 2m ν!Γ ν + α + 2 α− 1 (−1)ν(mγ) 2 zγ 2ν ∼ 1 , ν!Γ(ν + α + 2 ) 2 and

1 (α− 1 ,β− 1 ) z −(α− 2 ) 2 2 lim m Pmγ +γ cos m→∞ m 0 m mγm+γ0 −(α− 1 ) 1 X m 2 Γ(mγm + γ0 + α + β + ν)Γ mγm + γ0 + α + z ν = lim 2 − sin2 m→∞ ν!Γ ν + α + 1 Γ(mγ + γ + α + β)Γ(mγ + γ − ν + 1) 2m ν=0 2 m 0 m 0 ∞ ν α− 1 X (−1) γ 2 zγ 2ν = ν!Γ(ν + α + 1 ) 2 ν=0 2 − α− 1 z ( 2 ) = Jα− 1 (zγ), 2 2

where passing to the limit under the summation sign is valid, due to the existence of a dominant for the total sum which is readily derived. Indeed if m is large enough, then we

have

1 1 −(α− 2 ) 2ν m Γ(mγ + γ0 + α + β + ν)Γ mγ + γ0 + α + 2 z ν!(mγ + γ − ν)! 1 2m 0 Γ(mγ + γ0 + α + β)Γ ν + α + 2 (2mγ + 2γ + α + β)νΓ mγ + γ + α + 1 z2ν ≤ 0 0 2 1 1 α− 2 2ν ν!(mγ + γ0 − ν)!m Γ ν + α + 2 (2m) (mγ + γ )ν(2mγ + 2γ + α + β)νΓ mγ + γ + α + 1 z2ν ≤ 0 0 0 2 1 1 α− 2 2ν ν!(mγ + γ0)!m Γ ν + α + 2 (2m) = O(1).

29 5.2 Some Lemmas and Their Proofs

In order to prove Theorems 5.1.1 and 5.1.2 in the next section, we need some lemmas

involving the Jacobi polynomial and recurrent formulas related to bn in (5.1.2), besides Theorem 5.1.3.

For a nonnegative integer β, Lemma 5.2.1 expresses the n + β + 1 degree polynomial α− 1 ,β+ 1 α− 1 ,− 1 β+1 ( 2 2 ) ( 2 2 ) (1 + x) Pn (x) as a nonnegative combination of polynomials Pn+j (x), j = 0, 1, . . . , β + 1.

1 Lemma 5.2.1. For α > − 2 and β ∈ N0,

β+1 1 1 1 1 β+1 (α− 2 ,β+ 2 ) X α,β (α− 2 ,− 2 ) (1 + x) Pn (x) = aj,n Pn+j (x), x ∈ R, n ∈ N0, (5.2.1) j=0 where

β+1 3 α,β 2 (α + 2n + 2j)(n + j)!Γ(α + 2n + j)Γ(β + n + 2 )Γ(β + 2) aj,n = 1 . (5.2.2) n!j!Γ(α + β + 2n + j + 2)Γ(β − j + 2)Γ(n + j + 2 ) Proof. We will give two proofs of the Lemma. The ﬁrst one is by mathematical induction, in the second one we use some identities from the references.

Proof 1. The proof of (5.2.1) is by induction on β.

For β = 0, (5.2.1) appears as identity (22.7.16) of [1],

1 1 (α− 2 ,β+ 2 ) (2n + α + β + 1)(1 + x)Pn (x) 1 1 1 1 (5.2.3) (α− 2 ,β− 2 ) (α− 2 ,β− 2 ) = (2n + 2β + 1)Pn (x) + 2(n + 1)Pn+1 (x). or,

1 1 (α− 2 ,β+ 2 ) (1 + x)Pn (x) 2n + 2β + 1 (α− 1 ,β− 1 ) 2(n + 1) (α− 1 ,β− 1 ) = P 2 2 (x) + P 2 2 (x). 2n + α + β + 1 n 2n + α + β + 1 n+1

Taking β = 0, we have

1 1 (α− 2 , 2 ) (1 + x)Pn (x) 2n + 1 (α− 1 ,− 1 ) 2(n + 1) (α− 1 ,− 1 ) = P 2 2 (x) + P 2 2 (x), 2n + α + 1 n 2n + α + 1 n+1

30 where 2n + 1 aα,0 = 0,n 2n + α + 1 and 2(n + 1) aα,0 = . 1,n 2n + α + 1 Assume that (5.2.1) is true for β = λ − 1, i.e,

λ 1 1 1 1 λ (α− 2 ,λ− 2 ) X α,λ−1 (α− 2 ,− 2 ) (1 + x) Pn (x) = aj,n Pn+j (x), j=0 substituting n by n + 1,

λ 1 1 1 1 λ (α− 2 ,λ− 2 ) X α,λ−1 (α− 2 ,− 2 ) (1 + x) Pn+1 (x) = aj,n+1 Pn+1+j (x). j=0 Then for β = λ, by the identity (5.2.3)

1 1 λ+1 (α− 2 ,λ+ 2 ) (1 + x) Pn (x)

1 1 λ (α− 2 ,λ+ 2 ) = (1 + x)(1 + x) Pn (x) 2n + 2λ + 1 (α− 1 ,λ− 1 ) 2(n + 1) (α− 1 ,λ− 1 ) = (1 + x)λP 2 2 (x) + (1 + x)λP 2 2 (x) 2n + α + λ + 1 n 2n + α + λ + 1 n+1 λ λ 2n + 2λ + 1 X (α− 1 ,− 1 ) 2(n + 1) X (α− 1 ,− 1 ) = aα,λ−1P 2 2 (x) + aα,λ−1P 2 2 (x) 2n + α + λ + 1 j,n n+j 2n + α + λ + 1 j,n+1 n+1+j j=0 j=0 2n + 2λ + 1 (α− 1 ,− 1 ) 2(n + 1) (α− 1 ,− 1 ) = aα,λ−1P 2 2 (x) + aα,λ−1P 2 2 (x) 2n + α + λ + 1 0,n n 2n + α + λ + 1 λ,n+1 n+1+λ λ λ−1 2n + 2λ + 1 X (α− 1 ,− 1 ) 2(n + 1) X (α− 1 ,− 1 ) + aα,λ−1P 2 2 (x) + aα,λ−1P 2 2 (x) 2n + α + λ + 1 j,n n+j 2n + α + λ + 1 j,n+1 n+1+j j=1 j=0 2n + 2λ + 1 (α− 1 ,− 1 ) 2(n + 1) (α− 1 ,− 1 ) = aα,λ−1P 2 2 (x) + aα,λ−1P 2 2 (x) 2n + α + λ + 1 0,n n 2n + α + λ + 1 λ,n+1 n+1+λ λ λ 2n + 2λ + 1 X (α− 1 ,− 1 ) 2(n + 1) X (α− 1 ,− 1 ) + aα,λ−1P 2 2 (x) + aα,λ−1 P 2 2 (x) 2n + α + λ + 1 j,n n+j 2n + α + λ + 1 j−1,n+1 n+j j=1 j=1 2n + 2λ + 1 (α− 1 ,− 1 ) 2(n + 1) (α− 1 ,− 1 ) = aα,λ−1P 2 2 (x) + aα,λ−1P 2 2 (x) 2n + α + λ + 1 0,n n 2n + α + λ + 1 λ,n+1 n+1+λ λ 1 1 X 2n + 2λ + 1 2(n + 1) (α− ,− ) + aα,λ−1 + aα,λ−1 P 2 2 (x) 2n + α + λ + 1 j,n 2n + α + λ + 1 j−1,n+1 n+j j=1 λ+1 1 1 X α,λ (α− 2 ,λ− 2 ) = aj,n Pn+j (x) j=0

31 where

2n + 2λ + 1 aα,λ = aα,λ−1, 0,n 2n + α + λ + 1 0,n

2(n + 1) aα,λ = aα,λ−1, λ+1,n 2n + α + λ + 1 λ,n+1

and

2n + 2λ + 1 2(n + 1) aα,λ = aα,λ−1 + aα,λ−1 j,n 2n + α + λ + 1 j,n 2n + α + λ + 1 j−1,n+1

for j = 1, 2, ··· , λ.

Proof 2. Successively using this identity (5.2.3), the n + β + 1 degree polynomial (1 + α− 1 ,β+ 1 α− 1 ,− 1 β ( 2 2 ) ( 2 2 ) x) Pn (x) can be written as a linear combination of polynomials Pn+j (x), j = α,β 0, 1, . . . , β + 1. To verify (5.2.2) for aj,n in (5.2.1), we multiply both sides of (5.2.1) by α− 1 ,− 1 α− 1 − 1 ( 2 2 ) (1 − x) 2 (1 + x) 2 Pn+j (x) and then integrate from -1 and 1,

2αΓ(α + n + j + 1 )Γ(n + j + 1 ) 2 2 aα,β (n + j)!(α + 2n + 2j)Γ(α + n + j) j,n β+1 1 1 1 1 1 Z 1 1 X α,β (α− 2 ,− 2 ) (α− 2 ,− 2 ) α− 2 − 2 = ak,n (1 − x) (1 + x) Pn+j (x)Pn+k (x)dx k=0 −1 1 1 1 1 1 Z 1 1 (α− 2 ,− 2 ) (α− 2 ,β+ 2 ) α− 2 β+ 2 = (1 − x) (1 + x) Pn+j (x)Pn (x)dx −1 2α+β+1Γ(α + n + j + 1 )Γ(α + 2n + j)Γ(β + n + 3 )Γ(β + 2) = 2 2 , n!j!Γ(α + n + j)Γ(α + β + 2n + j + 2)Γ(β − j + 2)

where the ﬁrst equality is obtained from identity (7.391.1) of [13], and the last equality from

identity (7.391.9) of [13].

1 Suppose that g(ϑ) is a continuous function on [0, ∞) and g(ϑ) = 0, ϑ ≥ π. For α, β > − 2 , deﬁne

Z π 1 1 2α 2β (α,β) (α− 2 ,β− 2 ) ϑ ϑ ϑ bn,k = g(ϑ)Pn cos k sin k+1 cos k+1 dϑ, n ∈ N0, k ∈ N0, 0 2 2 2

32 (α,β) (α,β) and, for simplicity, write bn for bn,0 when k = 0. Thus,

Z π 1 1 2α 2β (α,β) (α,β) (α− 2 ,β− 2 ) ϑ ϑ bn = bn,0 = g(ϑ)Pn (cos ϑ) sin cos dϑ, 0 2 2

and, for bn in (5.1.2),

2α 1 2 Γ α + 2 Γ(n + 2α) (α,α) bn = 1 bn , n ∈ N0, Γ(2α)Γ n + α + 2 which is a direct result of (1.0.6).

1 Lemma 5.2.2. For α, β > − 2 ,

2n + 2β + 1 n + 1 b(α,β+1) = b(α,β) + b(α,β), n ∈ . (5.2.4) n 2(2n + α + β + 1) n 2n + α + β + 1 n+1 N0

1 If α > − 2 and β ∈ N0, then

β+1 (α,β+1) −(β+1) X α,β (α,0) bn,k = 2 aj,n bn+j,k, n ∈ N0, k ∈ N0, (5.2.5) j=0

and, in particular, for α ∈ N,

α (α,α) −α X α,α−1 (α,0) bn,k = 2 aj,n bn+j,k, n ∈ N0, k ∈ N0, (5.2.6) j=0 and α (α,α) −α X α,α−1 (α,0) bn = 2 aj,n bn+j , (5.2.7) j=0 α,β where aj,n ’s are positive constants deﬁned in (5.2.2).

Proof. Substituting x by cos ϑ in (5.2.3) yields

1 1 (α− 2 ,β+ 2 ) (2n + α + β + 1) (1 + cos ϑ) Pn (cos ϑ)

1 1 1 1 (α− 2 ,β− 2 ) (α− 2 ,β− 2 ) = (2n + 2β + 1)Pn (cos ϑ) + 2(n + 1)Pn+1 (cos ϑ) ,

or 1 1 ϑ (α− ,β+ ) 2(2n + α + β + 1) cos2 P 2 2 (cos ϑ) 2 n

33 1 1 1 1 (α− 2 ,β− 2 ) (α− 2 ,β− 2 ) = (2n + 2β + 1)Pn (cos ϑ) + 2(n + 1)Pn+1 (cos ϑ) .

ϑ 2α ϑ 2β Multiplying both sides of the last equation by g(ϑ) sin 2 cos 2 and integrating from 0 to π, we obtain (5.2.4).

ϑ To derive (5.2.5), we substitute x by cos( 2k ) in (5.2.1),

β+1 1 1 X (α− ,− ) ϑ aα,βP 2 2 cos j,n n+j 2k j=0

β+1 1 1 ϑ (α− ,β+ ) ϑ = 1 + cos P 2 2 cos 2k n 2k

2(β+1) 1 1 ϑ (α− ,β+ ) ϑ = 2β+1 cos P 2 2 cos , 2k+1 n 2k

and obtain

Z π 2(β+1) 1 1 2α (α,β+1) ϑ (α− 2 ,β+ 2 ) ϑ ϑ bn,k = g(ϑ) cos k+1 Pn cos k sin k+1 dϑ 0 2 2 2 β+1 π 1 1 2α X Z (α− ,− ) ϑ ϑ = 2−(β+1) aα,β g(ϑ)P 2 2 cos sin dϑ j,n n+j 2k 2k+1 j=0 0 β+1 −(β+1) X α,β (α,0) = 2 aj,n bn+j,k. j=0

1 Lemma 5.2.3. For α > − 2 and n, k ∈ N0, Γ 2n + α + 1 Γ(n + 1) b(α,α) = 2 b(α,0), (5.2.8) 2n,k+1 2α 1 n,k 2 Γ n + α + 2 Γ(2n + 1) and, in particular, Γ 2n + α + 1 Γ(n + 1) b(α,α) = 2 b(α,0). (5.2.9) 2n,1 2α 1 n 2 Γ n + α + 2 Γ(2n + 1) Proof. By identity (4.1.5) of [25],

1 α− 1 ,α− 1 Γ 2n + α + Γ(n + 1) α− 1 ,− 1 ( 2 2 ) 2 ( 2 2 ) 2 P2n (x) = 1 Pn (2x − 1), Γ n + α + 2 Γ(2n + 1)

34 ϑ which reads, after substituting x by cos 2k+1 ,

1 1 1 1 1 (α− ,α− ) ϑ Γ 2n + α + Γ(n + 1) (α− ,− ) ϑ P 2 2 cos = 2 P 2 2 cos . 2n 2k+1 1 n 2k Γ n + α + 2 Γ(2n + 1) From the last equality we obtain

Z π 1 1 2α (α,α) (α− 2 ,α− 2 ) ϑ ϑ ϑ b2n,k+1 = g(ϑ)P2n cos k+1 sin k+2 cos k+2 dϑ 0 2 2 2 Z π 1 1 2α −2α (α− 2 ,α− 2 ) ϑ ϑ = 2 g(ϑ)P2n cos k+1 sin k+1 dϑ 0 2 2 1 π 2α Γ 2n + α + Γ(n + 1) Z α− 1 ,− 1 2 ( 2 2 ) ϑ ϑ = g(ϑ)Pn cos sin dϑ 2α 1 2k 2k+1 2 Γ n + α + 2 Γ(2n + 1) 0 Γ 2n + α + 1 Γ(n + 1) = 2 b(α,0). 2α 1 n,k 2 Γ n + α + 2 Γ(2n + 1)

Lemma 5.2.4. For α ∈ N, k ∈ N, and n ∈ N0, α α (α,α) (k−1)α P P α,α−1 α,α−1 bn = 2 ... a ··· a k k−1 j0,n jk,2 (n+j0)+2 j1+...+2jk−1 j0=0 jk=0 (5.2.10) (α,0) ×c (j0, . . . , jk, n) b k k−1 , 2 (n+j0)+2 j1+...+jk,k where

c (j0, j1, . . . , jk, n) k−1 l l−1 1 l+1 l Y Γ 2 (n + j0) + 2 j1 + ... + jl + α + Γ 2 (n + j0) + 2 j1 + ... + 2jl + 1 = 2 Γ 2l+1(n + j ) + 2lj + ... + 2j + α + 1 Γ (2l(n + j ) + 2l−1j + ... + j + 1) l=0 0 1 l 2 0 1 l α,α−1 is a nonnegative constant and a k k−1 is deﬁned as in (5.2.2) which is also jk,2 (n+j0)+2 j1+...+2jk−1 nonegative.

Proof. The proof of (5.2.10) is by induction on k. For k = 1, (5.2.10) follows from identities

(5.2.7), (5.2.9) and (5.2.6)

(α,α) bn α X (α,0) = 2−α aα,α−1b j0,n n+j0 j0=0 α 2α 1 X 2 Γ n + j0 + α + Γ(2(n + j0) + 1) (α,α) = 2−α aα,α−1 2 b j0,n 1 2(n+j0),1 Γ 2(n + j0) + α + Γ(n + j0 + 1) j0=0 2

35 α 2α 1 α X 2 Γ n + j0 + α + Γ(2(n + j0) + 1) X (α,0) = 2−α aα,α−1 2 2−α aα,α−1 b j0,n 1 j1,2(n+j0) 2(n+j0)+j1,1 Γ 2(n + j0) + α + Γ(n + j0 + 1) j0=0 2 j1=0 α α 1 X X Γ n + j0 + α + Γ(2(n + j0) + 1) (α,0) = aα,α−1 aα,α−1 2 b . j0,n j1,2(n+j0) 1 2(n+j0)+j1,1 Γ 2(n + j0) + α + Γ(n + j0 + 1) j0=0 j1=0 2 Suppose that (5.2.10) is true for k = i, i.e.,

α α (α,α) (i−1)α P P α,α−1 α,α−1 b = 2 ... a ··· a i i−1 n j0,n ji,2 (n+j0)+2 j1+...+2ji−1 j0=0 ji=0 (α,0) ×c (j0, . . . , ji, n) b i i−1 , 2 (n+j0)+2 j1+...+ji,i then for k = i + 1, by (5.2.8) and (5.2.6)

(α,α) bn α α (i−1)α X X α,α−1 α,α−1 = 2 ... a ··· a i i−1 j0,n ji,2 (n+j0)+2 j1+...+2ji−1 j0=0 ji=0 (α,0) ×c (j0, . . . , ji, n) b i i−1 2 (n+j0)+2 j1+...+ji,i α α (i−1)α X X α,α−1 α,α−1 = 2 ... a ··· a i i−1 c (j0, . . . , ji, n) j0,n ji,2 (n+j0)+2 j1+...+2ji−1 j0=0 ji=0 22αΓ 2i(n + j ) + 2i−1j + ... + j + α + 1 Γ(2(2i(n + j ) + 2i−1j + ... + j ) + 1) × 0 1 i 2 0 1 i i i−1 1 i i−1 Γ 2(2 (n + j0) + 2 j1 + ... + ji) + α + 2 Γ(2 (n + j0) + 2 j1 + ... + ji + 1) (α,α) ×b i i−1 2(2 (n+j0)+2 j1+...+ji),i+1 α α (i−1)α X X α,α−1 α,α−1 = 2 ... a ··· a i i−1 c (j0, . . . , ji, n) j0,n ji,2 (n+j0)+2 j1+...+2ji−1 j0=0 ji=0 22αΓ 2i(n + j ) + 2i−1j + ... + j + α + 1 Γ(2(2i(n + j ) + 2i−1j + ... + j ) + 1) × 0 1 i 2 0 1 i i i−1 1 i i−1 Γ 2(2 (n + j0) + 2 j1 + ... + ji) + α + 2 Γ(2 (n + j0) + 2 j1 + ... + ji + 1) α −α X α,α−1 (α,0) ×2 a i i−1 b i i−1 ji+1,2(2 (n+j0)+2 j1+...+ji) 2(2 (n+j0)+2 j1+...+ji)+ji+1,i+1 ji+1=0 α α α iα X X X α,α−1 α,α−1 α,α−1 = 2 ... a ··· a i i−1 a i i−1 j0,n ji,2 (n+j0)+2 j1+...+2ji−1 ji+1,2(2 (n+j0)+2 j1+...+ji) j0=0 ji=0 ji+1=0

×c (j0, . . . , ji, n) Γ 2i(n + j ) + 2i−1j + ... + j + α + 1 Γ2i+1(n + j ) + 2ij + ... + 2j + 1) × 0 1 i 2 0 1 i i+1 i 1 i i−1 Γ 2 (n + j0) + 2 j1 + ... + 2ji + α + 2 Γ(2 (n + j0) + 2 j1 + ... + ji + 1) (α,0) ×b i+1 i 2 (n+j0)+2 j1+...+2ji+ji+1,i+1

36 α α iα X X α,α−1 α,α−1 = 2 ... a ··· a i i−1 c (j0, . . . , ji+1, n) j0,n ji+1,2(2 (n+j0)+2 j1+...+ji) j0=0 ji+1=0 (α,0) ×b i+1 i , 2 (n+j0)+2 j1+...+2ji+ji+1,i+1

where

c (j0, . . . , ji+1, n) i l l−1 1 l+1 l Y Γ 2 (n + j0) + 2 j1 + ... + jl + α + Γ 2 (n + j0) + 2 j1 + ... + 2jl + 1 = 2 . Γ 2l+1(n + j ) + 2lj + ... + 2j + α + 1 Γ (2l(n + j ) + 2l−1j + ... + j + 1) l=0 0 1 l 2 0 1 l

By the principle of mathematical induction, (5.2.10) holds for any k ∈ N.

A common and important feature of recurrent formulas (5.2.6)-(5.2.10) is that all coeﬃ-

cients are nonnegative. More importantly, k at the right-hand side of (5.2.10) is an arbitrary positive integer, making which large enough and combining with the following lemma help us to prove Theorems 5.1.1 and 5.1.2 in the next section.

1 Lemma 5.2.5. For α > − 2 and γ0 ∈ N0, if {γk, k ∈ N} is a sequence of bound positive k rational numbers such that 2 γk ∈ N and lim γk = γ > 0, then k→∞ Z π k α+ 1 (α,0) 1 1 ( 2 ) −α− 2 α+ 2 lim 2 b k = 2 g(ϑ)ϑ J 1 (γϑ)dϑ. (5.2.11) 2 γk+γ0,k α− 2 k→∞ 0

1 (α,0) k(α+ 2 ) Proof. Rewriting 2 b k as 2 γk+γ0,k

1 (α,0) k(α+ 2 ) 2 b k 2 γk+γ0,k !2α Z π 2α 1 1 ϑ ϑ 1 (α− ,− ) ϑ sin k+1 −k(α− 2 ) 2 2 2 = g(ϑ) 2 P k cos dϑ, 2 2 γk+γ0 2k ϑ 0 2k+1 we obtain (5.2.11) from Theorem 5.1.3, by letting k → ∞,

1 (α,0) k(α+ 2 ) lim 2 b k k→∞ 2 γk+γ0,k

37 !2α Z π 2α 1 1 ϑ ϑ 1 (α− ,− ) ϑ sin k+1 −k(α− 2 ) 2 2 2 = g(ϑ) lim 2 P k cos lim dϑ 2 k→∞ 2 γk+γ0 2k k→∞ ϑ 0 2k+1 π 2α −(α− 1 ) Z ϑ ϑ 2 = g(ϑ) J 1 (ϑγ)dϑ α− 2 0 2 2 Z π −(α+ 1 ) α+ 1 = 2 2 g(ϑ)ϑ 2 J 1 (ϑγ)dϑ, α− 2 0 where the exchange between the limit and integral at the ﬁrst equality is ensured by the dominated convergence theorem.

Lemma 5.2.6. For α ∈ N0, k ∈ N0, and n ∈ N0,

1 α− 1 Z π 1 (α,0) 2 2 (n + α − )! 1 Jα− (xϑ) 2α(k+1) 2 α+ 2 2 lim 2 b = lim ϑ 1 g(ϑ)dϑ. (5.2.12) n,k + α− k→∞ n! x→0 0 x 2

Proof. It can be proved by the following two limiting forms, namely,

2α π ϑ ! Z 1 1 sin 2α(k+1) (α,0) 2α (α− 2 ,− 2 ) ϑ 2k+1 lim 2 bn,k = g(ϑ)ϑ lim Pn (cos ) dϑ k→∞ k→∞ 2k ϑ 0 2k+1 1 1 Z π (α− 2 ,− 2 ) 2α = Pn (1) g(ϑ)ϑ dϑ, 0

(α− 1 ,− 1 ) 1 2 2 n+α− 2 where Pn (1) = n by (4.1.1) of [25], and

π π Z 1 Jα− 1 (xϑ) Z 1 Jα− 1 (xϑ) α+ 2 2 α+ 2 2 lim ϑ 1 g(ϑ)dϑ = ϑ lim 1 g(ϑ)dϑ + α− + α− x→0 0 x 2 0 x→0 x 2 1 −α Z π 2 2 2α = 1 g(ϑ)ϑ dϑ, Γ(α + 2 ) 0

zα which is a direct consequence of (10.7.3) of [19], Jα(z) ∼ 2αΓ(α+1) by ﬁxing α and letting z → 0.

5.3 Proof of Theorems 5.1.1 and 5.1.2

Since g(x) is continuous with compact support on [0, ∞) and g(kxk) is positive deﬁnite

in Rd, inequality (5.1.1) holds for every x ≥ 0, i.e.,

Z π α+ 1 u 2 J 1 (xu)g(u)du ≥ 0, x ≥ 0. α− 2 0

38 What needs a proof is inequality (5.1.2),i.e.,

Z π (α) 2α bn = g(ϑ)Pn (cos ϑ) sin ϑdϑ ≥ 0, n ∈ N0, 0

(α,α) d−1 or bn ≥ 0 for every n ∈ N0, where α = 2 is a nonnegative integer. While the case d = 1 or 3 is covered in [15], it suﬃces to consider d ≥ 5 or α ∈ N. (α,α) To prove bn ≥ 0 for each n ∈ N, from identity (5.2.10) we need to show

(α,0) b k k k−1 ≥ 0, ji = 0, 1, 2, . . . , α, i = 1, 2, . . . , k. 2 n+2 j0+2 j1+...+jk,k

Since k in (5.2.10) is an arbitrary positive integer, it can be chosen large enough. Noticing

that

k k k k−1 k 2 n ≤ 2 n + 2 j0 + 2 j1 + ... + jk ≤ 2 (n + 2α) − α,

k k k−1 we can write 2 n + 2 j0 + 2 j1 + ... + jk as

k k k−1 k 2 n + 2 j0 + 2 j1 + ... + jk = 2 γk + γ0,

where γ0 ∈ N0 and γk is a positive rational number bounded between n and n + 2α, so that

(α,0) (α,0) b k k k−1 = b k . 2 n+2 j0+2 j1+...+jk,k 2 γk+γ0,k

First, suppose that inequality (5.1.1) is strict, then the right-hand side of (5.2.11) is also

positive, i.e., Z π α+ 1 g(ϑ)ϑ 2 J 1 (γϑ)dϑ > 0. α− 2 0 This is due to (5.1.1) is strict and γ > 0. Then its left-hand side has to be positive.

Consequently, when k is large enough,

(α,0) (α,0) b k = b k k k−1 ≥ 0, 2 γk+γ0,k 2 n+2 j0+2 j1+...+jk,k

(α,α) and it implies bn ≥ 0 by (5.2.10). Moreover, since the left-hand side of (5.2.11) is positive,

(α,0) (α,α) there exists at least one number k such that b k k k−1 is positive, so that bn 2 n+2 j0+2 j1+...+jk,k must be positive.

39 k Now we consider the case n = 0, there are two possibilities for 2 γk + γ0. If it is

(α,0) unbounded, then, by Lemma 5.2.5, b k ≥ 0 for large k. If it is bounded, then it follows 2 γk+γ0,k (α,0) (α,α) from Lemma 5.2.6 that b k ≥ 0 when k is large enough. By (5.2.10), b ≥ 0. 2 γk+γ0,k 0 In a general case, (5.1.1) implies Z π α+ 1 ϑ 2 J 1 (γϑ)g(ϑ)dϑ ≥ 0, γ ≥ 0. α− 2 0 For an arbitrary > 0, we deﬁne a function

α+1 g(ϑ) = g(ϑ) + (π − ϑ)+ , ϑ ≥ 0.

By Theorem 7 of [11], Z π α+ 1 α+1 ϑ 2 J 1 (γϑ)(π − ϑ) dϑ > 0, γ ≥ 0, α− 2 + 0 so that Z π α+ 1 ϑ 2 J 1 (γϑ)g(ϑ)dϑ > 0, γ ≥ 0. α− 2 0 It implies

Z π 1 1 2α 2α (α− 2 ,α− 2 ) ϑ ϑ g(ϑ)Pn (cos ϑ) sin cos ≥ 0, n ∈ N. 0 2 2 (α,α) Let → 0, we obtain bn ≥ 0 for n ∈ N0. ∞ d−1 P ( 2 ) What remains is to verify that bnPn (1) converges. As is shown in [22], the con- n=0 ∞ d−1 P ( 2 ) tinuous function g(ϑ) possesses the expansion bnPn (cos ϑ), which is Abel-summable n=0 for every ϑ ∈ [0, π], and, in particular, for ϑ = 0. Thus,

k k ∞ d−1 d−1 d−1 X ( 2 ) X ( 2 ) X n ( 2 ) bn Pn (cos ϑ) ≤ bnPn (1) ≤ lim bnr Pn (1) = g(0). r→1− n=0 n=0 n=0 Let k d−1 X ( 2 ) Sk = bn Pn (cos ϑ) . n=0

Then sequence {Sk} is bounded and monotone, so it is convergent. Hence the sequence

k d−1 X ( 2 ) Sk = bnPn (cos ϑ). n=0 is absolutely and uniformly convergent for all ϑ ∈ [0, π], and converges to its Abel-sum g(ϑ).

40 CHAPTER 6

PLANS FOR FUTURE WORK

In this thesis, we have reviewed the classical characterizations of isotropic positive definite functions on Euclidean spaces and spheres, and we have applied them to develop a construction of isotropic positive deﬁnite functions on spheres. However, this construction only works when the dimension of the sphere is odd. Our next work is to develop isotropic positive deﬁnite functions on all spheres. It is stated as the following conjecture.

Conjecture. Suppose that g(x) is a continuous function on [0, ∞), and g(x) = 0, x ≥ π.

For a positive integer d, if g(kxk) is a positive deﬁnite function in Rd, then g(ϑ), ϑ ∈ [0, π], is a positive deﬁnite function on Sd.

41 REFERENCES

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