An Analogue of Bochner’s Theorem for Damek-Ricci Spaces

Abstract

We characterize the image of radial positive measures θ’s on a harmonic NA group S which satisfies ∫ _S ϕ ₀(x) dθ(x)<∞ under the spherical transform, where ϕ ₀ is the elementary spherical function.

1 Introduction

A continuous function f on ℝ is said to be positive definite if for any real numbers x ₁,…,x _m and complex numbers ξ ₁,…,ξ _m the following holds

This condition is equivalent to

$$\int_{\mathbb{R}}f(x) \bigl(\phi*\phi^{*}\bigr) (x)\,dx \geq0 $$

for all $\phi\in C_{c}^{\infty}(\mathbb{R})$, where $\phi^{\ast}(x)=\overline{\phi (-x)}$. A celebrated theorem of S. Bochner states that a positive definite function is the Fourier transform of some finite positive measure on ℝ. Therefore, it characterizes the image of finite positive measures under the Fourier transform. Also, it gives an integral representation of the positive definite functions. This theorem has been extended to locally compact Abelian groups. P. Graczyk and J.-J. Lœb characterize the image of K-biinvariant finite positive measures on G under the spherical transform when G is a connected, noncompact, complex semisimple Lie group with finite centre and K is a maximal compact subgroup of G (see [9]).

For any NA group (also known as Damek-Ricci space) S, we consider two types of problems:

• (1) The first one is to get an integral representation of the radially positive definite functions on S. We prove that such functions are given by positive measures on ℝ∪iℝ, such that the measure on ℝ is finite and the measure μ on iℝ satisfies ∫_ℝ e ^a|λ|  dμ(λ)<∞ for all a>0. This is an analogue of M.G. Krein’s theorem for evenly positive definite functions on ℝ (see [8, p. 196]).

• (2) In the second one, we characterize the image of radial positive measures θ’s on S which satisfies ∫ _S ϕ ₀(x) dθ(x)<∞ under the spherical transform. For this, first we show that, the spherical transform of such measures are even, continuous, bounded functions on ℝ and satisfy certain positive definite like condition. Conversely, we prove that any even, continuous, bounded function on ℝ which satisfies such positive definite like condition is the spherical transform of some radial, positive measure θ on S. This measure θ satisfies ∫ _S ϕ ₀(x) dθ(x)<∞. Then we prove that the image set is a subset of the positive definite functions on ℝ. This positive definite like condition can be stated alternatively as follows: The elements of the image space are positive linear functionals on the Banach algebra (L ¹(ℝ,|c(λ)|⁻²  dλ) _e ,⊙), whereas in the classical Bochner’s theorem the elements of the image space are positive linear functionals on the Banach algebra (L ¹(ℝ),∗). The condition on the measure θ (i.e. ∫ _S ϕ ₀(x) dθ(x)<∞) is due to a technical reason. If a measure θ is finite then it satisfies ∫ _S ϕ ₀(x) dθ(x)<∞. We guess the image space for finite, radial, positive measures under the spherical transform and write it as a conjecture. We refer [2, 3, 10, 12] for further study in this literature.

2 Preliminaries

In this section, we explain the required preliminaries for NA groups, mainly from [1] and the references therein. Let $\mathfrak{n}$ be a two step nilpotent Lie algebra with the inner product 〈  , 〉. Let $\mathfrak{z}$ be the centre of $\mathfrak{n}$ and $\mathfrak{p}$ be the orthogonal complement of $\mathfrak{z}$ in $\mathfrak{n}$. We call the Lie algebra $\mathfrak {n}$ an H-type algebra if for each $Z\in\mathfrak{z}$ the map $J_{Z}:\mathfrak{p}\rightarrow\mathfrak{p}$ defined by

$$\langle J_ZX,Y\rangle=\bigl\langle[X,Y],Z\bigr\rangle $$

satisfies $J_{Z}^{2}=-|Z|^{2}I_{\mathfrak{p}}$, where $I_{\mathfrak{p}}$ is the identity operator on $\mathfrak{p}$. A connected, simply connected Lie group N is called H-type group if its Lie algebra is an H-type Lie algebra. Since $\mathfrak{n}$ is nilpotent the exponential map is a diffeomorphism from $\mathfrak{n}$ to N. Therefore, any element n of N can be expressed as n=exp(X+Z) for some $X\in\mathfrak{p}$, $Z\in\mathfrak{z}$. Hence, we can parametrize elements of N by the pairs $(X,Z), X\in\mathfrak{p}, Z\in\mathfrak{z}$. It follows from Campbell-Baker-Hausdorff formula that the group law on N is given by

$$(X, Z) \bigl(X',Z'\bigr)=\biggl(X+X', Z+Z'+\frac{1}{2}\bigl[X,X'\bigr]\biggr). $$

Let A=ℝ⁺=(0,∞) and N an H-type group. For a∈A we define a dilation δ _a :N→N by δ _a ((X,Z))=(a ^1/2 X,aZ). Let S=NA be the semi direct product of N and A under the dilation. Thus the multiplication on S is given by

$$(X,Z,a) \bigl(X',Z',a'\bigr)= \biggl(X+a^{1/2}X', Z+aZ'+\frac{1}{2} a^{1/2}\bigl[X,X'\bigr], aa' \biggr). $$

Then S is a solvable, connected and simply connected Lie group having Lie algebra $\mathfrak{s}=\mathfrak{p}\oplus\mathfrak{z}\oplus \mathbb{R}$ with Lie bracket

$$\bigl[(X,Z,l),\bigl(X',Z',l'\bigr)\bigr]= \biggl(\frac{1}{2} lX'-\frac{1}{2} l'X, lZ'-l'Z+ \bigl[X,X'\bigr], 0 \biggr). $$

We note that for any $Z\in\mathfrak{z}$ with |Z|=1, we have $J_{Z}^{2}=-I_{\mathfrak{p}}$. Therefore, J _Z defines a complex structure on $\mathfrak{p}$. Hence $\dim\mathfrak{p}$ is even. Let $\dim \mathfrak {p}=m$ and dim$\mathfrak{z}=k$. Then $Q=\frac{m}{2} +k$ is called the homogeneous dimension of S. We also use the symbol ρ for $\frac{Q}{2}$ and n for $m+k+1=\dim\mathfrak{s}$.

The group S is equipped with the left-invariant Riemannian metric d induced by

$$\bigl\langle(X,Z,l), \bigl(X',Z',l'\bigr) \bigr\rangle=\bigl\langle X,X'\bigr\rangle+\bigl\langle Z,Z'\bigr\rangle+ll' $$

on $\mathfrak{s}$. The associated left invariant Haar measure dx on S is given by a ^−Q−1  dX  dZ  da where dX, dZ, da are the Lebesgue measures on $\mathfrak{p},\mathfrak{z}$ and ℝ⁺ respectively. Also, the following integral formula holds:

$$ \int_Sf(x)\,dx=C_1\int_\mathbb{R}\int_N f(na_t)e^{-2\rho t}\,dt\,dn. $$

(2.1)

The constant C ₁ depends on the normalization of the Haar measures involved.

The group S can also be realized as the unit ball

$$B(\mathfrak{s})=\bigl\{(X,Z,l)\in\mathfrak{s}\bigm{|}|X|^2 +|Z|^2+ l^2<1\bigr\} $$

via the Caley transform $C:S\rightarrow B(\mathfrak{s})$. The Caley transform is defined as follows:

$$C(X, Z, a)=\bigl(X', Z', l'\bigr) $$

where

For x∈S, we let

$$r(x)=d\bigl(C(x), 0\bigr)=d(x,e)=\log\frac{1+ \|C(x)\|}{1-\|C(x)\|}. $$

A function f on S is said to be radial if there exists a function f ₀ on [0,∞) such that f(x)=f ₀(r(x)) for all x∈S. For a suitable function f on S, its radial component is defined by

$$f^\sharp(x)=\int_{S_\nu} f(y)\,d\sigma_\nu(y)\quad \text{for all } x\in S $$

where ν=r(x) and dσ _ν is the surface measure induced by the left invariant Riemannian metric on the geodesic sphere S _ν ={y∈S∣r(y)=ν} normalized by $\int_{S_{\nu}}d\sigma_{\nu}(y)=1$. It is clear that if f is a radial function, then its radial component f ^♯(x)=f(x) for all x∈S. We shall identify often the radial functions f=f(x) on S with functions f=f(r) of geodesic distance to the origin r=d(x,e)∈[0,∞). The convolution between two radial functions f,g on S is given by (f∗g)(x)=∫ _S f(y)g(y ⁻¹ x) dy. It is easy to check that f∗g is a radial function. Also, for two suitable radial functions f and g we have f∗g=g∗f.

Let $\mathbb{D}(S)^{\sharp}$ be the algebra of invariant differential operators on S which are radial i.e., the operators which commutes with the operator f↦f ^♯. Then $\mathbb{D}(S)^{\sharp}$ is a polynomial algebra with a single generator, namely the Laplace-Beltrami operator $\mathcal{L}$ (related to the Riemannian metric of S). We refer to [4, p. 234] for an explicit expression of the Laplace-Beltrami operator. A function ϕ on S is called spherical function if ϕ(e)=1 and ϕ is a radial eigenfunction of $\mathcal{L}$. All spherical functions are given by (see [1])

$$\phi_\lambda(x)= \bigl(a(x)^{\rho-i\lambda} \bigr)^\sharp,\quad \lambda\in\mathbb{C} $$

where a(x)=e ^t if x=ne ^t. Moreover, it follows that

$$\mathcal{L}\phi_\lambda=-\bigl(\lambda^2+\rho^2 \bigr)\phi_\lambda\quad\text{for all }\lambda\in\mathbb{C}. $$

Also ϕ _λ (x)=ϕ _−λ (x), ϕ _λ (x)=ϕ _λ (x ⁻¹) and ϕ _λ (e)=1. The spherical functions are related to the Jacobi functions by:

$$\phi_\lambda(x)=\phi_{2\lambda}^{\alpha, \beta}\bigl(r(x)/2\bigr)\quad \text{with } \alpha=\frac{m+k-1}{2},\ \beta=\frac{k-1}{2}. $$

The spherical functions satisfy the basic estimate (see [1]):

$$\phi_{i(\frac{1}{p}-\frac{1}{2})Q}(x)\asymp \left \{ \begin{array}{l@{\quad}l} e^{-\frac{Q}{p'}r(x)} & \text{if } 1\leq p<2\\ (1+r(x))e^{-\rho r(x)} & \text{if } p=2. \end{array} \right . $$

Here A≍B means there exists positive constants C ₁,C ₂ such that C ₁ B≤A≤C ₂ B. Also we have

$$\bigl|\phi_\lambda(x)\bigr|\leq\phi_{i\mu}(x)\quad \text{for }|\Im\lambda | \leq\mu $$

where ℑλ denotes the imaginary part of λ and ϕ _iρ ≡1. Therefore if λ∈ℝ, then |ϕ _λ (x)|≤ϕ ₀(x) for all x∈S.

For a suitable radial function f on S, the spherical transform is defined by

$$\widehat{f}(\lambda)=\int_S f(x)\phi_\lambda(x) \,dx. $$

We have, $\widehat{f\ast g}(\lambda)=\widehat{f}(\lambda)\widehat {g}(\lambda)$.

Let $C_{c}^{\infty}(S)^{\sharp}$ be the set of compactly supported radial C ^∞ functions on S. The L ²-Schwartz space $\mathcal{C}^{2}(S)^{\sharp}$ is defined to be the set of all functions f∈C ^∞(S)^♯ such that

$$\rho_{M, N}(f)=\sup_{r\in[0,\infty)}(1+r)^N\biggl \vert \biggl(\frac {d}{dr} \biggr)^Mf(r)\biggr \vert e^{\rho r}< \infty $$

for all M,N∈ℕ∪{0}. We topologize $\mathcal{C}^{2}(S)^{\sharp}$ by the seminorms above. Then $C_{c}^{\infty}(S)^{\sharp}$ is a dense subspace of $\mathcal{C}^{2}(S)^{\sharp}$.

For R>0, the Paley-Wiener space PW _R (ℂ) is the set of all entire functions h:ℂ→ℂ satisfying for each N∈ℕ

$$\bigl|h(\lambda)\bigr|\leq C_{N}\bigl(1+|\lambda|\bigr)^{-N}e^{R|\Im\lambda|}\quad \text{for all } \lambda\in\mathbb{C} $$

for some constant C _N >0 depending on N. Let PW(ℂ)=⋃ _R>0 PW _R (ℂ). We shall denote the set of all even functions in PW(ℂ) by PW(ℂ) _e .

Let $\mathcal{S}(\mathbb{R})$ be the set of all functions of Schwartz class and let $\mathcal{S}(\mathbb{R})_{e}$ be the subspace of $\mathcal{S}(\mathbb{R})$ consisting of even functions. Also let $C_{c}^{\infty}(\mathbb{R} )_{e}, L^{1}(\mathbb{R},d\lambda)_{e}$ and L ¹(ℝ,|c(λ)|⁻²  dλ) _e be the subspaces of even functions of $C_{c}^{\infty}(\mathbb{R})$, L ¹(ℝ,dλ) and L ¹(ℝ,|c(λ)|⁻²  dλ) respectively. They are equipped with the subspace topologies. From the basic estimates of ϕ _λ it follows that the domain of spherical transform of a function in $C_{c}^{\infty}(S)^{\sharp}$ is ℂ and that of a function in $\mathcal{C}^{2}(S)^{\sharp}$ is ℝ. We have the following Paley-Wiener and L ²-Schwartz space isomorphism theorem (see [1, 5]).

Theorem 2.1

The map $f\mapsto \widehat{f}$ is a topological isomorphism between $C_{c}^{\infty}(S)^{\sharp}$ and PW(ℂ) _e and also between $\mathcal{C}^{2}(S)^{\sharp}$ and $\mathcal{S}(\mathbb{R})_{e}$.

For a function $f\in\mathcal{C}^{2}(S)^{\sharp}$, we have the following inversion formula

$$f(x)=c_0\int_\mathbb{R}\widehat{f}(\lambda) \phi_\lambda (x)\bigl|c(\lambda)\bigr|^{-2}\, d\lambda $$

where the Plancherel density |c(λ)|⁻² is given by

depending whether

$$\left \{ \begin{array}{l} k=1,\qquad m/2 \text{ is odd} \\ k \text{ is odd},\qquad m/2 \text{ is even}\\ k \text{ is even},\qquad m/2 \text{ is even}. \end{array} \right . $$

Here $c_{0}=2^{k-2}\pi^{-\frac{n}{2}-1}\varGamma(\frac{n}{2})$. We note that these three cases cover all NA groups of rank one.

This Plancherel density |c(λ)|⁻² satisfies the following estimate (see [11, Lemma 4.8]):

$$ \bigl|c(\lambda)\bigr|^{-2}\asymp\lambda^2 \bigl(1+ | \lambda|\bigr)^{m+ k-2}\quad \text{for } \lambda\in\mathbb{R}. $$

(2.2)

For a suitable radial function f on S, the Abel transform is defined by

$$\mathcal{A} f(t)= e^{-\rho t}\int_N f(na_t)\,dn\quad \text{where } a_t=e^t. $$

It satisfies the relation $\widehat{f}(\lambda)=\widetilde{\mathcal{A} f}(\lambda)$, where $\widetilde{\mathcal{A} f}$ is the Euclidean Fourier transform of $\mathcal{A} f$. Therefore it follows from Theorem 2.1 that the Abel transform $f\mapsto\mathcal{A} f$ is a topological isomorphism between $C_{c}^{\infty}(S)^{\sharp}$ and $C_{c}^{\infty}(\mathbb{R})_{e}$ and also between $\mathcal{C}^{2}(S)^{\sharp}$ and $\mathcal{S}(\mathbb{R})_{e}$.

3 Bochner’s Theorem

In this section, first we give an integral representation of the radially positive definite functions. Next, we characterize the image of radial positive measures θ’s on S which satisfies ∫ _S ϕ ₀(x) dθ(x)<∞ under the spherical transform.

We have the following theorem (see [8, Theorem 5, p. 226]).

Theorem 3.1

Let T be an evenly positive definite distribution on ℝ i.e. T(ϕ∗ϕ ^∗)≥0 for all $\phi\in C_{c}^{\infty}(\mathbb{R})_{e}$. Then there exists even positive measures μ ₁ and μ ₂ such that for all $\phi\in C_{c}^{\infty}(\mathbb{R})_{e}$,

$$T(\phi)=\int_\mathbb{R}\widetilde{\phi}(\lambda)\,d \mu_1(\lambda) + \int_\mathbb{R}\widetilde {\phi}(i \lambda)\,d\mu_2(\lambda) $$

where μ ₁ is a tempered measure and μ ₂ is such that ∫_ℝ e ^a|λ|  dμ ₂(λ)<∞ for all a>0.

Here by a tempered measure μ on ℝ we mean that the measure μ satisfies $\int_{\mathbb{R}}\frac{1}{(1+|\lambda|^{2})^{p}}\,d\mu(\lambda )<\infty$ for some p>0.

We call a continuous, radial function f on S radially positive definite if

$$\int_S f(x) \bigl(g\ast g^\ast\bigr) (x)\,dx \geq0,\quad \text{for all } g\in C_c^\infty (S)^\sharp \text{ where } g^\ast(x)=\overline{g\bigl(x^{-1}\bigr)}. $$

If the equation above is true for every $g\in C_{c}^{\infty}(S)$, we say that f is a positive definite function. Then it is clear that the set of positive definite radial functions is a subset of the set of radially positive definite functions.

Theorem 3.2

Let f be a radially positive definite function on S. Then there exists even positive measures μ ₁ and μ ₂ such that for all x∈S,

$$f(x)=\int_\mathbb{R}\phi_\lambda(x)\, d \mu_1(\lambda) + \int_\mathbb{R}\phi_{i\lambda }(x)\,d\mu_2(\lambda) $$

where μ ₁ is a finite measure and μ ₂ is such that ∫_ℝ e ^a|λ|  dμ ₂(λ)<∞ for all a>0.

Proof

We define a distribution $T_{f}: C_{c}^{\infty}(\mathbb{R})_{e}\rightarrow \mathbb{C}$ by

$$T_f(h)=\int_S f(x)\mathcal{A}^{-1}h(x)\,dx $$

for all $h\in C_{c}^{\infty}(\mathbb{R})_{e}$. By the continuity of f and isomorphism of Abel transform it is easy to check that the integral exists and T _f is continuous. Also, we can check easily that $\mathcal{A}^{-1}(h\ast h^{\ast})=\mathcal{A}^{-1}h \ast(\mathcal{A}^{-1}h)^{\ast}$. Since f is radially positive definite, for all $h\in C_{c}^{\infty}(\mathbb{R})_{e}$,

$$T_f\bigl(h\ast h^\ast\bigr)=\int_S f(x) \bigl(\mathcal{A}^{-1}h\ast\bigl(\mathcal{A}^{-1}h \bigr)^\ast \bigr) (x)\,dx\geq0. $$

That is T _f is an evenly positive definite distribution on ℝ. Hence, by Theorem 3.1, there exists even positive measures μ ₁ and μ ₂ such that for all $h\in C_{c}^{\infty}(\mathbb{R})_{e}$,

$$T_f(h)=\int_\mathbb{R}\widetilde{h}(\lambda)\, d \mu_1(\lambda) + \int_\mathbb{R}\widetilde {h}(i \lambda)\, d\mu_2(\lambda) $$

where μ ₁ is a tempered measure and μ ₂ is such that ∫_ℝ e ^a|λ|  dμ ₂(λ)<∞ for all a>0. Therefore, we have

$$\int_S f(x)g(x)\,dx=\int_\mathbb{R} \widehat{g}(\lambda)\, d\mu_1(\lambda) + \int_\mathbb{R}\widehat{g}(i\lambda)\, d\mu_2(\lambda)\quad\text{for all } g\in C_c^\infty (S)^\sharp. $$

Now we shall show that the measure μ ₁ is finite. For this, let {α _n } be a δ-sequence in $C_{c}^{\infty}(S)^{\sharp}$ and let $g_{n}=\alpha_{n}\ast\alpha_{n}^{\ast}$. Then {g _n } is a δ-sequence in $C_{c}^{\infty}(S)^{\sharp}$. Also, $\widehat{g_{n}}(\lambda)=|\widehat {\alpha_{n}}(\lambda)|^{2}\geq0$ for all λ∈ℝ∪iℝ and $\lim_{n\rightarrow\infty}\widehat{g_{n}}(\lambda)=\lim_{n\rightarrow \infty}\int_{S} g_{n}(x)\phi_{\lambda}(x)\,dx=\phi_{\lambda}(e)=1$. Now the equation

$$\int_S f(x)g_n(x)\,dx=\int _\mathbb{R}\widehat{g_n}(\lambda)\, d \mu_1(\lambda) + \int_\mathbb{R}\widehat{g_n}(i \lambda)\, d\mu_2(\lambda) $$

implies that $\int_{S} f(x)g_{n}(x)\,dx\geq\int_{\mathbb{R}}\widehat {g_{n}}(\lambda)\, d\mu_{1}(\lambda)$ (since $\widehat{g_{n}}(\lambda)\geq0$ for all λ∈ℝ∪iℝ and the measures μ ₁,μ ₂ are positive). Then using Fatou’s lemma we get that

$$f(e)=\lim_{n\rightarrow\infty}\int_S f(x)g_n(x) \,dx\geq\lim_{n\rightarrow\infty}\int_\mathbb{R}\widehat{g_n}( \lambda)\,d\mu_1(\lambda)\geq \int_\mathbb{R} \lim_{n\rightarrow\infty}\widehat{g_n}(\lambda)\, d\mu_1( \lambda). $$

This shows that

$$\int_\mathbb{R}\,d\mu_1(\lambda)\leq f(e). $$

Therefore the measure μ ₁ is finite. Then using Fubini’s theorem we get that

$$\int_S f(x)g(x)\, dx = \int_S g(x) \int_\mathbb{R}\phi_\lambda (x)\,d\mu_1(\lambda ) \,dx + \int_S g(x)\int_\mathbb{R} \phi_{i\lambda}(x)\, d\mu_2(\lambda)\,dx. $$

The equation above is true for every $g\in C_{c}^{\infty}(S)^{\sharp}$. Hence

$$f(x)=\int_\mathbb{R}\phi_\lambda(x)\, d \mu_1(\lambda) + \int_\mathbb{R}\phi_{i\lambda }(x)\,d\mu_2(\lambda)\quad \text{for all } x\in S $$

where μ ₁ is a finite positive even measure and μ ₂ is a positive even measure such that ∫_ℝ e ^a|λ|  dμ ₂(λ)<∞ for all a>0. □

Now we define an operation ⊙ on the suitable functions on ℝ which will make L ¹(ℝ,|c(λ)|⁻²  dλ) a Banach algebra (cf. [7]).

If λ∈ℝ then |ϕ _λ (r)|≤ϕ ₀(r)≤(1+r)e ^−ρr for all r∈[0,∞). Also for any M∈ℕ∪{0} we have $| (\frac{d}{dr} )^{M} \phi_{\lambda}(r)|\leq C (1+r) e^{-\rho r}$ for some positive constant C (which depends on M and λ) (see [6, Theorem 2]). Therefore for λ,μ∈ℝ we have $| (\frac{d}{dr} )^{M} \phi_{\lambda}(r)\phi_{\mu}(r)|\leq C' (1+r)^{t} e^{-2\rho r}$, where C′ is a positive constant (depends on λ,μ) and t is a nonnegative integer. This shows that

$$\sup_{r\in[0,\infty)} (1 +r)^N \biggl \vert \biggl(\frac{d}{dr} \biggr)^M (\phi_\lambda\phi_\mu) (r)\biggr \vert e^{\rho r}\leq C'\sup_{r\in[0,\infty )} (1 +r)^{N +t} e^{-\rho r}<\infty. $$

Therefore the function

$$f_{\lambda,\mu}:x\mapsto\phi_\lambda(x)\phi_\mu(x)\in \mathcal{C}^2(S)^\sharp\quad\text{for }\lambda, \mu\in\mathbb{R}. $$

Hence the spherical transform

$$\widehat{f_{\lambda, \mu}}(\nu)=\int_S f_{\lambda,\mu}(x)\phi_\nu(x)\, dx=\int_S \phi_\lambda(x)\phi_\mu(x)\phi_\nu(x)\,dx $$

exists on ℝ and it belongs to $\mathcal{S}(\mathbb{R})_{e}$. Let us denote $\widehat{f_{\lambda, \mu}}(\nu)$ by K(λ,μ,ν). It is easy to prove that for any even polynomial q, there exists constants C,r ₁,r ₂≥0 such that $|q(\lambda)K(\lambda, \mu, \nu)|\leq C (1 +|\mu |)^{r_{1}}(1+|\nu|)^{r_{2}}$ for all λ,μ,ν∈ℝ.

Using the inversion theorem we get that

$$\phi_\lambda(x)\phi_\mu(x)= c_0\int _\mathbb{R}K(\lambda,\mu,\nu )\phi_\nu (x)\bigl|c( \nu)\bigr|^{-2}\,d\nu. $$

Suppose $f, g\in\mathcal{C}^{2}(S)^{\sharp}$. Then by the inversion formula we have

For $A, B\in\mathcal{S}(\mathbb{R})$ we define

$$A\odot B(\nu)=c_0^2\int_\mathbb{R}\int _\mathbb{R}A(\lambda)B(\mu )K(\lambda,\mu,\nu )\bigl|c( \lambda)\bigr|^{-2}\bigl|c(\mu)\bigr|^{-2}\,d\lambda\,d\mu. $$

Then for $A, B\in\mathcal{S}(\mathbb{R})$, A⊙B exists on ℝ as the Plancherel density |c(λ)|⁻² has polynomial growth (see (2.2)). Therefore it follows that

$$f(x)g(x)=c_0\int_\mathbb{R} (\widehat{f}\odot \widehat{g} ) (\nu)\phi_\nu (x)\bigl|c(\nu)\bigr|^{-2}d\nu. $$

This shows that

$$ \widehat{f.g}(\lambda)=\widehat{f}\odot\widehat{g}(\lambda). $$

(3.1)

Also for a suitable radial function h on S, we have

$$ \int_S h(x) (f.g) (x)\,dx=\int _\mathbb{R}\widehat{h}(\lambda) (\widehat {f}\odot\widehat {g}) ( \lambda)\bigl|c(\lambda)\bigr|^{-2}\,d\lambda. $$

(3.2)

From [7, Theorem 4.4] and [1, (2.13), (2.14)] it follows that

$$K(\lambda, \mu, \nu)=\widehat{f_{\lambda,\mu }}(\nu)\geq 0\quad \mbox{for all } \lambda, \mu,\nu\in\mathbb{R}. $$

From this fact it is easy to check that L ¹(ℝ,|c(λ)|⁻²  dλ) is an algebra under the operation ⊙ and $\|f\odot g\|_{L^{1}(\mathbb{R},|c(\lambda )|^{-2}\,d\lambda )}\leq\|f\|_{L^{1}(\mathbb{R},|c(\lambda)|^{-2}\,d\lambda)}\* \|g\|_{L^{1}(\mathbb{R} ,|c(\lambda)|^{-2}\,d\lambda)}$.

Definition 3.3

For a radial positive measure θ on S, its spherical transform is defined by $\widehat{\theta}(\lambda)=\int_{S}\phi_{\lambda}(x)\, d\theta (x)$, whenever the integral exists.

Proposition 3.4

Let θ be a radial positive measure on S such that

$$\int_S\phi_0(x)\, d\theta (x)<\infty. $$

Then the following conditions are satisfied:

• (1) $\widehat{\theta}$ exists on ℝ and it is an even, continuous, bounded function on ℝ;

• (2) $\int_{\mathbb{R}}\widehat{\theta}(\lambda) (g\odot g^{\ast})(\lambda)|c(\lambda)|^{-2}\,d\lambda\geq0 \text{ \textit{for all} } g\in \mathcal{S}(\mathbb{R})_{e}$.

Proof

(1) Existence and boundedness of $\widehat{\theta}$ will follow from the fact that for λ∈ℝ, |ϕ _λ (x)|≤ϕ ₀(x) for all x∈S. Also ϕ _λ (x)=ϕ _−λ (x) for all x∈S,λ∈ℝ implies that $\widehat{\theta}$ is an even function. Continuity follows from the dominated convergence theorem.

(2) Let $\alpha\in\mathcal{C}^{2}(S)^{\sharp}$ be such that $\widehat{\alpha}(\lambda)=g(\lambda )$. Then

$$\int_\mathbb{R}\widehat{\theta}(\lambda) \bigl(g\odot g^\ast \bigr) (\lambda )\bigl|c(\lambda)\bigr|^{-2}\,d\lambda= \int _\mathbb{R}\widehat{\theta }(\lambda) \bigl(\widehat {\alpha} \odot(\widehat{\alpha})^\ast \bigr) (\lambda)\bigl|c(\lambda )\bigr|^{-2}\, d\lambda. $$

This is equal to $\int_{\mathbb{R}}\widehat{\theta}(\lambda)\widehat { (\alpha \overline{\alpha} )}(\lambda)|c(\lambda)|^{-2}\,d\lambda$. Then using the definition and the Fubini’s theorem we get that

$$\int_\mathbb{R}\widehat{\theta}(\lambda) \bigl(g\odot g^\ast \bigr) (\lambda )\bigl|c(\lambda)\bigr|^{-2}\,d\lambda=\int _S \int_\mathbb{R}\widehat{ (\alpha \overline {\alpha} )}(\lambda)\phi_\lambda(x)\bigl|c(\lambda)\bigr|^{-2}\, d\lambda\, d\theta(x). $$

Now we use the inversion formula to get

$$\int_\mathbb{R}\widehat{\theta}(\lambda) \bigl(g\odot g^\ast \bigr) (\lambda )\bigl|c(\lambda)\bigr|^{-2}\,d\lambda= \frac{1}{c_0}\int_S (\alpha .\overline{\alpha} ) (x)\,d \theta(x)= \frac{1}{c_0}\int_S \bigl|\alpha (x)\bigr|^2 \,d\theta(x)\geq0. $$

 □

In the following theorem, we characterize the image of such radial positive measures under the spherical transform.

Theorem 3.5

An even, continuous, bounded function h on ℝ is the spherical transform of a radial positive measure θ on S which satisfies ∫ _S ϕ ₀(x) dθ(x)<∞ if and only if h satisfies the condition ∫_ℝ h(λ)(g⊙g ^∗)(λ)|c(λ)|⁻²  dλ≥0 for all $g\in\mathcal{S}(\mathbb{R})_{e}$.

The classical Bochner’s theorem can be restated as follows: A continuous function p on ℝ is the Fourier transform of a finite positive measure on ℝ if and only if it is a positive linear functional on the algebra (L ¹(ℝ,dλ),∗).

Let p be an even, continuous, bounded function on ℝ which is a positive linear functional on the algebra (L ¹(ℝ,dλ) _e ,∗) i.e. p satisfies ∫_ℝ p(λ)(l∗l ^∗)(λ)≥0 for all l∈L ¹(ℝ,dλ) _e . Then by Krein’s theorem [8] there exists finite, positive, even measures ν ₁,ν ₂ on ℝ such that

$$p(\lambda)=\int_\mathbb{R}e^{i\lambda x}\,d \nu_1(x) + \int_{i\mathbb{R}} e^{i\lambda y}\,d \nu_2(y). $$

But the boundedness of p implies that p(λ)=∫_ℝ e ^iλx  dν ₁(x). Therefore p is a positive definite function. Conversely any even positive definite function can be considered as a positive linear functional on the algebra (L ¹(ℝ,dλ) _e ,∗).

Hence, Bochner’s theorem for even functions on ℝ can be stated as follows: An even, continuous, bounded function p on ℝ is the Fourier transform of a finite, positive, even measure on ℝ if and only if it is a positive linear functional on the algebra (L ¹(ℝ,dλ) _e ,∗).

Also, we can restate our theorem (Theorem 3.5) alternatively as follows: An even, continuous, bounded function h on ℝ is the spherical transform of a radial positive measure θ on S which satisfies ∫ _S ϕ ₀(x) dθ(x)<∞ if and only if it is a positive linear functional on the Banach algebra (L ¹(ℝ,|c(λ)|⁻²  dλ) _e ,⊙).

Therefore our theorem is analogous to the classical Bochner’s theorem.

We state the following corollary of the theorem:

Corollary 3.6

An even, continuous, bounded function h on ℝ which satisfies for all $g\in\mathcal{S}(\mathbb{R})_{e}$,

$$\int_\mathbb{R}h(\lambda) \bigl(g\odot g^\ast\bigr) ( \lambda)\bigl|c(\lambda)\bigr|^{-2}\, d\lambda \geq0 $$

is a positive definite function on ℝ.

Proof of Corollary

Suppose h is an even, continuous, bounded function on ℝ which satisfies the condition of the theorem above. So, by the Theorem 3.5 there exists a radial positive measure θ on S such that $\widehat{\theta}(\lambda)=h(\lambda)$ for all λ∈ℝ. The measure θ satisfies ∫ _S ϕ ₀(x) dθ(x)<∞. We consider the Abel transform $\mathcal{A}\theta$ of θ defined by

$$d\mathcal{A}\theta(a_t)= e^{-\rho t}\int_N \, d\theta(na_t)\quad \text{where } a_t=e^t. $$

Then $\mathcal{A}\theta$ is a positive measure on ℝ having the property that

$$ \widetilde{\mathcal{A}\theta}(\lambda):=\int_\mathbb{R}e^{-i\lambda t}\,d\mathcal{A}\theta(a_t)=\widehat{\theta}(\lambda). $$

(3.3)

From the condition ∫ _S ϕ ₀(x) dθ(x)<∞ and (3.3) it follows that the measure $\mathcal{A}\theta$ is finite and hence its Fourier transform $\widetilde{\mathcal{A}\theta}$ is a positive definite function on ℝ. But $\widetilde{\mathcal{A}\theta }(\lambda)=\widehat{\theta}(\lambda)=h(\lambda)$. Therefore h is a positive definite function on ℝ. □

Let $\mathcal{P}_{0}$ be the set of all even, continuous, bounded functions h on ℝ such that for all $g\in\mathcal{S}(\mathbb{R})_{e}$

$$\int_\mathbb{R}h(\lambda) \bigl(g\odot g^\ast\bigr) ( \lambda)\bigl|c(\lambda)\bigr|^{-2}\, d\lambda \geq0. $$

Also let $\mathcal{P}$ be the set of all positive definite functions on ℝ. Then using the theorem above we have the following partial informations about the set $\mathcal{P}_{0}$:

• (1) By the corollary above we have $\mathcal{P}_{0}\subseteq\mathcal{P}$.

• (2) For each fixed x ₀∈S, the function $\lambda\mapsto\phi_{\lambda}(x_{0})\in\mathcal{P}_{0}$. In particular $\mathcal{P}_{0}$ contains positive constants. Also for each fixed x ₀∈S, the function λ↦ϕ _λ (x ₀) is a positive definite function on ℝ, which can also be concluded from its integral representation.

• (3) Let us consider the heat kernel p _t on S. It is a radial, nonnegative function on S such that $\widehat{p_{t}}(\lambda )=e^{-t(\rho ^{2}+ \lambda^{2})}$ (see [1]). This fact together with the theorem above implies that for each t>0, the function $\lambda\mapsto e^{-t(\rho^{2} + \lambda^{2})}\in\mathcal{P}_{0}$.

• (4) If $\beta_{1}, \beta_{2}\in\mathcal{P}_{0}$ then it follows that $\beta_{1} +\beta_{2}, \beta_{1}\beta_{2}, c\beta_{1}\in\mathcal{P}_{0}$ for any positive constant c.

Remark 3.7

The condition in Theorem 3.5 on the measure θ (i.e. ∫ _S ϕ ₀(x) dθ(x)<∞) is due to a technical reason. A finite measure θ satisfies the condition ∫ _S ϕ ₀(x) dθ(x)<∞, since |ϕ ₀(x)|≤1 for all x∈S. Spherical transform of a finite positive measure exists on S ₁:={λ∈ℂ∣|ℑλ|≤ρ} as ϕ _λ ∈L ^∞ if and only if λ∈S ₁. Also, it is easy to prove that the spherical transform $\widehat{\theta}$ is analytic on $S_{1}^{0}$, continuous on S ₁ and satisfies the positive definite like condition (as stated in the Theorem 3.5). For the converse we conjecture the following:

An even, bounded function h, which is analytic on the interior of S ₁ and continuous on S ₁ is the spherical transform of a radial, finite, positive measure θ on S if and only if for all $g\in \mathcal{S}(\mathbb{R})_{e}$

$$\int_\mathbb{R}h(\lambda) \bigl(g\odot g^\ast\bigr) ( \lambda)\bigl|c(\lambda)\bigr|^{-2}\, d\lambda \geq0. $$

Proof of the Theorem 3.5

The necessity of the condition is proved in Proposition 3.4.

For the sufficiency we let h be an even, continuous, bounded function on ℝ which satisfies the condition above. We define a linear functional $T:\mathcal{S}(\mathbb{R})_{e}\rightarrow\mathbb{C}$ by

$$T(g)=c_0\int_\mathbb{R}h(\lambda)g(\lambda)\bigl|c( \lambda)\bigr|^{-2}\, d\lambda\quad\text{for all } g\in\mathcal{S}(\mathbb{R})_e. $$

This linear functional exists and continuous by the boundedness of h. Using this we also define a continuous linear functional $\widetilde {T}:\mathcal{C}^{2}(S)^{\sharp}\rightarrow\mathbb{C}$ by

$$\widetilde{T}(f)=T(\widehat{f}),\quad \text{for all } f\in\mathcal{C}^2(S)^\sharp. $$

This linear functional is well defined and continuous by the Schwartz space isomorphism theorem (Theorem 2.1). From the hypothesis we have

$$T\bigl(g\odot g^\ast\bigr)\geq0\quad \text{for all } g\in\mathcal{S}( \mathbb{R})_e. $$

This implies that $T(\widehat{\alpha}\odot(\widehat{\alpha})^{\ast})\geq0$ for all $\alpha\in\mathcal{C}^{2}(S)^{\sharp}$. That is $T(\widehat{\alpha\overline{\alpha}})\geq 0$ by (3.1), since $\widehat{\overline{\alpha }}(\lambda )=(\widehat{\alpha})^{\ast}(\lambda)$. This condition is equivalent to

$$\widetilde{T}(\alpha\overline{\alpha})\geq0\quad \text{for all } \alpha\in \mathcal{C}^2(S)^\sharp. $$

We claim that:

$$\widetilde{T}(\alpha)\geq0\quad \text{for all }\alpha\in\mathcal{C}^2(S)^\sharp\text{ with }\alpha\geq0. $$

To prove the claim, we first show that $\{\alpha\overline{\alpha }\mid \alpha\in C_{c}^{\infty}(S)^{\sharp}\}$ is dense in $\{\alpha\geq0\mid \alpha \in C_{c}^{\infty}(S)^{\sharp}\}$. For this we let ψ be a positive function in $C_{c}^{\infty}(S)^{\sharp}$ and suppose ψ(x)=0 for r(x)>a. Let γ be a compactly supported C ^∞ function on ℝ with γ(t)=1 for |t|≤a. We extend γ as a radial function to S. We define

$$\psi_m(x)=\gamma(x)\sqrt{\psi(x)+\frac{1}{m}}. $$

Then $\psi_{m}\in C_{c}^{\infty}(S)^{\sharp}$ and

$$\psi_m^2(x)=\gamma(x)^2 \biggl(\psi(x)+ \frac{1}{m} \biggr)\rightarrow \psi(x) $$

in the topology of $C_{c}^{\infty}(S)^{\sharp}$. Therefore $\widetilde {T}(\alpha )\geq0$ for all $\alpha\in C_{c}^{\infty}(S)^{\sharp}$ with α≥0. Now we let $\alpha\in\mathcal{C}^{2}(S)^{\sharp}$ such that α≥0. Then there exists a sequence $\alpha_{n}\in C_{c}^{\infty}(S)^{\sharp}$ with α _n ≥0 such that α _n →α in $\mathcal{C}^{2}(S)^{\sharp}$. Since each $\widetilde{T}(\alpha_{n})\geq0$ it follows that $\widetilde{T}(\alpha)\geq0$. Hence the claim is established.

Therefore $\alpha\mapsto\widetilde{T}(\alpha)$ is a positive linear functional on $\mathcal{C}^{2}(S)^{\sharp}$. By Riesz representation theorem there is a radial positive measure θ on S such that

$$\widetilde{T}(\alpha)=\int_S \alpha(x)\,d\theta(x)\quad \text{for all }\alpha \in C_c^\infty(S)^\sharp. $$

That is

$$c_0\int_\mathbb{R}h(\lambda)\widehat{\alpha}( \lambda)\bigl|c(\lambda )\bigr|^{-2}\,d\lambda =\int_S \alpha(x)\,d\theta(x)\quad\text{for all }\alpha\in C_c^\infty (S)^\sharp. $$

But h is such that the linear functional $\alpha\mapsto\int_{\mathbb{R}} h(\lambda)\widehat{\alpha}(\lambda)|c(\lambda)|^{-2}\,d\lambda$ extends to $\mathcal{C}^{2}(S)^{\sharp}$. Therefore

$$c_0\int_\mathbb{R}h(\lambda)\widehat{\alpha}( \lambda)\bigl|c(\lambda )\bigr|^{-2}\,d\lambda =\int_S \alpha(x)\,d\theta(x)\quad\text{for all }\alpha\in\mathcal{C}^2(S)^\sharp. $$

That is

(3.4)

We shall show that the measure θ satisfies ∫ _S ϕ ₀(x) dθ(x)<∞. For that we consider the heat kernel

$$p_t(x)=\int_\mathbb{R}e^{-t(\lambda^2 + \rho^2)} \phi_\lambda (x)\bigl|c(\lambda )\bigr|^{-2}\,d\lambda. $$

This is a radial function on S, which satisfies p _t (x)≥0 for all x∈S and ∫ _S p _t (x) dx=1. Let us define $\gamma_{n}(\lambda )=\frac{1}{p_{n}(e)}e^{-n(\lambda^{2} + \rho^{2})}$. Then it follows that $\int_{S} \frac{p_{n}(x)}{p_{n}(e)}\phi_{\lambda}(x)\,dx= \gamma_{n}(\lambda)$ and ∫_ℝ γ _n (λ)|c(λ)|⁻²  dλ=1. Now our claim is that for any β>0,

$$\int_{|\lambda|\ge\beta}\gamma_n(\lambda)\bigl|c( \lambda)\bigr|^{-2}\, d\lambda \rightarrow0\quad \text{as } n\rightarrow\infty. $$

Let β>0 fixed and choose α>0 such that β>α. We have $p_{n}(e)=\int_{\mathbb{R}}e^{-n(\lambda^{2} +\rho^{2})}\allowbreak |c(\lambda )|^{-2}\, d\lambda$. Then

This implies that $p_{n}(e)\ge C_{\alpha}\,e^{-n(\alpha^{2} +\rho^{2})}$ where C _α is a positive constant depends only on α. Now

where D _β is a positive constant depends only β. Therefore

$$\int_{|\lambda|\geq\beta}\gamma_n(\lambda)\bigl|c( \lambda)\bigr|^{-2}\, d\lambda \leq A e^{-n(\beta^2-\alpha^2)} $$

where A is a positive constant depends only on β,α. This establishes the claim.

Therefore since h is continuous and bounded we have,

$$\lim_{n\rightarrow\infty}\int_\mathbb{R}h(\lambda) \gamma_n(\lambda )\bigl|c(\lambda )\bigr|^{-2}\,d\lambda=h(0). $$

We apply the sequence {γ _n } to (3.4) and take the limit n→∞ and use the Fatou’s lemma to get

Now

Therefore we get

$$\int_S\phi_0(x)\,d\theta(x)\le h(0). $$

From (3.4) using Fubini’s theorem we get,

$$\int_\mathbb{R}h(\lambda)\widehat{\alpha}(\lambda)\bigl|c(\lambda )\bigr|^{-2}\,d\lambda=\int_\mathbb{R}\widehat{\alpha}( \lambda)\bigl|c(\lambda)\bigr|^{-2} \biggl(\int_S \phi_\lambda (x)\, d\theta(x) \biggr)\, d\lambda. $$

But the equation above is true for every $\alpha\in\mathcal{C}^{2}(S)^{\sharp}$. This implies that

$$h(\lambda)=\int_S \phi_\lambda(x)\,d\theta(x) \quad\text{for all }\lambda\in \mathbb{R}. $$

This completes the proof. □

Remark 3.8

(1) Let θ be a finite, positive, radial measure on S. Then its spherical transform $\widehat{\theta}$ is obviously analytic on $S_{1}^{0}$, continuous on S ₁ and satisfies the positive definite like condition (as stated in Theorem 3.5). Conversely if we start with an even function which is analytic on $S_{1}^{0}$, continuous on S ₁ and satisfies the positive definite like condition we can proceed as in the proof of the theorem to get a measure θ which satisfies (3.4) but from this we are unable to prove that the measure θ is finite. If we could prove that the measure θ is finite then we would get h(λ)=∫ _S ϕ _λ (x) dθ(x) for all λ∈ℝ. From analyticity and continuity the equality would hold on the strip S ₁.

(2) A Riemannian symmetric space X of noncompact type can be realized as a quotient space G/K where G is a connected noncompact semisimple Lie group with finite centre and K is a maximal compact subgroup of G. Also a symmetric space X is an NA group and radial functions of that NA group are K-biinvariant functions on G. Therefore the theorems proved in this article for radial functions on NA group is also true for K-biinvariant functions on the real rank one noncompact, connected, semisimple Lie group G with finite centre. Theorem 3.5 is new in the context of real rank one symmetric space case also.