A Unified View of Space–Time Covariance Functions Through Gelfand Pairs

Abstract

We give a characterization of positive definite integrable functions on a product of two Gelfand pairs as an integral of positive definite functions on one of the Gelfand pairs with respect to the Plancherel measure on the dual of the other Gelfand pair. In the very special case where the Gelfand pairs are Euclidean groups and the compact subgroups are reduced to the identity, the characterization is a much cited result in spatio-temporal statistics due to Cressie, Huang and Gneiting. When one of the Gelfand pairs is compact the characterization leads to results about expansions in spherical functions with positive definite expansion functions, thereby recovering recent results of the author in collaboration with Peron and Porcu. In the special case when the compact Gelfand pair consists of orthogonal groups, the characterization is important in geostatistics and covers a recent result of Porcu and White.

Appendix

We shall give an example showing that the functions $C({\mathbf {h}};u)$ constructed in Proposition 1.1 need not be integrable.

In the following ${\mathcal { F}} f$ denotes the Fourier transform of a function $f:{\mathbb {R}}\rightarrow {\mathbb {C}}$ given by

$$\begin{aligned} {\mathcal { F}} f(t)=\int _{-\infty }^\infty e^{-itx}f(x)\,dx,\quad t\in {\mathbb {R}}. \end{aligned}$$

Let $C_0(]-1,1[)$ denote the set of continuous functions $f:]-1,1[\rightarrow {\mathbb {C}}$ vanishing at “infinity”, i.e.,

$$\begin{aligned} \lim _{x\rightarrow -1}f(x)=\lim _{x\rightarrow 1} f(x)=0. \end{aligned}$$

It is a Banach space under the uniform norm

$$\begin{aligned} ||f||=\sup \{|f(x)|\mid -1<x<1\}. \end{aligned}$$

We proceed in a number of steps.

1: There exists $f\in C_0(]-1,1[)$ such that ${\mathcal { F}} f\notin L^1({\mathbb {R}})$.

This is a classical application of the Banach-Steinhaus Theorem to the continuous linear functionals on the Banach space $C_0(]-1,1[)$:

$$\begin{aligned} L_n(f)=\int _{-n\pi }^{n\pi } {\mathcal { F}} f(t)dt=\int _{-1}^1f(x)\frac{2\sin (n\pi x)}{x}dx,\quad f\in C_0(]-1,1[). \end{aligned}$$

In fact, assuming that $L_n, n\ge 0$ is pointwise bounded, we get that $||L_n||$ is bounded, which is a contradiction because

$$\begin{aligned} ||L_n||=\int _{-1}^1\left| \frac{2\sin (n\pi x)}{x}\right| dx=4\int _0^{n\pi }\frac{|\sin (u)|}{u}du \end{aligned}$$

tends to infinity for $n\rightarrow \infty$. This shows the existence of an $f\in C_0(]-1,1[)$ such that $(L_n(f))$ is an unbounded sequence, and in particular ${\mathcal { F}} f\notin L^1({\mathbb {R}})$.

2: There exists $f\in C_0^+(]-1,1[)$ with $\max f=1$ such that ${\mathcal { F}} f\notin L^1({\mathbb {R}})$.

This is an easy consequence of 1.

3: There exists $f\in C_c^+({\mathbb {R}})$ with $\max f=1$ and $f(x)>0$ for $x\in [0,1]$ such that ${\mathcal { F}} f\notin L^1({\mathbb {R}})$.

For f as in 2 let $x_0\in {\mathbb {R}}$ satisfy $f(x_0)=1$. Then there exists $\delta >0$ such that $f(x)>0$ for $x\in [x_0,x_0+\delta ]$, and $x\mapsto f(\delta x+x_0)$ satisfies 3.

4: There exists $f\in C({\mathbb {R}})\cap L^1({\mathbb {R}})$ such that $0<f(x)<1$ for all $x\in {\mathbb {R}}$ and ${\mathcal { F}} f\notin L^1({\mathbb {R}})$.

Let h have the properties of 3 and let $a_n>0, n\in {\mathbb {Z}}$ be such that $a_0=1/2$, $\sum _{n\in {\mathbb {Z}},n\ne 0} a_n=1/4$. Then

$$\begin{aligned} f(x)=\sum _{n\in {\mathbb {Z}}} a_n h(x-n),\quad x\in {\mathbb {R}}\end{aligned}$$

is continuous and has the properties $0<f(x)\le 3/4$ for $x\in {\mathbb {R}}$ and $\int f(x)dx=3/4\int h(x)dx<\infty$. Furthermore,

$$\begin{aligned} {\mathcal { F}} f(t)=\mathcal F h(t)\sum _{n\in {\mathbb {Z}}}a_n e^{-itn}, \end{aligned}$$

hence

$$\begin{aligned} |{\mathcal { F}} f(t)|=|\mathcal F h(t)|\left| \sum _{n\in {\mathbb {Z}}}a_n e^{-itn}\right| \ge |\mathcal F h(t)|\left( a_0-\sum _{n\in {\mathbb {Z}},n\ne 0}a_n\right) =1/4|\mathcal F h(t)|, \end{aligned}$$

showing that ${\mathcal { F}} f\notin L^1({\mathbb {R}})$.

Conclusion Define

$$\begin{aligned} g(\omega ;\tau )=\frac{1}{\sqrt{2\pi }}f(\omega )e^{-\tau ^2/2},\quad (\omega ,\tau )\in {\mathbb {R}}\times {\mathbb {R}}, \end{aligned}$$

where f satisfies 4. Then g is strictly positive, continuous and integrable and

$$\begin{aligned} h(\omega ;u):=\int _{-\infty }^\infty g(\omega ;\tau )e^{iu\tau }d\tau =f(\omega )e^{-u^2/2} \end{aligned}$$

satisfies (C1’) and (C2’) of Proposition 1.1 with $d=1$ and

$$\begin{aligned} C(h;u)=\int _{-\infty }^\infty e^{ih\omega }h(\omega ;u)d\omega =e^{-u^2/2}{\mathcal { F}} f(-h)\notin L^1({\mathbb {R}}^2). \end{aligned}$$