There are three main results in this note: Firstly, the basis function method is presented. The basis function approach starts with a fixed positive definite function \(\varphi\) on a space \(X\times X\), and approximates a function \(f\) by a linear combination of type \(\sum^N_{j=0} a_j\varphi (y,x_j)\) with constant coefficients \(a_1,\dots,a_N\) and \(y,x_j\in X\). Secondly, harmonic analysis on Gelfand pairs is discussed on a locally compact group with compact subgroup \(K\), including: (1) some properties of Gelfand pairs; (2) the spectrum of the convolution algebra; (3) a product formula for spherical functions; (4) the Gelfand transform for the commutative Banach algebra (or spherical Fourier transform); (5) the Bochner-Godement theorem for Gelfand pairs.
Finally, the reflection invariant functions from the point of view of the semi-direct product \(M_W=W_d \ltimes \mathbb{R}^d\) of \(W_d\) and \(\mathbb{R}^d\) are studied, where \(W_d\) is a finite reflection group acting on the Abelian group \(\mathbb{R}^d\), and the following result is obtained: (1) the pair \((M_W, W_d)\) is a Gelfand pair; (2) the positive definite spherical functions for the pair \((M_W, W_d)\) are given by \(J_{W_d} (\nu,x)= \frac{1}{|W_d|}\sum_{\sigma\in W_d} e^{i\nu^t (\sigma x)}\); (3) the set of \(W_d\)-invariant, positive definite, spherical functions on \(M_W\) is the set of functions \(\{J_{W_d}(\nu,\cdot): \nu\in W= \text{dom}_J\}\); (4) let \(\varphi\) be a continuous, \(W_d\)-invariant function on \(R^d\), then \(\varphi\) is positive definite if and only if it can be represented as \(\varphi(x)=\int_WJ_{W_d}(\nu,x)\,d\mu(\nu)\), \(x\in W\), and \(\mu\) is a non-negative Borel measure on \(W\).
For the entire collection see [Zbl 1061.41001].