Sampling in Reproducing Kernel Hilbert Space

An account of sampling in the setting of reproducing kernel spaces is given, the main point of which is to show that the sampling theory of Kluvánek, even though it is very general in some respects, is nevertheless a special case of the reproducing kernel theory. A Dictionary is provided as a handy summary of the essential steps. Starting with the classical formulation, the notion of band-limitation is a key feature in these settings. The present chapter is, by and large, self-contained and a specialist knowledge of reproducing kernel theory is not required. Here is one of Ramanujan’s beautiful Fourier integrals. Let denote Euler’s Gamma function as usual, and let ̨ C ˇ > 1. Then Z 1 1 e ixt . ̨ C t / .ˇ t / dt D 8̂ <̂ ˆ̂: f2 cos.x=2/g ̨Cˇ 2 . ̨ C ˇ 1/ e ix. ̨ ; jxj I 0; jxj : (2.1) One recognises qualities of simplicity and integrity in the nature of this remarkable formula. Simplicity appears first, in a left-hand side which involves only elementary functions and Gamma, the most basic transcendental function. By integrity I mean that the right-hand side stays within this regime. One might not have anticipated this! As well as giving rise to several interesting special cases [22, p. 187], the formula provides an example of a function whose Fourier transform has support on a compact set. Such functions are usually called band-limited. Here the compact set, or frequency band, or set of spectral support, is Œ ; . J.R. Higgins ( ) 4 rue du Bary, 11250, Montclar, France e-mail: [email protected] © Springer International Publishing Switzerland 2014 A.I. Zayed, G. Schmeisser (eds.), New Perspectives on Approximation and Sampling Theory, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-3-319-08801-3__2 23