All Routes to Sampling Expansions L

All Routes to Sampling Expansions Lead to Reproducing Kernels。The classical proof of the Shannon sampling theorem is based on the inverse of the Fourier integral and complex Fourier series. Suppose that the signal F defined on the real line is square integrable and bandlimitedsay to Fz. Then F can be expressed as the inverse of the Fourier transform FA, where now the integration is over Fz. We extend F periodically to the real line and expand the resulting periodic function by complex Fourier series. Finally we interchange the integration and summation, and the sampling theorem pops up. While this proof is very simple, it is not very revealing in two respects:(i) One does not get any insight from the interchange of integration and summation; in particular, one does not hear or feel the heart beat of the proof.