An RKHS analysis of sampling theorems for harmonic-limited signals

A T-periodic signal x(t) is said to be K-harmonic limited if, for some integer K > 0, its complex Fourier coefficients
\{c_{n}\}
satisfy
c_{n} = 0
for |n| > K. Such signals may be completely reconstructed from a finite number of uniformly spaced signal samples taken over a period, a property which facilitates the representation of two-dimensional (polar form) signals used in computerized tomography. By employing a reproducing kernel Hilbert space (RKHS) setting, a generalized theorem for sampling harmonic-limited signals is derived. All the special representations which have appeared in the literature with separate proofs, including three recent versions, [1]-[3], are shown to be special cases of the generalized reconstruction formula. Connections with Fourier series theory and Kramer's generalized sampling theorem are also discussed.