IRREGULAR SAMPLING THEOREMS AND SERIES EXPANSIONS OF BAND-LIMITED FUNCTIONS

We present a new approach to the problem of irregular sampling of band-limited functions that is based on the approximation and factorization of convolution operators. A special case of our main result is the following theorem: If Ω ⊆ Rn is compact, g ε{lunate} Lα1(Rn) a band-limited function with g ̂(t) ≠ 0 on Ω and (xi)i ε{lunate} I and (yj)j ε{lunate} J are two discrete sets in Rn which are "dense enough," then every band-limited function f ε{lunate} Lwp, 1 ≤ p < ∞, has a representation f = bEj ε{lunate} J cjLyjg as a norm-convergent series, where the coefficients cj are in lwp and can be calculated from the sampled values f(xi) of f alone. This is a far-reaching generalization of the classical Shannon-Whittacker sampling theorem and can be interpreted as an interpolation method of scattered data by band-limited functions. Our methods work (a) for a very general class of function spaces, not only weighted Lwp-spaces, (b) in all dimensions, even in general locally compact abelian groups, (c) to provide the correct behavior of the coefficients in contrast to the traditional estimates currently used by engineers. Furthermore the method is constructive and stable with respect to input errors such as round-off errors, truncation effors, jitter errors, or aliasing errors. © 1992.