Derivative Sampling Expansions in Shift-Invariant Spaces With Error Estimates Covering Discontinuous Signals

This paper is concerned with the problem of sampling and interpolation involving derivatives in shift-invariant spaces and the error analysis of the derivative sampling expansions for fundamentally large classes of functions. A new type of polynomials based on derivative samples is introduced, which is different from the Euler-Frobenius polynomials for the multiplicity
$r\gt 1$
. A complete characterization of uniform sampling with derivatives is given using Laurent operators. The rate of approximation of a signal (not necessarily continuous) by the derivative sampling expansions in shift-invariant spaces generated by compactly supported functions is established in terms of
$L^{p}$
- average modulus of smoothness. Finally, several typical examples illustrating the various problems are discussed in detail.