Sampling and Reconstruction of Bandlimited BMO-Functions

Functions of bounded mean oscillation (BMO) play an important role in complex function theory and harmonic analysis. In this paper a sampling theorem for bandlimited BMO-functions is derived for sampling points that are the zero sequence of some sine-type function. The class of sinetype functions is large and, in particular, contains the sine function, which corresponds to the special case of equidistant sampling. It is shown that the sampling series is locally uniformly convergent if oversampling is used. Without oversampling, the local approximation error is bounded. I. NOTATION Let f̂ denote the Fourier transform of a function f . L(R), 1 ≤ p <∞, is the space of all pth-power Lebesgue integrable functions on R, with the usual norm ‖ · ‖p, and L∞(R) is the space of all functions for which the essential supremum norm ‖ · ‖∞ is finite. For 0 < σ <∞ let Bσ be the set of all entire functions f with the property that for all > 0 there exists a constant C( ) with |f(z)| ≤ C( ) exp((σ + )|z|) for all z ∈ C. The Bernstein space B σ , 1 ≤ p ≤ ∞, consists of all functions in Bσ , whose restriction to the real line is in L(R). A function in B σ is called bandlimited to σ. I