Low-Pass Filters, Martingales, and Multiresolution Analyses

Abstract

Necessary and sufficient conditions for a trigonometric polynomial to be a low-pass filter have been given by A. Cohen (Ann. Inst. H. Poincaré Anal. Nonlinéaire7 (1990), 439-459) and W. Lawton (J. Math. Phys.31 (1990), 1898-1901). We give necessary and sufficient conditions for an arbitrary periodic function to be a low-pass filter, following the approach of Lawton. The technique also applies to functions that give rise to a Riesz basis. © 2000 Academic Press.