Variations of Cohen's Theorem

Abstract

A variation of Cohen's condition on a smooth low-pass filter mo, (Ka) There exists a compact set K congruent to K. modulo 2 pi for which \m(0)(2(-k)w )\ greater than or equal to A > 0 for any w is an element of K and any k is an element of N, where K-a = [a - pi, -2 pi /3] boolean OR[-2a, 2a] boolean OR [2 pi /3, pi - a] with pi /5 less than or equal to a less than or equal to pi /3, is also shown to be necessary and sufficient in order that the integer translates of the scaling function phi given by phi (w) = Pi (infinity)(k=1) m(0)(2(-k)w) form an orthonormal family. The set K. is a proper subset of [-pi, pi] which reduces to [-2 pi /3, 2 pi /3] when a = pi /3 and to [-4 pi /5,-2 pi /3]boolean OR[-2 pi /5, 2 pi /5]boolean OR [2 pi /3, 4 pi /5] of the smallest measure 16 pi /15 when a = pi /5.