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 a modulo 21r for which mo(2 -c w) > A > 0 for any w E K and any k E N, where Ka = [a -7r, -2ir/3] U[-2a, 2a] U[27r/3, Ir -a] with 7r/5 < a < 7r/3, is also shown to be necessary and sufficient in order that the integer translates of the scaling function ' given by O(w) = {J' 1 mo(2 -k w) form an orthonormal family. The set Ka is a proper subset of [-7r, 7r] which reduces to [-21r/3, 21r/3] when a = ir/3 and to [-47r/5, -27r/3] U[-27r/5, 2ir/5] U[27r/3, 4ir/5] of the smallest measure 16ir/15 when a = 7r/5.