Variations of Cohen’s theorem

Abstract

A variation of Cohen’s condition on a smooth low-pass filter m0, (Ka)There exists a compact set K congruent to Kamodulo 2πttfor which ¦m0(2−kw)¦ ≥A > 0for any w ∈K and any k ∈ ℕ,where Ka= [a- π, −2π/3] ∪[ 2a, 2a] ∪[2π/3, π−a] with π/5 ≤a ≤ π/3, is also shown to be necessary and sufficient in order that the integer translates of the scaling function given by ϕ(w)= Πk=1∞m0(2−kw) form an orthonormal family. The setKa is a proper subset of [−π,π] which reduces to [−2π/3, 2π/3] whena = π/3 and to [−4π/5, −2π/3]∪[−2π/5, 27π/5]∪[2π/3,4π/5] of the smallest measure 16π/15 whena = π/5.