Zeros of a Mask Function and the Orthogonality of the Related Scaling Function

Abstract

Suppose that m(ξ ) is a trigonometric polynomial with period 1 satisfying m(0) = 1 and |m is also an orthogonal function. A natural question, then, is whether this procedure can be extended to arbitrarily small intervals, i.e., whether for any ∈ (0, 1 2 ) there exists a finite set Z such that the orthogonality of ϕ(x) is ensured by the requirement that |m(ξ )| > 0 for |ξ | and for some ξ ∈ Z . In this paper, we show that this is true if and only if exceeds a strictly positive 0 which we derive explicitly.

Variations of Cohen’s theorem

“…Suppose that mo is a trigonometric polynomial with mo(0) = 1 satisfying Imo(w)1 2 + Imo(w + ir)j 2 = 1. Then {O(• -k) : k E Z} is an orthonormal family if and only if there exists a compact set K congruent to [-4IrAo, 4rrAo] U{±47ra n : n = 1, 2, • • • } such that infk>l inf,K Imo(2 -k w)l > 0, where an \ Ao = 0.0875... is a sequence constructed from CD A (j > 1) in [9].…”

“…For the case of trigonometric polynomial mo, the optimum size is provided by Zhou and Su in[9] using r-cycle method. Their result can be stated as follows:…”

“…We presented the previous version of this paper at the conference on International Conference on Wavelet Analysis and Its Applications (Guangzhou, November 1999) and had discussions with Professor R. Jia and X. Zhou who informed us of the papers [7,8,9]. We wish to express our sincere thanks to them for the valuable discussions we had with them.…”