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

✍ Scribed by Xingwei Zhou; Weifeng Su

Publisher: Elsevier Science
Year: 2000
Tongue: English
Weight: 80 KB
Volume: 9
Category: Article
ISSN: 1063-5203
DOI: 10.1006/acha.2000.0309

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✦ Synopsis

Suppose that m(ξ ) is a trigonometric polynomial with period 1 satisfying m(0) = 1 and |m(ξ

, is related to the zeros of m(ξ ). In 1995, A. Cohen and R. D. Ryan, "Wavelets and Multiscale Signal Processing," Chapman & Hall, proved that if m(ξ ) has no zeros in [-1 6 , 1 6 ], then ϕ(x) is an orthogonal function. In (X. Zhou and W. Su, Appl. Comput. Harmon. Anal. 8, 197-202 (2000)) we proved that if m(ξ ) has no zeros in [-1 10 , 1 10 ] and |m( 1 6 )| -|m(-1 6 )| > 0, then ϕ(x) 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.

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