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(ξ)∣2+∣m(ξ+ 12 )∣2=1 for all ξ in R. Let ϕ̂(ξ)=∏∞j=1m(2−jξ), ϕ(x)=∫+∞−∞ϕ̂(ξ)e2πixξdξ. The orthogonality of ϕ(x), i.e., ∫+∞−∞ϕ(x−m)ϕ(x−n) dx=δm,n, 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 [− 16 , 16 ], 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 [− 110 , 110 ] and ∣m( 16 )∣−∣m(− 16 )∣>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 Δ∈(0, 12 ) there exists a finite set ZΔ such that the orthogonality of ϕ(x) is ensured by the requirement that ∣m(ξ)∣>0 for ∣ξ∣≤Δ and for some ξ∈ZΔ. In this paper, we show that this is true if and only if Δ exceeds a strictly positive Δ0 which we derive explicitly.