On the preservation of stability under convolutions

Abstract

The preservation of stability under the convolution is shown to be related with the zero set of the Fourier transform of inducing stable function. For example, let phi be in the class Lambda(0) of all stable functions psi such that (ψ) over cap( 0) not equal 0 and (ψ) over cap as well as E-psi := Sigma\(ψ) over cap (w + 2pik)\(2) is continuous. Then Lambda(0) is preserved under the convolution by phi if and only if the zero set Z((φ) over cap) is contained in 2piZ\{0}. The condition can be transformed into the zero set of the inducing mask trigonometric polynomial in the class Lambda(#) of compactly supported refinable functions in Lambda(0). For example, our result shows that such phi must have its mask of the form m(phi)(w) = (1 + e(-iw)/2)(N)(1+e(-iw) + e(-i2w)/3)(M)Q(w), where integers N greater than or equal to 1 and M greater than or equal to 0, and Q(w) has no real zeros.