On the Preservation of Stability under Convolutions

AbstractThe 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 φ be in the class Λ0 of all stable functions ψ such that $$\widehat\psi \left( 0 \right) \ne 0{\text{ and }}\widehat\psi$$ as well as $$E_\psi : = \sum {\left| {\widehat\psi \left( {w + 2{\pi }k} \right)} \right|} ^2$$ is continuous. Then Λ0 is preserved under the convolution by φ if and only if the zero set $$Z\left( {\widehat\varphi } \right)$$ is contained in 2πZ\{0}. The condition can be transformed into the zero set of the inducing mask trigonometric polynomial in the class Λ# of compactly supported refinable functions in Λ0. For example, our result shows that such φ must have its mask of the form $$m_\varphi \left( w \right) = \left( {\frac{{1 + {\text{e}}^{{\text{ - i2}}w} }}{2}} \right)^N \left( {\frac{{1 + {\text{e}}^{{\text{ - i}}w} + {\text{e}}^{{\text{ - i2}}w} }}{3}} \right)^M Q\left( w \right),$$ where integers N≥1 and M≥0, and Q(w) has no real zeros.