More regular wavelets
If \(\psi\) is a wavelet with Fourier transform \(\widehat{\psi}\), then its Sobolev regularity exponent is \(\sigma=\sup\{s: | \xi| ^s\widehat{\psi}(\xi)\in L^2\}\). Let \(\phi\) be the scaling function in the corresponding multiresolution analysis. The Daubechies orthogonal compactly supported wavelets with support \(2N-1\) have regularity exponent \(\sigma_N\) with \(\lim_{N\to\infty}(\sigma_N/2N)=1/2-\ln 3/4\ln 2\). The proof of this result is revised, and then adapted to show that there is a class of wavelets introduced by Matzinger that have the same support as Daubechies wavelets, but have a better regularity exponent. The ``equality'' sign in the above limit can be replaced by a ``greater than'' sign. Thus the regularity/complexity ratio is improved. The Daubechies wavelets \(\psi_N\) have order of approximation \(N-1\), which is expressed by the fact that the scaling filter \(m_0(\xi)\) has a zero of order \(N-1\) in \(\pi\). In the case of Matzinger wavelets \(\psi_{L,M}\) this becomes a zero of order \(L\) in \(\pi\) and a zero of order \(2M\) in \(2\pi/3\). The Matzinger \(\psi_{L,M}\) should be compared with the Daubechies \(\psi_{L+2M}\) since they have the same filter length.
scientific article; zbMATH DE number 1124819