Periodic wavelet frames
The wavelet system associated to a function \(\psi\in L^2(\mathbb{R})\) consists of the functions \(\psi_{mn,}(x)=2^{m/2}\psi(2^mx-n), \;m,n\in Z\). Assume that the Fourier transform \(\hat{\psi}\) is twice differentiable, has support in \([-\pi, -\epsilon]\cup [\epsilon, \pi]\) for some \(\epsilon>0\), and satisfies that \(D(\gamma):=\sum_{m\in \mathbb{Z}} | \hat{\psi} (\gamma +m)| ^2>0\) for a.e. \(\gamma\). Define \(\tilde{\psi}\) by \(\hat{\tilde{\psi}}(\gamma)= \frac{\hat{\psi}(\gamma)}{D(\gamma)}\). Defining the periodization of a function \(f\) by \(f^{per}(t)=\sum_{k\in Z} f(t+k)\), it is proved that the functions in \(\Psi^{per}:= \{1, \psi_{m,n}^{per}, m\geq 0, n=0,1,\dots,2^m-1 \}\) and the corresponding functions in \(\tilde{\Psi}^{per}\) constitute dual frames for \(L^2(0,1)\). The optimal frame bounds for \(\Psi\) and \(\tilde{\Psi}\) are expressed in terms of the infimum and the supremum of the function \(D\).
scientific article; zbMATH DE number 2140538