A general Hilbert space approach to framelets (Q949068)

Given a Hilbert space \(\mathcal{H}\) of real-valued functions defined on a domain \(\Sigma\subset \mathbb{R}^d\), and an orthonormal basis \(\{U_n\}\) for \(\mathcal{H}\) one can define the convolution of \(F\in \mathcal{H}\) with a kernel \(\Gamma: \Sigma\times \Sigma \to \mathbb{R}\) as \[ \Gamma\ast F (x) = \sum \widehat{\Gamma}(n) \widehat{F}(n) U_n (x), \] where \(\widehat{F}(n)=\langle F, U_n\rangle\) with respect to the inner product on \(\mathcal{H}\). The author considers special \(\mathcal{H}\)-admissible kernel families \(\Phi_J\) having the properties (i) \(\lim_{J\to\infty} (\widehat{\Phi_J}(n))^2=1\) for each \(n=1,2,\dots\) and (ii) \((\widehat{\Phi_J}(n))^2\) is increasing in \(J\) for each \(n\). Under these hypothesis it is shown that \(\{\Phi_J\}\) defines an approximate identity in the sense that \[ \lim_{J\to\infty} \|(\Phi_J\ast F)\ast \Phi_J -F\|_{\mathcal{H}} =0 \] for any \(F\in \mathcal{H}\). Here one thinks of \(J\) as a {\textit{scale}}. Then \(\{\Phi_J\}\) can be regarded as a {\textit{scaling family}}. One then factors \[ (\widehat{\Phi_{J+1}}(n))^2 -(\widehat{\Phi_{J}}(n))^2 \equiv \widehat{\Psi_J}(n) \widehat{\tilde{\Psi}_J}(n) \] for each \(J=1,2,\dots\) (and defining \(\widehat{\Psi_{0}}=\widehat{\widetilde{\Psi_{0}}}\)). One considers the kernel \(\Psi_J\) as defining a \(J\)-scale wavelet transform via \(F\mapsto \text{WT} (F)(x,J) =\langle \Psi_J(x,\cdot),F\rangle\) and \(\widetilde{\Psi}_J\) as defining a corresponding dual \(J\) scale wavelet transform by the symbol \((\widetilde{\Psi}_J)^\wedge (n) =(\Psi_j)^{\wedge}(n)/\sum_J ((\Psi_j)^{\wedge}(n)^2\). The main purpose of this work is to extend this generalized multiresolution analysis to frames. In this context, a family \(\Gamma_J\) of \(\mathcal{H}\)-admissible kernels is called an \(\mathcal{H}\)-frame if there are constants \(A,B\) such that \[ A\|f\|^2 \leq \|\sum\Gamma_J\ast F\|\leq B\|f\|^2 \] for all \(F\in \mathcal{H}\). This frame condition holds if and only if the kernel symbols satisfy \[ A\leq \sum_J \widehat{\Gamma_J}(n)^2\leq B \] uniformly in \(n\). The main result provides conditions in terms of partial sums defining the wavelet symbols in order that certain primal and dual \(\mathcal{H}\)-framelet identities hold. A specific example is provided for \(L^2(\Omega)\) where \(\Omega \) is the unit sphere in \(\mathbb{R}^3\). Framelets are built in terms of localized spherical harmonics and used to analyze an Earth gravity model.