Discretization of continuous frame (Q692360)

Continuous frames for the subspaces of a Hilbert space \(H\) are considered. The concept of unconditional continuous atomic resolution of the identity \(\left\{\left\{T_{x} \right\}_{x\in X},\, \omega \left(x\right)\right\}\), \(\left\{T_{x} \right\}_{x\in X} \subset L\left(H\right)\), with respect to a measurable mapping \(\omega :X\to \left(0,\infty \right)\) defined on a locally compact Hausdorff space \(X\) with positive Radon or discontinuous measure, is introduced. Set \(\overline{T_{x} \left(H\right)}=W_x\). Conditions that make \(\left\{\left\{W_{x} \right\}_{x\in X} ,\, \omega \right\}\) a continuous frame for subspaces of \(H\) are obtained. For the continuous Bessel frame of subspaces in \(H\), the conditions under which \(\left\{T_{x} \right\}_{x\in X} \) satisfies the frame inequality are found. Using the continuous unconditional atomic resolution of the identity \(\left\{\left\{T_{x} \right\}_{x\in X},\, \omega(x) \right\}\) in \(H\), for a given frame \(\left\{f_{i} \right\}\), the authors find a countable subset of indices \(J\subset X\) such that the sequence \(\left\{\omega \left(j\right)\sqrt{\mu \left\{j\right\}} T_{j}^{*} \left(f_{i} \right)\right\}\), \(j\in J\), generates a representation with a frame operator \(S\). The rest of the paper is dedicated to the study of the stability of the unconditional atomic resolution of the identity in \(H\). The concept of \(\left(\lambda _{1} ,\lambda _{2} ,\varphi \right)\)-perturbation is introduced, where \(\lambda _{1} ,\lambda _{2} \in \left[0,1\right)\) are some numbers, and \(\varphi :X\to \left[0,\infty \right)\) is a measurable mapping. Let \(\left\{T_{x} \right\}_{x\in X} \) be an unconditional resolution of the identity in \(H\) and \(\left\{\left\{S_{x} \right\}_{x\in X} ,\, \omega(x) \right\}\) be a \(\left(\lambda _{1} ,\lambda _{2} ,\varphi \right)\)-perturbation of it with \(\varphi \left(x\right)\neq 0\) almost everywhere. Then conditions under which \(\left\{S_{x} S^{-1}, \omega(x) \right\}_{x\in X} \) is an unconditional resolution of the identity in \(H\) are found.