Discretization of linear operators on \(L^p(\mathbb{R}^N)\)
We study conditions on a pair of functions \(\varphi\) and \(\phi\) in \(L^2(\mathbb{R}^N)\) so that the following condition holds: Any linear operator \(T\) is bounded on \(L^p(\mathbb{R}^N)\) if and only if the operators \(\widetilde T_k\) associated to the matrices \[ (\langle 2^{kN/p}T(\varphi(2^k\cdot- n)), 2^{kN/p'}\phi(2^k\cdot- m)\rangle)_{n,m}, \] are bounded on \(\ell^p(\mathbb{Z}^N)\) uniformly in \(k\in\mathbb{Z}\). We work under the restriction \(\text{supp }\widehat\varphi\) and \(\text{supp }\widehat\phi\subset (-1,1)^N\) and \(1< p<\infty\). We will also study some connections with multiresolution analysis on \(L^p(\mathbb{R}^N)\).
scientific article; zbMATH DE number 1190783