Existence of frames due to nonhomogeneous Calderón reproducing formula (Q2747442)

Let \(\phi, \psi\in L^2(R^n)\) and \(a>1\). Via Calderón-Zygmund theory it is proved that certain decay conditions on \(\psi\) together with the inequalities NEWLINE\[NEWLINE0<A \leq |\hat{\phi}(\xi)|^2 + \sum_{j=1}^\infty |\hat{\psi}(a^{-j}\xi)|^2 \leq B<\infty\tag{1}NEWLINE\]NEWLINE imply that \(\{\phi(x-kb)\}_{k\in Z^n}\cup \{a^{nj/2}\psi(a^jx-kb)\}_{j\in N, k\in Z^n}\) is a frame for \(L^2(R^n)\) for all \(b>0\) sufficiently close to \(0\). In the special case where NEWLINE\[NEWLINE |\hat{\phi}(\xi)|^2 + \sum_{j=1}^\infty |\hat{\psi}(a^{-j}\xi)|^2 =1NEWLINE\]NEWLINE the obtained frame decomposition converges for all \(f\in L^p, 1<p< \infty\). Furthermore, it is proved that the condition (1) is necessary for \(\{\phi(x-kb)\}_{k\in Z^n}\cup \{a^{nj/2}\psi(a^jx-kb)\}_{j\in N, k\in Z^n}\) to be a frame for \(L^2(R^n)\).