On the construction of frames for spaces of distributions (Q837076)

The present paper aims to construct frames with prescribed properties for different spaces of distributions. More precisely, given a separable Hilbert space \(H\) of functions, a locally convex space (of test funcions) \({\mathcal S}\) with \({\mathcal S}\subset H \subset {\mathcal S} ^\prime\) and a quasi-Banach space of distributions \(L\subset {\mathcal S} ^\prime\), Section 2 presents a general method, based on a perturbation argument, for the construction of frames in the quasi-Banach space \(L\) with some prescribed features. The perturbation argument is related to the method of \textit{O. Christensen} and \textit{C. Heil} [Math. Nachr. 185, 33--47 (1997; Zbl 0868.42013)] for the construction of atomic decompositions. The obtained results are applied in Section 3 to obtain frames for Triebel-Lizorkin and Besov spaces on the sphere of \({\mathbb R}^{n+1}\) with the property that the frame elements are supported on small shrinking caps.