Frames, their relatives and reproducing kernel Hilbert spaces

Michael Speckbacher, Peter Balazs

Published 2017-04-10Version 1

This paper is devoted to three aspects of the relation between reproducing kernel Hilbert spaces (RKHS) and stable analysis/synthesis processes: First, we analyze the structure of the reproducing kernel of a RKHS using frames and reproducing pairs. Second, we give a new proof for the result that finite redundancy of a continuous frame implies atomic structure of the underlying measure. Our proof relies on the RKHS structure of the range of the analysis operator. With this result we show that every continuous Riesz basis can in fact be identified with a discrete Riesz basis, making the continuous concept therefore superfluos. Finally, we show how the range of the analysis operators of a reproducing pair can be equipped with a RKHS structure.