Codes and Expansions (CodEx) Seminar

Ole Christensen (Technical University of Denmark):
Frames and redundancy

Redundancy is one of the key issues in frame theory. In this talk we will consider redundancy properties for two recent classes of frames. First, we consider the so-called Carleson frames constructed by Aldroubi et al. in 2016; they have the form \(\{T^k \varphi \}_{k=0}^\infty \) for a bounded linear operator on the underlying Hilbert space. We show that a subclass of these frames has a number of additional remarkable features that have not been identified for any other frames in the literature. Most importantly, the subfamily obtained by selecting each \(N\)th element from the frame is itself a frame, regardless of the choice of \(N\in \mathbb{N} .\) Furthermore, the frame property is kept upon removal of an arbitrarily finite number of elements. The second class of frames to be considered is based on a method that allows to construct an overcomplete frame for Hilbert spaces of the form \(L^2(-r,r)\) or \(L^2(0,r),\) starting with a Riesz basis for the same space. When applied to standard orthogonal polynomials the construction yields polynomial frames with attractive features: the frames are linearly independent, have infinite excess, the frame decomposition is simple, and the functions in the frame are “very close” to the functions in the given orthogonal system.

The first part of the talk is joint work with M. Hasannasab, F. M. Philipp, and D. Stoeva; the second part is joint work with H. O. Kim and R. Y. Kim.