Frames and Semi-Frames

TL;DR

This paper introduces the concept of semi-frames in Hilbert spaces, focusing on upper semi-frames, and explores their properties, duality, and conditions under which reconstruction remains feasible.

Contribution

It defines semi-frames, analyzes their properties, especially upper semi-frames, and discusses reconstruction methods and duality in these generalized frame settings.

Findings

Upper semi-frames have bounded frame operators with unbounded inverses.

Reconstruction is possible under certain conditions for semi-frames.

The paper extends frame theory to semi-frames, broadening applicability.

Abstract

Loosely speaking, a semi-frame is a generalized frame for which one of the frame bounds is absent. More precisely, given a total sequence in a Hilbert space, we speak of an upper (resp. lower) semi-frame if only the upper (resp. lower) frame bound is valid. Equivalently, for an upper semi-frame, the frame operator is bounded, but has an unbounded inverse, whereas a lower semi-frame has an unbounded frame operator, with bounded inverse. We study mostly upper semi-frames, both in the continuous case and in the discrete case, and give some remarks for the dual situation. In particular, we show that reconstruction is still possible in certain cases.