Biframes and some of their properties

TL;DR

This paper introduces the concept of biframes in Hilbert spaces, explores their properties, classifications, and a new class called b-Riesz bases, extending the theory of frames and their generalizations.

Contribution

It defines biframes, investigates their properties, classifies them, and introduces b-Riesz bases, expanding the framework of frame theory in Hilbert spaces.

Findings

Biframes are a generalization of pair frames and controlled frames.

B-Riesz bases form a proper subset of Riesz bases.

Biframes with one sequence as an orthonormal basis are studied.

Abstract

Recently, frame multipliers, pair frames, and controlled frames have been investigated to improve the numerical efficiency of iterative algorithms for inverting the frame operator and other applications of frames. In this paper, the concept of biframe is introduced for a Hilbert space. A biframe is a pair of sequences in a Hilbert space that applies to an inequality similar to frame inequality. Also, it can be regarded as a generalization of controlled frames and a special kind of pair frames. The basic properties of biframes are investigated based on the biframe operator. Then, biframes are classified based on the type of their constituent sequences. In particular, biframes for which one of the constituent sequences is an orthonormal basis $\{e_k\}_{k=1}^\infty$ are studied. Then, a new class of Riesz bases denoted by $[\{e_k\}]$, is introduced and is called b-Riesz bases. An interesting result is also proved, showing that the set of all b-Riesz bases is a proper subset of the set of all Riesz bases. More precisely, b-Riesz bases induce an equivalence relation on $[\{e_k\}]$.