Some equalities and inequalities of g-continuous frames

Abstract

In this paper, we derive some equalities and inequalities about g-continuous frames, which are an extension to g-frames and continuous frames We also discuss the stability of the perturbation of a g-continuous frameNational Natural Science Foundation of China [10571145]; Natural Science Foundation of Fujian Province of China [2010J01012]; Science and Technology Foundation of Xiamen City of China [20083012

Continuous g ‐frame and g ‐Riesz sequences in Hilbert spaces

“…due to  ∈ range( ) and ∈ range( ) = (ker( * )) ⟂ by (24). This yields that range( ) is closed by (24) and (26). The proof is completed.…”

“…Based on this, the theory of continuous g-frames has interested many mathematicians. Xiao and Zeng 26 derived some equalities and inequalities of continuous g-frames and discussed their stabilities. Mehdi and Akbar 27 presented some properties for continuous g-Bessel sequences and studied the stabilities of continuous g-frames and their dual frames.…”

Some New Inequalities for Dual Continuous g-Frames

“…The research of inequalities and equalities relevant to Parseval frames was done by Balan, Casazza et al in [14]. It has interested some engineers and mathematicians (see [15][16][17][18][19][20][21][22][23][24][25][26]). In particular, Balan et al and Gȃvruţa obtained the following two inequalities: Proposition 1 ([14]).…”

“…In 2010, Xiao and Zeng in [18] generalized Proposition 1 to continuous generalized frames (see [18], Theorem 3.2). In 2018, Zhang and Li in [17] generalized Proposition 2 to continuous generalized frames (see [17], Theorem 3.4), and gave some bi-directional inequalities for continuous generalized frames with more general forms.…”

TightK-g-Frame and Its Novel Characterizations via Atomic Systems

“…Today, frame theory has been widely used in filter theory [3], image processing [4], numerical analysis, and other areas. We refer to [5][6][7][8] for an introduction to frame theory in Hilbert space and its application.…”