• Kyungpook Mathematical Journal
• /
• v.54 no.1
• /
• pp.87-94
• /
• 2014

In this paper, we introduce ultra g-Bessel sequences and study some properties of this kind of sequences in Hilbert spaces. We also show that every g-frame for a finite dimensional Hilbert space is an ultra g-Bessel sequence.

• Honam Mathematical Journal
• /
• v.27 no.2
• /
• pp.233-241
• /
• 2005

A sequence $\{f_i\}^{\infty}_{i=1}$ in a Hilbert space H is said to be exponentially localized with respect to a Riesz basis $\{g_i\}^{\infty}_{i=1}$ for H if there exist positive constants r < 1 and C such that for all i, $j{\in}N$, ${\mid}{\mid}{\leq}Cr^{{\mid}i-j{\mid}}$ and ${\mid}{\mid}{\leq}Cr^{{\mid}i-j{\mid}}$ where $\{{\tilde{g}}_i\}^{\infty}_{i=1}$ is the dual basis of $\{g_i\}^{\infty}_{i=1}$. It can be shown that such sequence is always a Bessel sequence. We present an additional condition which guarantees that $\{f_i\}^{\infty}_{i=1}$ is a frame for H.

• Journal of the Korean Mathematical Society
• /
• v.53 no.6
• /
• pp.1331-1345
• /
• 2016

A K-g-frame is a generalization of a g-frame. It can be used to reconstruct elements from the range of a bounded linear operator K in Hilbert spaces. K-g-frames have a certain advantage compared with g-frames in practical applications. In this paper, the interchangeability of two g-Bessel sequences with respect to a K-g-frame, which is different from a g-frame, is discussed. Several construction methods of K-g-frames are also proposed. Finally, by means of the methods and techniques in frame theory, several results of the stability of K-g-frames are obtained.

• Journal of the Korean Mathematical Society
• /
• v.48 no.5
• /
• pp.899-912
• /
• 2011

We introduce ${\varphi}$-frames in $L^2$(G), as a generalization of a-frames defined in [8], where G is a locally compact Abelian group and ${\varphi}$ is a topological automorphism on G. We give a characterization of ${\varphi}$-frames with regard to usual frames in $L^2$(G) and show that ${\varphi}$-frames share several useful properties with frames. We define the associated ${\varphi}$-analysis and ${\varphi}$-preframe operators, with which we obtain criteria for a sequence to be a ${\varphi}$-frame or a ${\varphi}$-Bessel sequence. We also define ${\varphi}$-Riesz bases in $L^2$(G) and establish equivalent conditions for a sequence in $L^2$(G) to be a ${\varphi}$-Riesz basis.

• Kyungpook Mathematical Journal
• /
• v.53 no.1
• /
• pp.135-141
• /
• 2013

In this paper we show that a $g$-frame for a Hilbert space $\mathcal{H}$ can be written as a linear combination of two $g$-orthonormal bases for $\mathcal{H}$ if and only if it is a $g$-Riesz basis for $\mathcal{H}$. Also, we show that every $g$-frame for a Hilbert space $\mathcal{H}$ is a multiple of a sum of three $g$-orthonormal bases for $\mathcal{H}$.