• Kyungpook Mathematical Journal
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• v.54 no.1
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• pp.87-94
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• 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
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• v.27 no.2
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• pp.233-241
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• 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.

• Communications of the Korean Mathematical Society
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• v.21 no.1
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• pp.101-115
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• 2006

Localization of sequences with respect to Riesz bases for Hilbert spaces are comparable with perturbation of Riesz bases or frames. Grochenig first introduced the notion of localization. We introduce more general definition of localization and show that exponentially localized sequences and polynomially localized sequences with respect to Riesz bases are Bessel sequences. Furthermore, they are frames provided some additional conditions are satisfied.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.11 no.3
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• pp.1-5
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• 2007

We give an elementary proof of a necessary and sufficient condition for integer translates {${\phi}(t-{\alpha})\;:\;{\alpha}{\in}{\mathbb{Z}}^d$} of ${\phi}$(t) in $L^2({\mathbb{R}}^d)$ to be a frame sequence.

• Journal of the Korean Mathematical Society
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• v.53 no.6
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• pp.1331-1345
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• 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
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• v.48 no.5
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• pp.899-912
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• 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
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• v.53 no.1
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• pp.135-141
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• 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 $\mathcal{H}$.