• 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 $\mathcal{H}$.

• 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.

• Kyungpook Mathematical Journal
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• v.61 no.2
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• pp.371-381
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• 2021

In this paper, we attempt to obtain sufficient conditions for the existence of tight wavelet frame sets in locally compact abelian groups. The condition is generated by modulating a collection of characteristic functions that correspond to a generalized shift-invariant system via the Fourier transform. We present two approaches (for stationary and non-stationary wavelets) to construct the scaling function for L²(G) and, using the scaling function, we construct an orthonormal wavelet basis for L²(G). We propose an open problem related to the extension principle for Riesz wavelets in locally compact abelian groups.

• East Asian mathematical journal
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• v.25 no.3
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• pp.229-245
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• 2009

G. Cantor gave a deep influence to the society of mathematics in many ways, especially in the set theory. It is important for gifted and talented high school students in mathematics to understand the Euler constant and the fractal dimension of the Cantor set in a heuristic sense. On the historic basis of mathematics and the standard of high school students, we give the teaching method for the talented high school student to understand them better. Further we introduce the Riesz-N$\acute{a}$gy-Tak$\acute{a}$cs distribution and its first moment. We hope that from these topics, the gifted and talented students in mathematics will have insight in the analysis of mathematics.