• 제목/요약/키워드: Riesz frame

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THE STABILITY OF HILBERT SPACE FRAMELETS AND RIESZ FRAMES

  • LEE, JEONG-GON;LEE, DONG-MYUNG
    • 호남수학학술지
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    • 제27권4호
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    • pp.621-629
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    • 2005
  • We consider the stability of Hilbert space framelets and related Riesz frames. Our results are in spirit close to classical results for orthonormal bases, due to Mazur and Schauder.

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GENERATING NEW FRAMES IN $L^2(\mathbb{R})$ BY CONVOLUTIONS

  • Kwon, Kil-Hyun;Lee, Dae-Gwan;Yoon, Gang-Joon
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • 제15권4호
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    • pp.319-328
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    • 2011
  • Let $\mathbf{c}=\{c_n\}_{n{\in}\mathbb{Z}}\in{\ell}^1(\mathbb{Z})$ and $\{f_n\}_{n{\in}\mathbb{Z}}$ be a frame (Riesz basis, respectively) of $L^2(\mathbb{R})$. We obtain necessary and sufficient conditions of $\mathbf{c}$ under which $\{\mathbf{c}{\ast}_{\lambda}f_n\}_{n{\in}\mathbb{Z}}$ becomes a frame (Riesz basis, respectively) of $L^2(\mathbb{R})$, where ${\lambda}$ > 0 and $(\mathbf{c}{\ast}_{\lambda}f)(t)\;:=\;{\sum}_{n{\in}\mathbb{Z}}c_nf(t-n{\lambda})$. When $\{\mathbf{c}{\ast}_{\lambda}f_n\}_{n{\in}\mathbb{Z}}$ becomes a frame of $L^2(\mathbb{R})$, we present its frame operator and the canonical dual frame in a simple form. Some interesting examples are included.

PERTURBATION OF WAVELET FRAMES AND RIESZ BASES I

  • Lee, Jin;Ha, Young-Hwa
    • 대한수학회논문집
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    • 제19권1호
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    • pp.119-127
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    • 2004
  • Suppose that $\psi{\;}\in{\;}L^2(\mathbb{R})$ generates a wavelet frame (resp. Riesz basis) with bounds A and B. If $\phi{\;}\in{\;}L^2(\mathbb{R})$ satisfies $$\mid$\^{\psi}(\xi)\;\^{\phi}(\xi)$\mid${\;}<{\;}{\lambda}\frac{$\mid$\xi$\mid$^{\alpha}}{(1+$\mid$\xi$\mid$)^{\gamma}}$ for some positive constants $\alpha,{\;}\gamma,{\;}\lambda$ such that $1{\;}<1{\;}+{\;}\alpha{\;}<{\;}\gamma{\;}and{\;}{\lambda}^2M{\;}<{\;}A$, then $\phi$ also generates a wavelet frame (resp. Riesz basis) with bounds $A(1{\;}-{\;}{\lambda}\sqrt{M/A})^2{\;}and{\;}B(1{\;}+{\;}{\lambda}\sqrt{M/A})^2$, where M is a constant depending only on $\alpha,{\;}\gamma$ the dilation step a, and the translation step b.

Riesz and Tight Wavelet Frame Sets in Locally Compact Abelian Groups

  • Sinha, Arvind Kumar;Sahoo, Radhakrushna
    • Kyungpook Mathematical Journal
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    • 제61권2호
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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.

LOCALIZATION PROPERTY AND FRAMES II

  • HA YOUNG-HWA;RYU JU-YEON
    • 대한수학회논문집
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    • 제21권1호
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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.

φ-FRAMES AND φ-RIESZ BASES ON LOCALLY COMPACT ABELIAN GROUPS

  • Gol, Rajab Ali Kamyabi;Tousi, Reihaneh Raisi
    • 대한수학회지
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    • 제48권5호
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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.

G-frames as Sums of Some g-orthonormal Bases

  • Abdollahpour, Mohammad Reza;Najati, Abbas
    • Kyungpook Mathematical Journal
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    • 제53권1호
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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 $\mathcal{H}$.

FRAMES BY INTEGER TRANSLATIONS

  • Kim, J.M.;Kwon, K.H.
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • 제11권3호
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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.

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LOCALIZATION PROPERTY AND FRAMES

  • HA, YOUNG-HWA;RYU, JU-YEON
    • 호남수학학술지
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    • 제27권2호
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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.

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