• Honam Mathematical Journal
• /
• v.27 no.4
• /
• pp.621-629
• /
• 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.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.15 no.4
• /
• pp.319-328
• /
• 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.

• Communications of the Korean Mathematical Society
• /
• v.19 no.1
• /
• pp.119-127
• /
• 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.

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

• Communications of the Korean Mathematical Society
• /
• v.21 no.1
• /
• pp.101-115
• /
• 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 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 $\mathcal{H}$.

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

• 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.

• Bulletin of the Korean Mathematical Society
• /
• v.50 no.6
• /
• pp.1841-1846
• /
• 2013

In this paper we establish some new results on sums of Hilbert space frames and Riesz bases. We also provide a correction to some recently established results in [2].