Nagoya Mathematical Journal

Annihilator, completeness and convergence of wavelet system

Kwok-Pun Ho

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Abstract

We show that if $\{ \varphi \}_{Q \in \mathcal{Q}} \in \bigcap \mathcal{M}_{\alpha}(\mathbb{R}^{n})$ is a frame and $\{ \psi_{Q} \}_{Q \in \mathcal{Q}} \in \bigcap \mathcal{M}_{\alpha}(\mathbb{R}^{n})$ is its dual frame (for the definition of $\mathcal{M}_{\alpha}(\mathbb{R}^{n})$, see Definition 2.1), where $\mathcal{Q}$ is the collection of dyadic cubes, then for any $f \in \mathcal{S}'(\mathbb{R}^{n})$, there exists a sequence of polynomials, $P_{L, L', L''}$, such that

\lim_{L, L', L'' \to \infty} \biggl\{ \sum_{-L' \le i \le L} \sum_{|k| \le \delta(i)2^{L''}} \langle f, \psi_{Q_{i, k}} \rangle \varphi_{Q_{i, k}}-P_{L, L', L''} \biggr\} = f

in the topology of $\mathcal{S}'(\mathbb{R}^{n})$, where $\delta(i) = \max(2^{i}, 1)$. We prove this result by explicitly constructing the polynomials $P_{L, L', L''}$. Furthermore, using the above result, we assert that the linear span of the one-dimensional wavelet system is dense in a function space if and only if the dual space of this function space has an trivial intersection with the set of polynomials. This is proved by using the annihilator of the one-dimensional wavelet system.

Article information

Source
Nagoya Math. J., Volume 188 (2007), 59-105.

Dates
First available in Project Euclid: 17 December 2007

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1197908744

Mathematical Reviews number (MathSciNet)
MR2371769

Zentralblatt MATH identifier
1137.42010

Subjects
Primary: 42B35: Function spaces arising in harmonic analysis 42C15: General harmonic expansions, frames 42C40: Wavelets and other special systems
Secondary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 47B38: Operators on function spaces (general)

Keywords
Frames Wavelets Annihilator and Schwartz distribution

Citation

Ho, Kwok-Pun. Annihilator, completeness and convergence of wavelet system. Nagoya Math. J. 188 (2007), 59--105. https://projecteuclid.org/euclid.nmj/1197908744


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