A note on constructing affine systems for L 2

H. Q. Bui, N. Kaiblinger, Richard S Laugesen

Research output: Contribution to journalLetter

Abstract

Assume ψ ∈ L 2 (R d ) has Fourier transform continuous at the origin, with over(ψ, ̂) (0) = 1, and that ∑ l ∈ Zd | over(ψ, ̂) (ξ - l) | 2 is bounded as a function of ξ ∈ R d . Then every function f ∈ L 2 (R d ) can be represented by an affine series f = ∑ j > 0k ∈ Zd c j, k ψ j, k for some coefficients satisfying{norm of matrix} c {norm of matrix} ℓ1 (ℓ2) = under(∑, j > 0) (under(∑, k ∈ Z d ) | c j, k | 2 ) 1 / 2 < ∞ . Here ψ j, k (x) = | det a j | 1 / 2 ψ (a j x - k) and the dilation matrices a j expand, for example a j = 2 j I. The result improves an observation by Daubechies that the linear combinations of the ψ j, k are dense in L 2 (R d ).

Original languageEnglish (US)
Pages (from-to)400-406
Number of pages7
JournalApplied and Computational Harmonic Analysis
Volume25
Issue number3
DOIs
StatePublished - Nov 1 2008

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Affine Systems
Norm
Dilation
Expand
Linear Combination
Fourier transform
Fourier transforms
Series
Coefficient

Keywords

  • Completeness
  • Multiresolution analysis
  • Quasi-interpolation

ASJC Scopus subject areas

  • Applied Mathematics

Cite this

A note on constructing affine systems for L 2 . / Bui, H. Q.; Kaiblinger, N.; Laugesen, Richard S.

In: Applied and Computational Harmonic Analysis, Vol. 25, No. 3, 01.11.2008, p. 400-406.

Research output: Contribution to journalLetter

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N2 - Assume ψ ∈ L 2 (R d ) has Fourier transform continuous at the origin, with over(ψ, ̂) (0) = 1, and that ∑ l ∈ Zd | over(ψ, ̂) (ξ - l) | 2 is bounded as a function of ξ ∈ R d . Then every function f ∈ L 2 (R d ) can be represented by an affine series f = ∑ j > 0 ∑ k ∈ Zd c j, k ψ j, k for some coefficients satisfying{norm of matrix} c {norm of matrix} ℓ1 (ℓ2) = under(∑, j > 0) (under(∑, k ∈ Z d ) | c j, k | 2 ) 1 / 2 < ∞ . Here ψ j, k (x) = | det a j | 1 / 2 ψ (a j x - k) and the dilation matrices a j expand, for example a j = 2 j I. The result improves an observation by Daubechies that the linear combinations of the ψ j, k are dense in L 2 (R d ).

AB - Assume ψ ∈ L 2 (R d ) has Fourier transform continuous at the origin, with over(ψ, ̂) (0) = 1, and that ∑ l ∈ Zd | over(ψ, ̂) (ξ - l) | 2 is bounded as a function of ξ ∈ R d . Then every function f ∈ L 2 (R d ) can be represented by an affine series f = ∑ j > 0 ∑ k ∈ Zd c j, k ψ j, k for some coefficients satisfying{norm of matrix} c {norm of matrix} ℓ1 (ℓ2) = under(∑, j > 0) (under(∑, k ∈ Z d ) | c j, k | 2 ) 1 / 2 < ∞ . Here ψ j, k (x) = | det a j | 1 / 2 ψ (a j x - k) and the dilation matrices a j expand, for example a j = 2 j I. The result improves an observation by Daubechies that the linear combinations of the ψ j, k are dense in L 2 (R d ).

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