TY - JOUR

T1 - A note on constructing affine systems for L2

AU - Bui, H. Q.

AU - Kaiblinger, N.

AU - Laugesen, R. S.

N1 - Funding Information:
This research began at meetings in Strobl, Austria, organized in May 2005 and June 2007 by the Numerical Harmonic Analysis Group at the University of Vienna. Laugesen gratefully acknowledges support from the Scholars’ Travel Fund, University of Illinois. H.-Q. Bui acknowledges the hospitality of the Erwin Schrödinger International Institute, Vienna, during the Special Semester on “Modern Methods of Time–Frequency Analysis” in 2005. We thank M.B. Erdogan for enlightening discussions on the bounded Gramian condition Gψ and its relation to the periodization of |ψ| belonging to L2.

PY - 2008/11

Y1 - 2008/11

N2 - Assume ψ ∈ L2 (Rd) 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 ∈ Rd. Then every function f ∈ L2 (Rd) can be represented by an affine series f = ∑j > 0 ∑k ∈ Zd cj, k ψj, k for some coefficients satisfying{norm of matrix} c {norm of matrix}ℓ1 (ℓ2) = under(∑, j > 0) (under(∑, k ∈ Zd) | cj, k |2)1 / 2 < ∞ . Here ψj, k (x) = | det aj |1 / 2 ψ (aj x - k) and the dilation matrices aj expand, for example aj = 2j I. The result improves an observation by Daubechies that the linear combinations of the ψj, k are dense in L2 (Rd).

AB - Assume ψ ∈ L2 (Rd) 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 ∈ Rd. Then every function f ∈ L2 (Rd) can be represented by an affine series f = ∑j > 0 ∑k ∈ Zd cj, k ψj, k for some coefficients satisfying{norm of matrix} c {norm of matrix}ℓ1 (ℓ2) = under(∑, j > 0) (under(∑, k ∈ Zd) | cj, k |2)1 / 2 < ∞ . Here ψj, k (x) = | det aj |1 / 2 ψ (aj x - k) and the dilation matrices aj expand, for example aj = 2j I. The result improves an observation by Daubechies that the linear combinations of the ψj, k are dense in L2 (Rd).

KW - Completeness

KW - Multiresolution analysis

KW - Quasi-interpolation

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U2 - 10.1016/j.acha.2008.04.001

DO - 10.1016/j.acha.2008.04.001

M3 - Letter

AN - SCOPUS:53049095359

SN - 1063-5203

VL - 25

SP - 400

EP - 406

JO - Applied and Computational Harmonic Analysis

JF - Applied and Computational Harmonic Analysis

IS - 3

ER -