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 -