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 > 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} ).
Original language | English (US) |
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Pages (from-to) | 400-406 |
Number of pages | 7 |
Journal | Applied and Computational Harmonic Analysis |
Volume | 25 |
Issue number | 3 |
DOIs | |
State | Published - Nov 1 2008 |
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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 journal › Letter
}
TY - JOUR
T1 - A note on constructing affine systems for L 2
AU - Bui, H. Q.
AU - Kaiblinger, N.
AU - Laugesen, Richard S
PY - 2008/11/1
Y1 - 2008/11/1
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 ).
KW - Completeness
KW - Multiresolution analysis
KW - Quasi-interpolation
UR - http://www.scopus.com/inward/record.url?scp=53049095359&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=53049095359&partnerID=8YFLogxK
U2 - 10.1016/j.acha.2008.04.001
DO - 10.1016/j.acha.2008.04.001
M3 - Letter
AN - SCOPUS:53049095359
VL - 25
SP - 400
EP - 406
JO - Applied and Computational Harmonic Analysis
JF - Applied and Computational Harmonic Analysis
SN - 1063-5203
IS - 3
ER -