Wavelet bases in L p ( ℝ )

Gustaf Gripenberg

Studia Mathematica (1993)

  • Volume: 106, Issue: 2, page 175-187
  • ISSN: 0039-3223

Abstract

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It is shown that an orthonormal wavelet basis for L 2 ( ℝ ) associated with a multiresolution is an unconditional basis for L p ( ℝ ) , 1 < p < ∞, provided the father wavelet is bounded and decays sufficiently rapidly at infinity.

How to cite

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Gripenberg, Gustaf. "Wavelet bases in $L^{p}(ℝ)$." Studia Mathematica 106.2 (1993): 175-187. <http://eudml.org/doc/216011>.

@article{Gripenberg1993,
abstract = {It is shown that an orthonormal wavelet basis for $L^2(ℝ)$ associated with a multiresolution is an unconditional basis for $L^p(ℝ)$, 1 < p < ∞, provided the father wavelet is bounded and decays sufficiently rapidly at infinity.},
author = {Gripenberg, Gustaf},
journal = {Studia Mathematica},
keywords = {basis; $L^p$; multiresolution; unconditional; wavelet; orthonormal wavelet basis; unconditional basis; father wavelet},
language = {eng},
number = {2},
pages = {175-187},
title = {Wavelet bases in $L^\{p\}(ℝ)$},
url = {http://eudml.org/doc/216011},
volume = {106},
year = {1993},
}

TY - JOUR
AU - Gripenberg, Gustaf
TI - Wavelet bases in $L^{p}(ℝ)$
JO - Studia Mathematica
PY - 1993
VL - 106
IS - 2
SP - 175
EP - 187
AB - It is shown that an orthonormal wavelet basis for $L^2(ℝ)$ associated with a multiresolution is an unconditional basis for $L^p(ℝ)$, 1 < p < ∞, provided the father wavelet is bounded and decays sufficiently rapidly at infinity.
LA - eng
KW - basis; $L^p$; multiresolution; unconditional; wavelet; orthonormal wavelet basis; unconditional basis; father wavelet
UR - http://eudml.org/doc/216011
ER -

References

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  1. [1] A. Cohen, Ondelettes, analyses multirésolutions et filtres miroirs en quadrature, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), 439-459. Zbl0736.42021
  2. [2] I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), 909-996. Zbl0644.42026
  3. [3] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992. 
  4. [4] C. E. Heil and D. F. Walnut, Continuous and discrete wavelet transforms, SIAM Rev. 31 (1989), 628-666. Zbl0683.42031
  5. [5] R. C. James, Bases in Banach spaces, Amer. Math. Monthly 89 (1982), 625-640. Zbl0506.46006
  6. [6] P. G. Lemarié, Analyse multi-échelles et ondelettes à support compact, in: Les Ondelettes en 1989, P. G. Lemarié (ed.), Lecture Notes in Math. 1438, Springer, Berlin 1990, 26-38. 
  7. [7] S. G. Mallat, Multiresolution approximations and wavelet orthonormal bases of L 2 ( ℝ ) , Trans. Amer. Math. Soc. 315 (1989), 69-87. 
  8. [8] Y. Meyer, Ondelettes et Opérateurs I, Hermann, Paris 1990. Zbl0694.41037
  9. [9] I. Singer, Bases in Banach Spaces, Vol. I, Springer, Berlin 1970. 
  10. [10] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton 1970. Zbl0207.13501
  11. [11] G. Strang, Wavelets and dilation equations: a brief introduction, SIAM Rev. 31 (1989), 614-627. Zbl0683.42030
  12. [12] A. Zygmund, Trigonometric Series, Vol. I, Cambridge University Press, Cambridge 1959. Zbl0085.05601

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