Abstract
We construct an unconditional basis in the Banach space L p(Ω) for p 1 by using the refinement equation and the basic operation of translation and scale, where Ω is a compact subset in ℝn. We also give an algorithm of how to construct an unconditional basis in L p(Ω p). At the end of this paper, we give the characterization of the functions in L p(Ω p) by using the wavelet coefficients.
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References
Micchelli, C. A., Xu, Y., Using the matrix refinement equation for the construction of wavelets on invariant sets, Appl. Comp. Harmoni Anal, 1994, 1: 391–401.
Micchelli, C. A., Xu, Y., Using the matrix refinement equation for the construction of wavelets II, Smooth wavelets on [0,1], International Series of Numerical Mathematics, 1994, 119: 433–457.
Micchelli, C. A., Xu, Y., Reconstruction and decomposition algorithms for biorthogonal multi- wavelets, Multidimensional System and Signal Processing, 1997, 8: 31–69.
Alpert, B. K., A class of bases in L 2 for sparse repressentation of integral operators, J. Math. Anal., 1993, 24: 246–262.
Beylkin, G., Coifman R, Rokhlin V, Fast wavelet transforms and numerical algorithms I, Comm. Pure Appli. Math., 1991, 44: 141–183.
Micchelli, C. A., Xu, Y., Zhao, Y., Wavelet methods for multidimensional integral equations, J. Comp. Appl. Math., 1997, 86: 251–270.
Hutchinson, J. E., Fractals and self similarity, Indiana Univ. Math. J., 1981, 30: 713–747.
Meyer, Y., Wavelets and Operators, London: Cambridge University Press, 1993.
Donoho, D. L., Unconditional bases are optimal bases for data compression and for statistical estimation, Appl. Comp. Harmoni. Anal., 1993, 1: 100–115.
Royden, H. L., Real Analysis, New York: MacMillan, 1963.
SingerI, Bases in Banach Space I, Berlin-Heidelberg: Springer-Verlag, 1970.
Burkholder, D. L., A nonlinear partial differential equation and the uncoditional constant of the Haar system in L p, Bull. Amer. Math. Soc. N. S., 1982, 7: 591–595.
Burkholder, D. L., Exploration in martingale theory and its applications, in Ecole d’Eté de Probabilités de Saint-Flour XIX-1989, Volume 1464 of Lecture Notes in Mathematics, Berlin: Springer- Verlag, 1991, 1–66.
Burkholder, D. L., An elementary proof of an inequality of R. E. A. C. Paley, Bull. London Math. Soc., 1985, 17: 474–478.
Long Ruiling, H p Martingale Theory (in chinese), Beijing: Beijing University Press, 1985.
Zygmund, A., Trigonometric Series, 2nd ed., London: Cambridge University Press, 1959.
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Bingzheng, L., Jun, L. Unconditional basis recursively generated in L p on compact sets. Sci. China Ser. A-Math. 48, 1707–1720 (2005). https://doi.org/10.1360/022005-023
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DOI: https://doi.org/10.1360/022005-023