×

Pointwise convergence of wavelet expansions. (English) Zbl 0788.42014

Summary: In this note we announce that under general hypotheses, wavelet-type expansions (of functions in \(L^ p\), \(1\leq p\leq\infty\), in one or more dimensions) converge pointwise almost everywhere, and identify the Lebesgue set of a function as a set of full measure on which they converge. It is shown that unlike the Fourier summation kernel, wavelet summation kernels \(P_ j\) are bounded by radial decreasing \(L^ 1\) convolution kernels. As a corollary it follows that best \(L^ 2\) spline approximations on uniform meshes converge pointwise almost everywhere. Moreover, summation of wavelet expansions is partially insensitive to order of summation.
We also give necessary and sufficient conditions for given rates of convergence of wavelet expansions in the sup norm. Such expansions have order of convergence \(s\) if and only if the basic wavelet \(\psi\) is in the homogeneous Sobolev space \(H^{-s-d/2}_ h\). We also present equivalent necessary and sufficient conditions on the scaling function. The above results hold in one and in multiple dimensions.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
40A30 Convergence and divergence of series and sequences of functions
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Guy Battle, A block spin construction of ondelettes. I. Lemarié functions, Comm. Math. Phys. 110 (1987), no. 4, 601 – 615.
[2] C. de Boor, R. DeVore, and A. Ron, Approximation from shift-invariant subspaces of \( {L^2}({\textbf{R}^d})\), preprint. · Zbl 0790.41012
[3] C. de Boor and A. Ron, Fourier analysis of the approximation power of principal shiftinvariant subspaces, preprint.
[4] Lennart Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135 – 157. · Zbl 0144.06402 · doi:10.1007/BF02392815
[5] Ingrid Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), no. 7, 909 – 996. · Zbl 0644.42026 · doi:10.1002/cpa.3160410705
[6] Ingrid Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. · Zbl 0776.42018
[7] A. Grossmann and J. Morlet, Decomposition of Hardy functions into square integrable wavelets of constant shape, SIAM J. Math. Anal. 15 (1984), no. 4, 723 – 736. · Zbl 0578.42007 · doi:10.1137/0515056
[8] Alfred Haar, Zur Theorie der orthogonalen Funktionensysteme, Math. Ann. 69 (1910), no. 3, 331 – 371 (German). · JFM 41.0469.03 · doi:10.1007/BF01456326
[9] Richard A. Hunt, On the convergence of Fourier series, Orthogonal Expansions and their Continuous Analogues (Proc. Conf., Edwardsville, Ill., 1967) Southern Illinois Univ. Press, Carbondale, Ill., 1968, pp. 235 – 255.
[10] P. G. Lemarié, Ondelettes á localisation exponentielle, J. Math. Pures Appl. (to appear).
[11] P. G. Lemarié and Y. Meyer, Ondelettes et bases hilbertiennes, Rev. Mat. Iberoamericana 2 (1986), no. 1-2, 1 – 18 (French). · Zbl 0657.42028 · doi:10.4171/RMI/22
[12] Susan E. Kelly, Mark A. Kon, and Louise A. Raphael, Pointwise convergence of wavelet expansions, Bull. Amer. Math. Soc. (N.S.) 30 (1994), no. 1, 87 – 94. · Zbl 0838.42010
[13] Stephane G. Mallat, Multiresolution approximations and wavelet orthonormal bases of \?²(\?), Trans. Amer. Math. Soc. 315 (1989), no. 1, 69 – 87. · Zbl 0686.42018
[14] Yves Meyer, Ondelettes et opérateurs. I, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1990 (French). Ondelettes. [Wavelets]. · Zbl 0694.41037
[15] G. Strang and G. Fix, A Fourier analysis of the finite element variational method, Constructive Aspects of Functional Analysis, Edizioni Cremonese, Rome, 1973. · Zbl 0278.65116
[16] Jan-Olov Strömberg, A modified Franklin system and higher-order spline systems on \?\(^{n}\) as unconditional bases for Hardy spaces, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981) Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 475 – 494.
[17] Gilbert G. Walter, Approximation of the delta function by wavelets, J. Approx. Theory 71 (1992), no. 3, 329 – 343. · Zbl 0766.41020 · doi:10.1016/0021-9045(92)90123-6
[18] -, Pointwise convergence of wavelet expansions, preprint, 1992.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.