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Multiresolution analysis for compactly supported interpolating tensor product wavelets

Department of Physics, Tampere University of Technology, P. O. Box 692, FI-33101 Tampere, Finland

https://doi.org/10.1142/S0219691315500101Cited by:2 (Source: Crossref)

Abstract

We construct multidimensional interpolating tensor product multiresolution analyses (MRA's) of the function spaces C₀(ℝⁿ, K), K = ℝ or K = ℂ, consisting of real or complex valued functions on ℝⁿ vanishing at infinity and the function spaces C_u(ℝⁿ, K) consisting of bounded and uniformly continuous functions on ℝⁿ. We also construct an interpolating dual MRA for both of these spaces. The theory of the tensor products of Banach spaces is used. We generalize the Besov space norm equivalence from the one-dimensional case to our n-dimensional construction.

Keywords:

AMSC: 46A32, 46B28, 15A69, 46E10

References

A. Almeida , J. Math. Anal. Appl. 304 , 198 ( 2005 ) . Crossref, Web of Science, Google Scholar
C. Bennett and R. Sharpley , Interpolation of Operators ( Academic Press , London , 1988 ) . Google Scholar
P. Brenner , Besov Spaces and Applications to Difference Methods for Initial Value Problems ( Springer , Berlin , 1975 ) . Crossref, Google Scholar
J. Brodzki and G. A. Niblo, Rapid decay and the metric approximation property, preprint (2004) , arXiv:math.GR/0403423v2 . Google Scholar
C. K. Chui and C. Li , Appl. Comput. Harmon. Anal. 1 , 368 ( 1994 ) . Crossref, Google Scholar
C. K. Chui and C. Li, SIAM J. Math. Anal. 27(3), 865 (1996). Crossref, Web of Science, Google Scholar
S. Dahlke, K. Gröchenig and P. Maass, Appl. Anal. 72(4), 485 (1999). Crossref, Google Scholar
I. Daubechies , Commun. Pure Appl. Math. XLI , 909 ( 1988 ) . Crossref, Web of Science, Google Scholar
I. Daubechies , Ten Lectures on Wavelets ( SIAM , 1992 ) . Crossref, Google Scholar
M. Daws, Dual Banach algebras: Representations and injectivity, preprint (2006) , arXiv:math.FA/0604372v1 . Google Scholar
A. Defant and K. Floret , Tensor Norms and Operator Ideals ( North-Holland , Amsterdam , 1993 ) . Google Scholar
G. Deslauriers and S. Dubuc, Constr. Approx. 5(1), 49 (1989). Crossref, Web of Science, Google Scholar
R. A. DeVore, B. Jawerth and B. J. Lucier, IEEE Trans. Inform. Theory 38(2), 719 (1992). Crossref, Web of Science, Google Scholar
R. A. DeVore and V. A. Popov, Trans. Amer. Math. Soc. 305(1), 397 (1988). Crossref, Web of Science, Google Scholar
P. Domański and M. Lindström , Bull. London Math. Soc. 28 , 617 ( 1996 ) . Crossref, Web of Science, Google Scholar
P. Domański, M. Lindström and G. Schlüchtermann, Proc. Amer. Math. Soc. 125(8), 2285 (1997). Crossref, Web of Science, Google Scholar
D. L. Donoho, Interpolating wavelet transforms, tech. report, Department of Statistics, Stanford University (1992) . Google Scholar
S. Dubuc , J. Math. Anal. Appl. 114 , 185 ( 1986 ) . Crossref, Web of Science, Google Scholar
H. Glöckner, Tensor products in the category of topological vector spaces are not associative, preprint (2004) , arXiv:math.FA/0307041v2 . Google Scholar
S. Goedecker , Wavelets and Their Application for the Solution of Partial Differential Equations in Physics ( Presses Polytechniques et Universitaires Romandes , Lausanne , 1998 ) . Google Scholar
A. Grothendieck , Mem. Amer. Math. Soc. 16 , ( 1955 ) . Google Scholar
B. Han , Linear Algebra Appl. 353 , 207 ( 2002 ) . Crossref, Web of Science, Google Scholar
B. Han and R.-Q. Jia, SIAM J. Numer. Anal. 36(1), 105 (1998). Crossref, Web of Science, Google Scholar
W. Härdle, G. Kerkyacharian, D. Picard and A. Tsybakov, Wavelets, Approximation, and Statistical Applications, http://www.quantlet.com/mdstat/scripts/wav/html/ . Google Scholar
W. He and M. Lai, Bivariate box spline wavelets in Sobolev spaces, SPIE's Int. Symp. on Optical Science, Engineering, and Instrumentation (International Society for Optics and Photonics, 1998) pp. 56–66. Google Scholar
T. Höynälänmaa, Multiresolution analysis for compactly supported interpolating tensor product Wavelets (long version), preprint (2010), [math.FA] , arXiv:1005.3371 . Google Scholar
H. Ji and Z. Shen, Adv. Comput. Math. 11(1), 81 (1999). Crossref, Web of Science, Google Scholar
R. Q. Jia and C. A. Micchelli , Curves Surfaces 209 ( 1991 ) . Crossref, Google Scholar
S. Karakaz'yan, M. Skopina and M. Tchobanou, Int. J. Wavelets, Multiresolut. Inf. Process. 7(3), 1 (2008). Web of Science, Google Scholar
J. Kovačević and W. Sweldens, IEEE Trans. Image Process. 9(3), 480 (2000). Crossref, Web of Science, Google Scholar
A. Krivoshein and M. Skopina, Appl. Comput. Harmon. Anal. 31(3), 410 (2011). Crossref, Web of Science, Google Scholar
J. Kustermans and S. Vaes, Proc. Natl. Acad. Sci. USA 97(2), 547 (2000). Crossref, Web of Science, Google Scholar
G. Kyriazis and P. Petrushev, Trans. Amer. Math. Soc. 354(2), 749 (2002). Crossref, Web of Science, Google Scholar
H. E. Lacey , The Isometric Theory of Classical Banach Spaces ( Springer-Verlag , 1974 ) . Crossref, Google Scholar
W. Lawton, S. L. Lee and Z. Shen, SIAM J. Math. Anal. 28(4), 999 (1997). Crossref, Web of Science, Google Scholar
R. M. Lewis , J. Approx. Theory 76 , 177 ( 1994 ) . Crossref, Web of Science, Google Scholar
W. A. Light and E. W. Cheney , Approximation Theory in Tensor Product Spaces ( Springer-Verlag , 1985 ) . Crossref, Google Scholar
Y. Meyer , Wavelets and Operators ( Cambridge University Press , 1992 ) . Google Scholar
P. W. Michor , Functors and Categories of Banach Spaces ( Springer-Verlag , 1978 ) . Crossref, Google Scholar
O. I. Reinov, On Banach spaces without the approximation property, preprint (2002) , arXiv:math.FA/0203146v1 . Google Scholar
R. A. Ryan , Introduction to Tensor Products of Banach Spaces ( Springer , New York , 2002 ) . Crossref, Google Scholar
H. H. Schaefer , Topological Vector Spaces ( The Macmillan Company , New York , 1966 ) . Google Scholar
P.-L. Shui, Z. Bao and X.-D. Zhang, IEEE Trans. Signal Process. 49(8), 1704 (2001). Crossref, Web of Science, Google Scholar
H. Triebel , Theory of Function Spaces ( Birkhäuser-Verlag , Basel , 1983 ) . Crossref, Google Scholar
P. Wojtaszczyk , A Mathematical Introduction to Wavelets ( Cambridge University Press , 1997 ) . Crossref, Google Scholar