Existence of multiwavelets in ℝⁿ

Cabrelli, Carlos; Gordillo, María

doi:10.1090/S0002-9939-01-06223-2

Existence of multiwavelets in $\mathbb {R}^n$
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by Carlos A. Cabrelli and María Luisa Gordillo PDF

Proc. Amer. Math. Soc. 130 (2002), 1413-1424 Request permission

Abstract:

For a $q$-regular Multiresolution Analysis of multiplicity $r$ with arbitrary dilation matrix $A$ for a general lattice $\Gamma$ in $\mathbb {R}^n$, we give necessary and sufficient conditions in terms of the mask and the symbol of the vector scaling function in order that an associated wavelet basis exists. We also show that if $2r(m-1) \geq n$ where $m$ is the absolute value of the determinant of $A$, then these conditions are always met, and therefore an associated wavelet basis of $q$-regular functions always exists. This extends known results to the case of multiwavelets in several variables with an arbitrary dilation matrix $A$ for a lattice $\Gamma$.

References

Akram Aldroubi, Oblique and hierarchical multiwavelet bases, Appl. Comput. Harmon. Anal. 4 (1997), no. 3, 231–263. MR 1454403, DOI 10.1006/acha.1997.0211
Bradley K. Alpert, A class of bases in $L^2$ for the sparse representation of integral operators, SIAM J. Math. Anal. 24 (1993), no. 1, 246–262. MR 1199538, DOI 10.1137/0524016
Ryuichi Ashino and Makoto Kametani, A lemma on matrices and a construction of multi-wavelets, Math. Japon. 45 (1997), no. 2, 267–287. MR 1441499
C. Cabrelli, C. Heil and U. Molter, Self-Similarity and Multiwavelets in Higher Dimensions, preprint (1999).
A. Calogero, Wavelets on general lattices, associated with general expanding maps of $\mathbf R^n$, Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 1–10. MR 1667201, DOI 10.1090/S1079-6762-99-00054-2
Di-Rong Chen, On the existence and constructions of orthonormal wavelets on $L_2(\mathbf R^s)$, Proc. Amer. Math. Soc. 125 (1997), no. 10, 2883–2889. MR 1396974, DOI 10.1090/S0002-9939-97-03876-8
Albert Cohen, Ingrid Daubechies, and Gerlind Plonka, Regularity of refinable function vectors, J. Fourier Anal. Appl. 3 (1997), no. 3, 295–324. MR 1448340, DOI 10.1007/BF02649113
Ingrid Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), no. 7, 909–996. MR 951745, DOI 10.1002/cpa.3160410705
L. De Michele and P. M. Soardi, On multiresolution analysis of multiplicity $d$, Monatsh. Math. 124 (1997), no. 3, 255–272. MR 1476365, DOI 10.1007/BF01298247
Jeffrey S. Geronimo, Douglas P. Hardin, and Peter R. Massopust, Fractal functions and wavelet expansions based on several scaling functions, J. Approx. Theory 78 (1994), no. 3, 373–401. MR 1292968, DOI 10.1006/jath.1994.1085
T. N. T. Goodman, S. L. Lee, and W. S. Tang, Wavelets in wandering subspaces, Trans. Amer. Math. Soc. 338 (1993), no. 2, 639–654. MR 1117215, DOI 10.1090/S0002-9947-1993-1117215-0
Karlheinz Gröchenig, Analyse multi-échelles et bases d’ondelettes, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 1, 13–15 (French, with English summary). MR 902279
K. Gröchenig and W. R. Madych, Multiresolution analysis, Haar bases, and self-similar tilings of $\textbf {R}^n$, IEEE Trans. Inform. Theory 38 (1992), no. 2, 556–568. MR 1162214, DOI 10.1109/18.119723
Christopher Heil and David Colella, Matrix refinement equations: existence and uniqueness, J. Fourier Anal. Appl. 2 (1996), no. 4, 363–377. MR 1395770
Christopher Heil, Gilbert Strang, and Vasily Strela, Approximation by translates of refinable functions, Numer. Math. 73 (1996), no. 1, 75–94. MR 1379281, DOI 10.1007/s002110050185
John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI 10.1512/iumj.1981.30.30055
Rong-Qing Jia, Sherman D. Riemenschneider, and Ding-Xuan Zhou, Smoothness of multiple refinable functions and multiple wavelets, SIAM J. Matrix Anal. Appl. 21 (1999), no. 1, 1–28. MR 1709723, DOI 10.1137/S089547989732383X
Stephane G. Mallat, Multiresolution approximations and wavelet orthonormal bases of $L^2(\textbf {R})$, Trans. Amer. Math. Soc. 315 (1989), no. 1, 69–87. MR 1008470, DOI 10.1090/S0002-9947-1989-1008470-5
Yves Meyer, Wavelets and operators, Cambridge Studies in Advanced Mathematics, vol. 37, Cambridge University Press, Cambridge, 1992. Translated from the 1990 French original by D. H. Salinger. MR 1228209
A. Potiopa, A problem of Lagarias and Wang, Master’s thesis, Siedlce University, Siedlce, Poland June (1997), (Polish).
P. Wojtaszczyk, A mathematical introduction to wavelets, London Mathematical Society Student Texts, vol. 37, Cambridge University Press, Cambridge, 1997. MR 1436437, DOI 10.1017/CBO9780511623790