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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Multivariate periodic wavelets

Authors: I. E. Maksimenko and M. A. Skopina
Translated by: the authors
Original publication: Algebra i Analiz, tom 15 (2003), nomer 2.
Journal: St. Petersburg Math. J. 15 (2004), 165-190
MSC (2000): Primary 42C40
Published electronically: January 26, 2004
MathSciNet review: 2052130
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Abstract | References | Similar Articles | Additional Information

Abstract: A general construction of a multiresolution analysis with a matrix dilation for periodic functions is described, together with a method of finding wavelet biorthogonal bases. The convergence of expansions with respect to these bases is studied.

References [Enhancements On Off] (What's this?)

  • 1. I. Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conf. Ser. in Appl. Math., vol. 61, SIAM, Philadelphia, PA, 1992. MR 93e:42045
  • 2. S. Mallat, Multiresolution approximations and wavelet orthonormal bases of $L^2(R)$, Trans. Amer. Math. Soc. 315 (1989), 69-87. MR 90e:42046
  • 3. Y. Meyer, Ondelettes et opérateurs. I. Ondelettes, Hermann, Paris, 1990. MR 93i:42002
  • 4. C. de Boor, R. DeVore, and A. Ron, On the construction of multivariate (pre)wavelets, Constr. Approx. 9 (1993), 123-166. MR 94k:41048
  • 5. R. Q. Jia and C. A. Micchelli, Using the refinement equations for the construction of pre-wavelets. II. Powers of two, Curves and Surfaces (Chamonix-Mont-Blanc, 1990) (P. J. Laurent, A. Le Mhauté and L. L. Schumaker, eds.), Academic Press, Boston, MA, 1991, pp. 209-246. MR 93e:65024
  • 6. -, Using the refinement equation for the construction of pre-wavelets. V. Extensibility of trigonometric polynomials, Computing 48 (1992), no. 1, 61-72. MR 94a:42049
  • 7. S. D. Riemenschneider and Z. W. Shen, Box splines, cardinal series and wavelets, Approximation Theory and Functional Analysis (College Station, TX, 1990) (C. K. Chui, ed.), Academic Press, Boston, MA, 1991, pp. 133-149. MR 92g:41001
  • 8. -, Wavelets and pre-wavelets in low dimensions, J. Approx. Theory 71 (1992), no. 1, 18-38. MR 94d:42046
  • 9. -, Construction of compactly supported biorthogonal wavelets in $L_2(\mathbb R^s)$, Preprint, 1997.
  • 10. H. Ji, S. D. Riemenschneider, and Z. W. Shen, Multivariate compactly supported fundamental refinable functions and biorthogonal wavelets, Preprint.
  • 11. R. Q. Jia and Z. Shen, Multiresolution and wavelets, Proc. Edinburgh Math. Soc. (2) 37 (1994), no. 2, 271-300. MR 95h:42035
  • 12. C. K. Chui and H. N. Mhaskar, On trigonometric wavelets, Constr. Approx. 9 (1993), no. 2-3, 167-190. MR 94c:42002
  • 13. C. K. Chui and J. Wang, A general framework of compactly supported splines and wavelets, J. Approx. Theory 71 (1992), no. 3, 263-304. MR 94a:42043
  • 14. V. A. Zheludev, Periodic splines and wavelets, Mathematical Analysis, Wavelets, and Signal Processing (Cairo, 1994), Contemp. Math., vol. 190, Amer. Math. Soc., Providence, RI, 1995, pp. 339-354. MR 96d:41001
  • 15. A. P. Petukhov, Periodic wavelets, Mat. Sb. 188 (1997), no. 10, 69-94; English transl., Sb. Math. 188 (1997), no. 10, 1481-1506. MR 99b:42042
  • 16. S. S. Goh, S. L. Lee, Z. Shen, and W. S. Tang, Construction of Schauder decomposition on Banach spaces of periodic functions, Proc. Edinburgh Math. Soc. (2) 41 (1998), no. 1, 61-91. MR 99b:46033
  • 17. V. A. Sadovnichii, Theory of operators, 3rd ed., ``Vyssh. Shkola'', Moscow, 1999; English transl. of 2nd. ed., Consultants Bureau, New York, 1991. MR 88b:47001
  • 18. P. Wojtaszczyk, A mathematical introduction to wavelets, London Math. Soc. Stud. Texts, vol. 37, Cambridge Univ. Press, Cambridge, 1997. MR 98j:42025
  • 19. M. Skopina, Multiresolution analysis of periodic functions, East J. Approx. 3 (1997), no. 2, 203-224. MR 99h:42066
  • 20. S. E. Kelly, M. A. Kon, and L. A. Raphael, Local convergence for wavelet expansions, Preprint.
  • 21. I. E. Maksimenko, Biorthogonality of scaling multivariate functions, Contemporary Problems of Approximation Theory, 2002 (to appear). (Russian)
  • 22. -, Sufficient conditions of a biorthogonality of scaling multivariate functions, Optimization of Finite Element Approximations, Splines and Wavelets (2nd Internat. Conf. OFEA'2001, St. Petersburg, Russia, 2001). Vol. 2, S.-Peterburg Univ., St. Petersburg, 2002, pp. 80-91. (Russian)
  • 23. A. Zygmund, Trigonometric series. Vol. 2, Cambridge Univ. Press, New York, 1959. MR 21:6498
  • 24. A. Petukhov, Trigonometric rational wavelet bases, Self-Similar Systems (Internat. Workshop July 30-August 7, 1998, Dubna, Russia), JINR, Dubna, 1999, pp. 116-119 (JINR, E5-99-38). MR 2001j:00031
  • 25. E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Math. Ser., No. 32, Princeton Univ. Press Princeton, NJ, 1971. MR 46:4102

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Additional Information

I. E. Maksimenko
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ pr. 28, St. Petersburg 198504, Russia

M. A. Skopina
Affiliation: Department of Applied Mathematics and Control Processes, St. Petersburg State University, Universitetskiĭ pr. 28, St. Petersburg 198504, Russia

Keywords: Multiresolution analysis, wavelet bases, matrix dilation
Received by editor(s): July 10, 2002
Published electronically: January 26, 2004
Additional Notes: Supported by RFBR (grant no. 3-01-00373)
Article copyright: © Copyright 2004 American Mathematical Society

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