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Homogeneous approximation property for wavelet frames with matrix dilations, II

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Abstract

The homogeneous approximation property (HAP) states that the number of building blocks involved in a reconstruction of a function up to some error is essentially invariant under time-scale shifts. In this paper, we show that every wavelet frame with nice wavelet function and arbitrary expansive dilation matrix possesses the HAP. Our results improve some known ones.

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References

  1. Christensen, O.: An Introduction to Frames and Riesz Bases, Birkhäuser, Boston, 2003

    MATH  Google Scholar 

  2. Young, R. M.: An Introduction to Non-Harmonic Fourier Series, Academic Press, New York, 1980

    Google Scholar 

  3. Balan, R., Casazza, P. G., Heil, C., et al.: Density, overcompleteness, and localization of frames, I. Theory. J. Fourier Anal. Appl., 12, 105–143 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chui, C. K., He, W., Stöckler, J.: Compactly supported tight and sibling frames with maximum vanishing moments. Appl. Comput. Harmon. Anal., 13, 224–262 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chui, C. K., Shi, X. L.: Inequalities of Littlewood-Paley type for frames and wavelets. SIAM. J. Math. Anal., 24, 263–277 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chui, C. K., Shi, X. L.: Orthonormal wavelets and tight frames with arbitrary dilations. Appl. Comput. Harmon. Anal., 9, 243–264 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Daubechies, I.: Ten Lectures on Wavelets, SIAM, Philadelphia, 1992

    Book  MATH  Google Scholar 

  8. Daubechies, I., Han, B., Ron, A., et al.: Framelets: MRA-based construction of wavelet frames. Appl. Comput. Harmon. Anal., 14, 1–46 (2004)

    Article  MathSciNet  Google Scholar 

  9. Gröchenig, K.: Localization of frames, Banach frames, and the invertibility of the frame operator. J. Fourier Anal. Appl., 10, 105–132 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gröchenig, K.: Localization of frames. Adv. Comput. Math., 18, 149–157 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Heil, C., Walnut, D.: Continuous and discrete wavelet transforms. SIAM Review, 31, 628–666 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ron, A., Shen, Z.: Affine systems in L 2(ℝd): the analysis of the analysis operator. J. Funct. Anal., 148, 408–447 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ron, A., Shen, Z.: Generalized shift-invariant systems. Constr. Approx., 22, 1–45 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Sun, Q.: Wiener’s lemma for infinite matrices. Trans. Amer. Math. Soc., 359, 3099–3123 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gröchenig, K.: The homogeneous approximation property and the comparison theorem for coherent frames. Sampl. Theory Signal Image Process., 7, 271–279 (2008)

    MathSciNet  MATH  Google Scholar 

  16. Heil, C., Kutyniok, G.: The homogeneous approximation property for wavelet frames. J. Approx. Theory, 147, 28–46 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kutyniok, G.: Affine Density in Wavelet Analysis, Lecture Notes in Mathematics, 1914, Springer-Verlag, Berlin, 2007

    MATH  Google Scholar 

  18. Sun, W.: Homogeneous approximation property for wavelet frames. Monatshefte für Mathematik, 159, 289–324 (2010)

    Article  MATH  Google Scholar 

  19. Sun, W.: Homogeneous approximation property for wavelet frames with matrix dilations. Math. Nachr., 283, 1488–1505 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Wen Chang Sun.

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Supported partially by National Natural Science Foundation of China (Grant Nos. 10971105 and 10990012) and Natural Science Foundation of Tianjin (Grant No. 09JCYBJC01000)

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Zhao, Z.J., Sun, W.C. Homogeneous approximation property for wavelet frames with matrix dilations, II. Acta. Math. Sin.-English Ser. 29, 183–192 (2013). https://doi.org/10.1007/s10114-012-0546-9

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  • DOI: https://doi.org/10.1007/s10114-012-0546-9

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