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Higher Dimensional Continuous Wavelet Transform in Wiener Amalgam Spaces

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Topics in Mathematical Analysis and Applications

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Abstract

Norm convergence and convergence at Lebesgue points of the inverse wavelet transform are obtained for functions from the L p and Wiener amalgam spaces.

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References

  1. Ashurov, R.: Convergence of the continuous wavelet transforms on the entire Lebesgue set of L p -functions. Int. J. Wavelets Multiresolution Inf. Process. 9, 675–683 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Butzer, P.L., Nessel, R.J.: Fourier Analysis and Approximation. Birkhäuser Verlag, Basel (1971)

    Book  MATH  Google Scholar 

  3. Christensen, O.: An Introduction to Frames and Riesz Bases. Birkhäuser Verlag, Basel (2003)

    Book  MATH  Google Scholar 

  4. Chui, C.K.: An Introduction to Wavelets. Academic, Boston (1992)

    MATH  Google Scholar 

  5. Daubechies, I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992)

    Book  MATH  Google Scholar 

  6. Feichtinger, H.G., Weisz, F.: The Segal algebra \(\mathbf{S}_{0}(\mathbb{R}^{d})\) and norm summability of Fourier series and Fourier transforms. Monatshefte Math. 148, 333–349 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Feichtinger, H.G., Weisz, F.: Wiener amalgams and pointwise summability of Fourier transforms and Fourier series. Math. Proc. Camb. Philos. Soc. 140, 509–536 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gát, G.: Pointwise convergence of cone-like restricted two-dimensional (C, 1) means of trigonometric Fourier series. J. Appr. Theory 149, 74–102 (2007)

    Article  MATH  Google Scholar 

  9. Goginava, U.: The maximal operator of the Marcinkiewicz-Fejér means of d-dimensional Walsh-Fourier series. East J. Appr. 12, 295–302 (2006)

    MathSciNet  Google Scholar 

  10. Grafakos, L.: Classical and Modern Fourier Analysis. Pearson Education, Upper Saddle River (2004)

    MATH  Google Scholar 

  11. Gröchenig, K.: Foundations of Time-Frequency Analysis. Birkhäuser, Boston (2001)

    Book  MATH  Google Scholar 

  12. Holschneider, M., Tchamitchain, P.: Pointwise analysis of Riemann’s “nondifferentiable” function. Invent. Math. 105, 157–175 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kelly, S.E., Kon, M.A., Raphael, L.A.: Local convergence for wavelet expansions. J. Funct. Anal. 126, 102–138 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kelly, S.E., Kon, M.A., Raphael, L.A.: Pointwise convergence of wavelet expansions. Bull. Am. Math. Soc. 30, 87–94 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  15. Li, K., Sun, W.: Pointwise convergence of the Calderon reproducing formula. J. Fourier Anal. Appl. 18, 439–455 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Rao, M., Sikic, H., Song, R.: Application of Carleson’s theorem to wavelet inversion. Control Cybern. 23, 761–771 (1994)

    MathSciNet  MATH  Google Scholar 

  17. Rubin, B., Shamir, E.: Carlderon’s reproducing formula and singular integral operators on a real line. Integral Equ. Operator Theory 21, 78–92 (1995)

    Article  MathSciNet  Google Scholar 

  18. Saeki, S.: On the reproducing formula of Calderon. J. Fourier Anal. Appl. 2, 15–28 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  19. Simon, P.: (C, α) summability of Walsh-Kaczmarz-Fourier series. J. Appr. Theory 127, 39–60 (2004)

    Article  MATH  Google Scholar 

  20. Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton (1971)

    MATH  Google Scholar 

  21. Szarvas, K., Weisz, F.: Almost everywhere and norm convergence of the inverse continuous wavelet transform in Pringsheim’s sense (preprint)

    Google Scholar 

  22. Trigub, R.M., Belinsky, E.S.: Fourier Analysis and Approximation of Functions. Kluwer Academic, Dordrecht/Boston/London (2004)

    Book  MATH  Google Scholar 

  23. Weisz, F.: Convergence of the inverse continuous wavelet transform in Wiener amalgam spaces (preprint)

    Google Scholar 

  24. Weisz, F.: Inverse continuous wavelet transform in Pringsheim’s sense in Wiener amalgam spaces (preprint)

    Google Scholar 

  25. Weisz, F.: Pointwise convergence in Pringsheim’s sense of the summability of Fourier transforms on Wiener amalgam spaces. Monatshefte Math. (to appear)

    Google Scholar 

  26. Weisz, F.: Summability of Multi-Dimensional Fourier Series and Hardy Spaces. Mathematics and Its Applications. Kluwer Academic, Dordrecht/Boston/London (2002)

    Book  Google Scholar 

  27. Weisz, F.: Summability of multi-dimensional trigonometric fourier series. Surv. Approximation Theory 7, 1–179 (2012)

    MathSciNet  MATH  Google Scholar 

  28. Weisz, F.: Inversion formulas for the continuous wavelet transform. Acta Math. Hungar. 138, 237–258 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Weisz, F.: Orthogonality relations for continuous wavelet transforms. Ann. Univ. Sci. Budapest. Sect. Comput. 41, 361–368 (2013)

    MathSciNet  MATH  Google Scholar 

  30. Wilson, M.: Weighted Littlewood-Paley Theory and Exponential-Square Integrability. Lecture Notes in Mathematics, vol. 1924. Springer, Berlin (2008)

    Google Scholar 

  31. Wilson, M.: How fast and in what sense(s) does the Calderon reproducing formula converge? J. Fourier Anal. Appl. 16, 768–785 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zayed, A.: Pointwise convergence of a class of non-orthogonal wavelet expansions. Proc. Am. Math. Soc. 128, 3629–3637 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Ferenc Weisz .

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Weisz, F. (2014). Higher Dimensional Continuous Wavelet Transform in Wiener Amalgam Spaces. In: Rassias, T., Tóth, L. (eds) Topics in Mathematical Analysis and Applications. Springer Optimization and Its Applications, vol 94. Springer, Cham. https://doi.org/10.1007/978-3-319-06554-0_33

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