Skip to main content
SpringerLink
Log in

Multi-dimensional Summability Theory and Continuous Wavelet Transform

  • Chapter
  • First Online:
Current Topics in Summability Theory and Applications

Abstract

The connection between multi-dimensional summability theory and continuous wavelet transform is investigated. Two types of \(\theta \)-summability of Fourier transforms are considered, the circular and rectangular summability. Norm and almost everywhere convergence of the \(\theta \)-means are shown for both types. The inversion formula for the continuous wavelet transform is usually considered in the weak sense. Here, the inverse wavelet transform is traced back to summability means of Fourier transforms. Using the results concerning the summability of Fourier transforms, norm and almost everywhere convergence of the inversion formula are obtained for functions from the \(L_p\) and Wiener amalgam spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Bownik, M.: Boundedness of operators on Hardy spaces via atomic decompositions. Proc. Am. Math. Soc. 133, 3535–3542 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  4. Carleson, L.: On convergence and growth of partial sums of Fourier series. Acta Math. 116, 135–157 (1966)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  6. Chui, C.K.: An Introduction to Wavelets. Academic Press, Boston, MA (1992)

    MATH  Google Scholar 

  7. Cordero, E., Okoudjou, K.A.: Dilation properties for weighted modulation spaces. J. Funct. Spaces Appl. 29 pp, Art. ID 145491 (2012)

    Google Scholar 

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

    Book  MATH  Google Scholar 

  9. Feichtinger, H.G., Weisz, F.: The Segal algebra \({\mathbf{S}}_0({\mathbb{R}}^d)\) and norm summability of Fourier series and Fourier transforms. Monatsh. Math. 148, 333–349 (2006)

    Google Scholar 

  10. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  13. Grafakos, L.: Classical and Modern Fourier Analysis. Pearson Education, New Jersey (2004)

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Hunt, R.A.: On the convergence of Fourier series. In: Orthogonal Expansions and their Continuous Analogues, Proc. Conf. Edwardsville, Ill, 1967, pp. 235–255. Illinois University Press Carbondale (1968)

    Google Scholar 

  17. 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 

  18. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Marcinkiewicz, J., Zygmund, A.: On the summability of double Fourier series. Fund. Math. 32, 122–132 (1939)

    MATH  Google Scholar 

  21. Meyer, M., Taibleson, M., Weiss, G.: Some functional analytic properties of the spaces \(B_q\) generated by blocks. Indiana Univ. Math. J. 34, 493–515 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  22. Navarro, J., Arellano-Balderas, S.: Singularities of the continuous wavelet transform in \(L^p({\mathbb{R}}^n)\). J. Concr. Appl. Math. 13, 198–208 (2015)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  24. Riesz, M.: Sur la sommation des séries de Fourier. Acta Sci. Math. (Szeged) 1, 104–113 (1923)

    MATH  Google Scholar 

  25. Rubin, B., Shamir, E.: Carlderon’s reproducing formula and singular integral operators on a real line. Integral Equat. Oper. Th. 21, 78–92 (1995)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Stein, E.M.: Harmonic Analysis: Real-Variable Methods. Orthogonality and Oscillatory Integrals. Princeton University Press, Princeton, N.J. (1993)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  30. Sugimoto, M., Tomita, N.: The dilation property of modulation spaces and their inclusion relation with Besov spaces. J. Funct. Anal. 248, 79–106 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Szarvas, K., Weisz, F.: Almost everywhere and norm convergence of the inverse continuous wavelet transform in Pringsheim’s sense. Acta Sci. Math. (Szeged) (to appear)

    Google Scholar 

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

    Book  Google Scholar 

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

    Book  MATH  Google Scholar 

  34. Weisz, F.: Boundedness of operators on Hardy spaces. Acta Sci. Math. (Szeged) 78, 541–557 (2012)

    MathSciNet  MATH  Google Scholar 

  35. Weisz, F.: Summability of multi-dimensional trigonometric Fourier series. Surv. Approx. Theory 7, 1–179 (2012)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  38. Weisz, F.: Pointwise convergence in Pringsheim’s sense of the summability of Fourier transforms on Wiener amalgam spaces. Monatsh. Math. 175, 143–160 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  39. Weisz, F.: Convergence of the inverse continuous wavelet transform in Wiener amalgam spaces. Analysis 35, 33–46 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  40. Weisz, F.: Inverse continuous wavelet transform in Pringsheim’s sense on Wiener amalgam spaces. Acta Math. Hungar. 145, 392–415 (2015)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  42. 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 

  43. 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 

Download references

Acknowledgments

This research was supported by the Hungarian Scientific Research Funds (OTKA) No K115804.

Author information

Authors and Affiliations

  1. Department of Numerical Analysis, Eötvös L. University, Pázmány P. sétány 1/C., Budapest, 1117, Hungary

    Ferenc Weisz

Authors
  1. Ferenc Weisz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ferenc Weisz .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Gauhati University, Guwahati, Assam, India

    Hemen Dutta

  2. Department of Mathematics, Indiana University, Bloomington, Indiana, USA

    Billy E. Rhoades

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer Science+Business Media Singapore

About this chapter

Cite this chapter

Ferenc Weisz (2016). Multi-dimensional Summability Theory and Continuous Wavelet Transform. In: Dutta, H., E. Rhoades, B. (eds) Current Topics in Summability Theory and Applications. Springer, Singapore. https://doi.org/10.1007/978-981-10-0913-6_6

Download citation

Publish with us

Policies and ethics

Search

Navigation