Skip to main content
SpringerLink
Log in

Metric convolution and frames

  • Published:
Periodica Mathematica Hungarica Aims and scope Submit manuscript

Abstract

The fractal convolution of two mappings is a binary operation in some space of functions. In previous papers we extracted the main properties of this association and defined a new type of inner operations in metric spaces, not necessarily linked to fractal theory. This operation has been called metric convolution, though it does not agree with the classical convolution of functions. In this paper we develop a further insight into this association, deducing additional properties. When the metric space framework is substituted by a normed space setting, we address the definition of bases and frames composed of convolution elements, different from those of other articles. We study also the dynamics of two maps linked to the operation.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data availability

The manuscript does not contain any data from third party. All of the material is owned by the authors and/or no permissions are required.

References

  1. O. Christensen, Frame perturbations. Proc. Am. Math. Soc. 123(4), 1217–1220 (1995)

    Article  MathSciNet  Google Scholar 

  2. O. Christensen, Frames and Bases: An Introductory Course (Birkhauser, Boston, 2008)

    Book  Google Scholar 

  3. R.J. Duffin, A.C. Schaeffer, A class of nonharmonic Fourier series. Trans. Am. Math. Soc. 72, 341–366 (1952)

    Article  MathSciNet  Google Scholar 

  4. G.B. Folland, A Course in Abstract Harmonic Analysis (CRC Press, Boca Raton, 1995)

    Google Scholar 

  5. A. Khosravi, M.S. Asgari, Frames and bases in tensor product of Hilbert spaces. Int. Math. J. 4, 527–537 (2003)

    MathSciNet  Google Scholar 

  6. M.A. Navascués, Fractal bases of \({\cal{L} }^p\) spaces. Fractals 20(2), 141–148 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  7. M.A. Navascués, Fractal functions of discontinuous approximation. J. Basic Appl. Sci. 10, 173–176 (2014)

    Article  Google Scholar 

  8. M.A. Navascués, A.K.B. Chand, Fundamental sets of fractal functions. Acta Appl. Math. 100, 247–261 (2008)

    Article  MathSciNet  Google Scholar 

  9. M.A. Navascués, P. Massopust, Fractal convolution: a new operation between functions. Fract. Calc. Appl. Anal. 22(3), 619–643 (2019)

    Article  MathSciNet  Google Scholar 

  10. M.A. Navascués, R. Mohapatra, A.K.B. Chand, Some properties of the fractal convolution of functions. Fract. Calc. Appl. Anal. 24(6), 1735–1757 (2021)

    Article  MathSciNet  Google Scholar 

  11. M.A. Navascués, R. Pasupathi, A.K.B. Chand, Binary operations in metric spaces satisfying side inequalities. Mathematics 10(1), 11, 1–17 (2022)

  12. M.A. Navascués, P. Viswanathan, R. Mohapatra, Convolved fractal bases and frames. Adv. Oper. Theory 6(42), 1–23 (2021)

    MathSciNet  Google Scholar 

  13. I. Singer, Bases in Banach Spaces I (Springer, New York, 1970)

    Book  Google Scholar 

  14. J. Von Neumann, Mathematical Foundations of Quantum Mechanics. Translated from the German and with a preface by Robert T. Beyer. Edited and with a preface by Nicholas A. Wheeler. Princeton University Press, Princeton (2018)

  15. R. Young, An Introduction to Nonharmonic Fourier Series (Academic Press, New York, 1980)

    Google Scholar 

Download references

Acknowledgements

The first and third authors would like to thank Ministry of Education for the IoE Research Project: SB20210848MAMHRD008558. The authors are thankful to the anonymous reviewer for his/her constructive suggestions to improve the presentation of the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Madras, Chennai, 600036, India

    R. Pasupathi & A. K. B. Chand

  2. Departmento de Matemática Aplicada, Escuela de Ingeniería y Arquitectura, Universidad de Zaragoza, 50018, Zaragoza, Spain

    M. A. Navascués

  3. Department of Mathematical Sciences, University of Central Florida, Orlando, FL, 32816, USA

    R. N. Mohapatra

Authors
  1. R. Pasupathi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. A. Navascués
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. K. B. Chand
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. R. N. Mohapatra
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. K. B. Chand.

Ethics declarations

Competing interest

It is declared that the authors have no competing interests as defined by Springer, or other interests that might be perceived to influence the results and/or discussion reported in this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pasupathi, R., Navascués, M.A., Chand, A.K.B. et al. Metric convolution and frames. Period Math Hung 88, 243–265 (2024). https://doi.org/10.1007/s10998-023-00550-5

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10998-023-00550-5

Keywords

Mathematics Subject Classification

Search

Navigation