Home Titchmarsh’s theorem with moduli of continuity in Laguerre hypergroup
Article
Licensed
Unlicensed Requires Authentication

Titchmarsh’s theorem with moduli of continuity in Laguerre hypergroup

  • Larbi Rakhimi ORCID logo EMAIL logo and Radouan Daher
Published/Copyright: January 3, 2024
Journal of Applied Analysis
From the journal Journal of Applied Analysis Volume 30 Issue 1

Abstract

In this paper, we prove the Titchmarsh theorem for Laguerre hypergroup K = [ 0 , + ∞ [ × R , via moduli of continuity of higher orders.

Keywords: Laguerre hypergroup; Fourier–Laguerre transform; Titchmarsh’s theorem; modulus of continuity
MSC 2010: 26A16; 43A30; 26D10

Acknowledgements

The authors express their gratitude to the reviewer for several valuable discussions, reading the manuscript of this paper and providing suggestions on how to improve it.

References

[1] W. O. Bray, Growth and integrability of Fourier transforms on Euclidean space, J. Fourier Anal. Appl. 20 (2014), no. 6, 1234–1256. 10.1007/s00041-014-9354-1Search in Google Scholar

[2] W. O. Bray and M. A. Pinsky, Growth properties of Fourier transforms via moduli of continuity, J. Funct. Anal. 255 (2008), no. 9, 2265–2285. 10.1016/j.jfa.2008.06.017Search in Google Scholar

[3] R. Daher, A. Fernandez and J. E. Restrepo, Characterising extended Lipschitz type conditions with moduli of continuity, Results Math. 76 (2021), no. 3, Paper No. 125. 10.1007/s00025-021-01433-2Search in Google Scholar

[4] S. El Ouadih and R. Daher, Lipschitz conditions in Damek-Ricci spaces, C. R. Math. Acad. Sci. Paris 359 (2021), 675–685. 10.5802/crmath.211Search in Google Scholar

[5] A. Fernandez, J. E. Restrepo and D. Suragan, Lipschitz and Fourier type conditions with moduli of continuity in rank 1 symmetric spaces, Monatsh. Math. 197 (2022), no. 2, 353–364. 10.1007/s00605-021-01621-wSearch in Google Scholar

[6] T. Jordão, Decay of Fourier transforms and generalized Besov spaces, Constr. Math. Anal. 3 (2020), no. 1, 20–35. 10.33205/cma.646557Search in Google Scholar

[7] V. Kokilashvili, A. Meskhi, H. Rafeiro and S. Samko, Integral Operators in Non-Standard Function Spaces. Vol. 1: Variable Exponent Lebesgue and Amalgam Spaces, Oper. Theory Adv. Appl. 248, Birkhäuser/Springer, Cham, 2016. 10.1007/978-3-319-21015-5_1Search in Google Scholar

[8] M. M. Nessibi and M. Sifi, Laguerre hypergroup and limit theorem, Lie Groups and Lie Algebras Their Representations, Generalizations and Applications, Kluwer Academic, Dordrecht (1998), 133–145. 10.1007/978-94-011-5258-7_9Search in Google Scholar

[9] M. M. Nessibi and K. Trimèche, Inversion of the Radon transform on the Laguerre hypergroup by using generalized wavelets, J. Math. Anal. Appl. 208 (1997), no. 2, 337–363. 10.1006/jmaa.1997.5299Search in Google Scholar

[10] L. Rakhimi and R. Daher, Equivalence of K-functionals and modulus of smoothness for Laguerre type operator, J. Pseudo-Differ. Oper. Appl. 12 (2021), no. 4, Paper No. 53. 10.1007/s11868-021-00424-9Search in Google Scholar

[11] L. Rakhimi and R. Daher, Boas-type theorems for Laguerre type operator, J. Pseudo-Differ. Oper. Appl. 13 (2022), no. 3, Paper No. 42. 10.1007/s11868-022-00472-9Search in Google Scholar

[12] L. Rakhimi and R. Daher, An analog of Titchmarsh’s theorem for the Laguerre-Bessel transform, Arab. J. Math. Sci. (2023), 10.1108/AJMS-04-2022-0101. 10.1108/AJMS-04-2022-0101Search in Google Scholar

[13] E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, 2nd ed., Chelsea, New York, 1948. Search in Google Scholar

[14] M. S. Younis, Fourier transforms of Dini–Lipschitz functions on Vilenkin groups, Internat. J. Math. Math. Sci. 15 (1992), no. 3, 609–612. 10.1155/S0161171292000784Search in Google Scholar
Received: 2023-04-04
Revised: 2023-11-06
Accepted: 2023-11-08
Published Online: 2024-01-03
Published in Print: 2024-06-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. New analytical technique to solve fractional-order Sharma–Tasso–Olver differential equation using Caputo and Atangana–Baleanu derivative operators
  3. Nondensely defined partial neutral functional integrodifferential equations with infinite delay under the light of integrated resolvent operators
  4. On the number of limit cycles coming from a uniform isochronous center with continuous and discontinuous quartic perturbations
  5. On ℐ2(𝒮θ p,r )-summability of double sequences in neutrosophic normed spaces
  6. Earthquake convexity and some new related inequalities
  7. On λ-statistically φ-convergence
  8. Multivalued relation-theoretic weak contractions and applications
  9. Titchmarsh’s theorem with moduli of continuity in Laguerre hypergroup
  10. Application of Touchard wavelet to simulate numerical solutions to fractional pantograph differential equations
  11. Existence and linear independence theorem for linear fractional differential equations with constant coefficients
  12. A study on δ‐ℐ‐compactness in a mixed fuzzy ideal topological space
  13. Lie symmetry, exact solutions and conservation laws of time fractional Black–Scholes equation derived by the fractional Brownian motion
  14. On statistical convergence of order α of sequences in gradual normed linear spaces
  15. Uniqueness of meromorphic functions and their powers in set sharing
  16. Multiplicative 𝔪-metric space, fixed point theorems with applications in multiplicative integrals equation and numerical results
  17. Note on bounds on the coefficients of a subclass of m-fold symmetric bi-univalent functions
  18. Multiple solitons, periodic solutions and other exact solutions of a generalized extended (2 + 1)-dimensional Kadomstev--Petviashvili equation
https://doi.org/10.1515/jaa-2023-0035
Keywords for this article
Laguerre hypergroup; Fourier–Laguerre transform; Titchmarsh’s theorem; modulus of continuity

Articles in the same Issue

  1. Frontmatter
  2. New analytical technique to solve fractional-order Sharma–Tasso–Olver differential equation using Caputo and Atangana–Baleanu derivative operators
  3. Nondensely defined partial neutral functional integrodifferential equations with infinite delay under the light of integrated resolvent operators
  4. On the number of limit cycles coming from a uniform isochronous center with continuous and discontinuous quartic perturbations
  5. On ℐ2(𝒮θ p,r )-summability of double sequences in neutrosophic normed spaces
  6. Earthquake convexity and some new related inequalities
  7. On λ-statistically φ-convergence
  8. Multivalued relation-theoretic weak contractions and applications
  9. Titchmarsh’s theorem with moduli of continuity in Laguerre hypergroup
  10. Application of Touchard wavelet to simulate numerical solutions to fractional pantograph differential equations
  11. Existence and linear independence theorem for linear fractional differential equations with constant coefficients
  12. A study on δ‐ℐ‐compactness in a mixed fuzzy ideal topological space
  13. Lie symmetry, exact solutions and conservation laws of time fractional Black–Scholes equation derived by the fractional Brownian motion
  14. On statistical convergence of order α of sequences in gradual normed linear spaces
  15. Uniqueness of meromorphic functions and their powers in set sharing
  16. Multiplicative 𝔪-metric space, fixed point theorems with applications in multiplicative integrals equation and numerical results
  17. Note on bounds on the coefficients of a subclass of m-fold symmetric bi-univalent functions
  18. Multiple solitons, periodic solutions and other exact solutions of a generalized extended (2 + 1)-dimensional Kadomstev--Petviashvili equation
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 21.11.2025 from https://www.degruyterbrill.com/document/doi/10.1515/jaa-2023-0035/html
Scroll to top button