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Discrete Fourier-Jacobi Transform and Generalized Lipschitz Classes

Abstract

In this paper, we use the methods of Fourier-Jacobi harmonic analysis to generalize Boas-type results. We give necessary and sufficient conditions in terms of the Fourier-Jacobi coefficients of a function f in order to ensure that it belongs either to one of the generalized Lipschitz classes \({H}_{\alpha }^{m}\) and \({h}_{\alpha }^{m}\) for α > 0.

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References

  1. Askey, R., Wainger, S.: A convolution structure for Jacobi series. Am. J. Math. 91, 463–485 (1969)

    MathSciNet  Article  Google Scholar 

  2. Boas, R.P. Jr.: Integrability Theorems for Trigonometric Transforms. Springer-Verlag, New York (1967)

    Book  Google Scholar 

  3. Erdélyi, A., Magnus, W., Oberttinger, F., Tricomi, F. G.: Higher Transcendental Functions, vol. II, Mc-Graw-Hill, New York-Toronto-London, 1953, Russian transl., Nauka, Moscow (1974)

  4. Moricz, F.: Absolutely convergent Fourier integrals and classical function spaces. Arch. Math. 91(1), 49–62 (2008)

    MathSciNet  Article  Google Scholar 

  5. Moricz, F.: Absolutely convergent Fourier series and function classes. J. Math. Anal. Appl. 324(2), 1168–1177 (2006)

    MathSciNet  Article  Google Scholar 

  6. Moricz, F.: Higher order Lipschitz classes of functions and absolutely convergent Fourier series. Acta Math. Hungar. 120(4), 355–366 (2008)

    MathSciNet  Article  Google Scholar 

  7. Moricz, F.: Absolutely convergent Fourier series, classical function spaces and Paley’s theorem. Anal. Math. 34(4), 261–276 (2008)

    MathSciNet  Article  Google Scholar 

  8. El Hamma, M., Daher, R.: Equivalence of K-functionalsand modulus of smoothness constructed by generalized Jacobi transform. Integral Transforms Spec. Funct. 30(12), 1018–1024 (2019)

    MathSciNet  Article  Google Scholar 

  9. Platonov, S.S.: Fourier–jacobi harmonic analysis and some problems of approximation of functions on the half-axis in l2 metric: Nikol’skii–besov type function spaces. Integral Transforms Spec Funct. 31(4), 281–298 (2020)

    MathSciNet  Article  Google Scholar 

  10. Platonov, S.S.: Fourier-jacobi harmonic analysis and approximation of functions Izvestiya. Mathematics 78(1), 106–153 (2014)

    MathSciNet  MATH  Google Scholar 

  11. Szegö, G.: Orthogonal Polynomials. Am. Math. Soc. Colloq. Publ., vol. 23, Am. Math. Soc., Providence, RI, 1959, Russian transl., Fizmatgiz, Moscow (1962)

  12. Elgargati, A., Loualid, E.M., Daher, R.: Generalization of Titchmarsh theorem in the deformed Hankel setting. Ann. Univ. Ferrara 67, 243–252 (2021). https://doi.org/10.1007/s11565-021-00379-1

    MathSciNet  Article  MATH  Google Scholar 

  13. Achak, A., Daher, R., Dhaouadi, L., et al.: An analog of Titchmarsh’s theorem for the q-Bessel transform. Ann. Univ. Ferrara 65, 1–13 (2019). https://doi.org/10.1007/s11565-018-0309-3

    MathSciNet  Article  MATH  Google Scholar 

  14. Volosivets, S.S.: Fourier transforms and generalized Lipschitz classes in uniform metric. J. Math. Anal. Appl. 383, 344–352 (2011)

    MathSciNet  Article  Google Scholar 

  15. Volosivets, S.S.: Multiple fourier coefficients and generalized Lipschitz classes in uniform metric. J. Math. Anal. Appl. https://doi.org/10.1016/j.jmaa.2015.02.011 (2015)

  16. Berkak, E.M., Loualid, E.M., Daher, R.: Boas-type theorems for the q-Bessel Fourier transform. Anal. Math. Phys. 11, 102 (2021)

    MathSciNet  Article  Google Scholar 

  17. Loualid, E.M., Elgargati, A., Daher, R.: Quaternion Fourier transform and generalized Lipschitz classes. Adv. Appl. Clifford Algebras 31(1), Paper No. 14 15 (2021)

  18. Loualid, E.M., Elgargati, A., Berkak, E.M., Daher, R.: Boas-type theorems for the Bessel transform. Rev. R. Acad. Cienc. Exactas fís. Nat. Ser. A Mat. RACSAM 115(3), Paper No. 141 12 pp (2021)

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Correspondence to El Mehdi Loualid.

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Loualid, E.M., Elgargati, A. & Daher, R. Discrete Fourier-Jacobi Transform and Generalized Lipschitz Classes. Acta Math Vietnam (2022). https://doi.org/10.1007/s40306-022-00478-x

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  • DOI: https://doi.org/10.1007/s40306-022-00478-x

Keywords

  • Discrete Fourier-Jacobi transform
  • Boas-type theorems
  • Generalized Lipschitz classes
  • Generalized Zygmund classes

Mathematics Subject Classification (2010)

  • 42B35
  • 42A38
  • 26A16
  • 42A16