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Bernstein’s inequalities and Jackson’s inverse theorems in the Laguerre hypergroup

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Abstract

In this research, we investigate some approximation theorems for \( L^{2} \)-space on the Laguerre hypergroup which is the fundamental manifold of the radial functions space for the Heisenberg group. An analogue of Bernstein’s theorem is shown. Some inverse theorems of Jackson–Stechkin in terms of best approximations for the moduli of smoothness defined using generalized translation operators on the Laguerre hypergroup are obtained. To prove these theorems, we use functions of bounded spectrum as a tool of approximation.

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References

  1. Akhiezer, N.I.: Lectures on the Theory of Approximation. Ungar, New York (1956)

    Google Scholar 

  2. Bernstein, S.N.: On the best approximation of continuous functions by polynomials of given degree, 1912, In: Collected Works 1, Acad. Nauk SSSR, Moscow, 11-104 1952 (in Russian)

  3. Bloom, W., Heyer, H.: Harmonic Analysis of Probability Measures on Hypergroups, de Gruyter Stud. Math., 20, Berlin/New York (1994)

  4. Daher, R., Tyr, O.: Modulus of smoothness and theorems concerning approximation in the space \(L^{2}_{q,\alpha }(\mathbb{R}_{q})\) with power weight. Mediterr. J. Math. 18(69), 1–16 (2021)

    Google Scholar 

  5. Faraut, J., Harzallah, K.: Deux cours d’Analyse Harmonique. In: Ecole d’été d’Analyse Harmonique de Tunis. Birkhäuser, Boston (1984)

  6. Jackson, D.: Über die Genauigkeit der Annäherung stetiger Funktionen durch ganze rationale Funktionen gegebenen Grades und trigonometrische Summen gegebener Ordnung. Göttingen, Thesis (1911)

    Google Scholar 

  7. Lebedev, N.N.: Special Functions and Their Applications. Dover Publications, New York (1972)

    Google Scholar 

  8. Li, I.P, Su, C.M., Ivanov, V.I.: Some Problems of Approximation Theory in the Spaces \( L_{p} \) on the Line with Power Weight. Mat. Zametki 90(3), 362–383 (2011); English transl., Math. Notes 90(3), 344–364 (2011)

  9. Negzaoui, S.: Lipschitz conditions in Laguerre hypergroup. Mediterr. J. Math. 14(191), 1–12 (2017)

    MathSciNet  Google Scholar 

  10. Nessibi, M.M., Selmi, B.: A Wiener-Tauberian and a Pompeiu type theorems on the Laguerre hypergroup. J. Math. Anal. Appl. 351, 232–243 (2009)

    Article  MathSciNet  Google Scholar 

  11. Nessibi, M.M., Sifi, M.: Laguerre hypergroup and limit theorem. In: Komrakov, B.P., Krasilshchink, I.S., Litvinov, G.L., Sossink, A.B. (eds.) Lie Groups and Lie Algebras Their Representations, Generalizations and Applications, pp. 133–145. Kluwer Academic Publishers, Dordrecht (1998)

    Google Scholar 

  12. Nessibi, M.M., Trimèche, K.: Inversion of the Radon Transform on the Laguerre hypergroup by using generalized wavelets. J. Math. Anal. Appl. 208, 337–363 (1997)

    Article  MathSciNet  Google Scholar 

  13. Platonov, S.S.: Bessel generalized translations and some problems of approximation theory for functions on the half-line. Sib. Math. J. 50(1), 123–140 (2009)

    Article  MathSciNet  Google Scholar 

  14. Platonov, S.S.: Fourier-Jacobi harmonic analysis and some problems of approximation of functions on the half-axis in \( L^{2} \) metric: Jackson’s type direct theorems. Integral Transform. Spec. Funct. 30(4), 264–281 (2019)

    Article  Google Scholar 

  15. Stechkin, S.B.: On the order of best approximation of continuous functions. Izv. Akad. Nauk SSSR Ser. Mat. 15(3), 219–242 (1951)

    MathSciNet  Google Scholar 

  16. Szego, G.: Orthogonal Polynomials. American Mathematical Society, Providence, RI (1939)

    Google Scholar 

  17. Timan, A.F.: Theory of Approximation of Functions of a Real Variable. Fizmatgiz, Moscow (1960). (English transl., Pergamon Press, Oxford-New York (1963))

  18. Timan, A.F., Timan, M.F.: Generalized modulus of continuity and best approximation in the mean. Dokl. Akad. Nauk SSSR 71(1), 17–20 (1950). (in Russian)

    MathSciNet  Google Scholar 

  19. Timan, M.F.: Approximation and Properties of Periodic Functions. Nauk. dumka, Kiev (2009). (in Russian)

    Google Scholar 

  20. Tyr, O., Daher, R.: Abilov’s Estimates for the Clifford-Fourier Transform in Real Clifford Algebras Analysis. Ann. Univ, Ferrara (2022)

  21. Tyr, O., Daher, R.: A note of some approximation theorems of functions on the Laguerre hypergroup. Filomat 36(7), 1959–1974 (2023)

    Article  MathSciNet  Google Scholar 

  22. Tyr, O., Daher, R., El Ouadih, S., El Fourchi, O.: On the Jackson-type inequalities in approximation theory connected to the \( q \)-Dunkl operators in the weighted space \(L^{2}_{q,\alpha }(\mathbb{R} _{q},|x|^{2\alpha +1}d_{q}x)\). Bol. Soc. Mat. Mex. 27(51), 1–21 (2021)

    Google Scholar 

  23. Tyr, O., Daher, R.: Jackson’s inequalities in Laguerre hypergroup. J. Pseudo-Differ. Oper. Appl. 13, 54 (2022). https://doi.org/10.1007/s11868-022-00487-2

    Article  MathSciNet  Google Scholar 

  24. Tyr, O., Daher, R.: Jackson’s inequalities in Mellin’s analysis. Ann. Univ. Ferrara (2023). https://doi.org/10.1007/s11565-023-00462-9

    Article  Google Scholar 

  25. Tyr, O., Daher, R.: On the Jackson-Stechkin theorems for the best approximations of functions in Clifford algebras. Adv. Appl. Clifford Algebras 33, 11 (2023). https://doi.org/10.1007/s00006-023-01261-3

    Article  MathSciNet  Google Scholar 

  26. Tyr, O., Daher, R.: Some direct and inverse theorems of approximation of functions in Jacobi-Dunkl discrete harmonic analysis. J. Math. Sci. 266, 534–553 (2022)

    Article  MathSciNet  Google Scholar 

  27. Zygmund, A.: On the continuity module of the sum of the series conjugate to a Fourier series. Prace Mat.-Fiz. 33, 25–132 (1924). (in Polish)

    Google Scholar 

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  1. Laboratory of Fundamental and Applied Mathematics, Department of Mathematics, Faculty of Sciences Aïn Chock, University Hassan II, Casablanca, Morocco

    Othman Tyr

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Tyr, O. Bernstein’s inequalities and Jackson’s inverse theorems in the Laguerre hypergroup. Anal.Math.Phys. 14, 11 (2024). https://doi.org/10.1007/s13324-023-00868-w

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