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On the Jackson-type inequalities in approximation theory connected to the q-Dunkl operators in the weighted space \(L^{2}_{q,\alpha }({\mathbb {R}}_{q}, |x|^{2\alpha +1}d_{q}x )\)

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Abstract

In this paper, using some elements of the q-harmonic analysis associated to the q-Dunkl operator introduced in Bettaibi and Bettaieb (Tamsui Oxf J Math Sci 25(2):117–205, 2007), for fixed \(q\in ]0,1[\), we look at problems in the theory of approximation of functions on the space \(L^{2}_{q,\alpha }({\mathbb {R}}_{q})\) with power weight, we prove analogues of direct and some inverse theorems of Jackson in terms of best approximations of functions and the moduli of smoothness of all orders constructed by the q-Dunkl generalized translations.

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Data Availability Statement

All data generated or analysed during the current study are available from the corresponding author on reasonable request.

References

  1. Akhiezer, N.I.: Lectures on Approximation Theory, 2nd ed. Nauka, Moscow (1965) (Engl. transl. of the 1st ed., (1947): Theory of Approximation, Ungar, New York, (1956)), (in Russian)

  2. Bettaibi, N., Bettaieb, R.H.: q-Analogue of the Dunkl transform on the real line. Tamsui Oxf. J. Math. Sci. 25(2), 117–205 (2007)

    MathSciNet  MATH  Google Scholar 

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

  4. Butzer, P., Nessel, R.: Fourier Analysis and Approximation. One-Dimensional Theory. Birkhäuser, Basel (1971)

  5. Daher, R., El Ouadih, S., El Hamma, M.: Direct and inverse theorems of approximation theory in \(L^{2}({\mathbb{R}}^{d}, w_{l}(x)dx)\). Matematika 33(2), 177–189 (2017)

    Article  MathSciNet  Google Scholar 

  6. Daher, R., Tyr, O.: An analog of Titchmarsh’s theorem for the q-Dunkl transform in the space \(L^{2}_{q,\alpha }({\mathbb{R}}_{q})\). J. Pseudo-Differ. Oper. Appl. (2020). https://doi.org/10.1007/s11868-020-00330-6

  7. 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)

  8. El Ouadih, S., Daher, R., Tyr, O., Saadi, F.: Equivalence of K-functionals and moduli of smoothness generated by the Beltrami–Laplace operator on the spaces \(S^{(p,q)}(\sigma ^{m-1})\). Rend. Circ. Mat. Palermo II Ser (2021)

  9. Gasper, G., Rahman, M.: Basic Hypergeometric Series, Encyclopedia of Mathematics and Its Applications, vol. 35. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  10. Jackson, F.H.: On a q-definite integrals. J. Pure Appl. Math. 41, 193–203 (1910)

    MATH  Google Scholar 

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

  12. Kac, V.G., Cheung, P.: Quantum Calculus. Universitext. Springer, New York (2002)

    Book  Google Scholar 

  13. Nikol’skii, S.M.: A generalization of an inequality of S. N. Bernstein. Dokl. Akad. Nauk. SSSR 60(9), 1507–1510 (1948). (in Russian)

    MATH  Google Scholar 

  14. Nikol’skii, S.M.: Approximation of Functions in Several Variables and Embedding Theorems. Nauka, Moscow (1977). (in Russian)

    Google Scholar 

  15. Platonov, S.S: Fourier–Jacobi harmonic analysis and approximation of functions. Izv. RAN. Ser. Mat. 78(1), 106–153 (2014)

  16. 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 

  17. Rubin, R.L.: A \(q^{2}\)-analogue operator for \(q^{2}\)-analogue Fourier analysis. J. Math. Anal. Appl. 212, 571–582 (1997)

    Article  MathSciNet  Google Scholar 

  18. Rubin, R.L.: Duhamel: solutions of non-homogenous \(q^{2}\)-analogue wave equations. Proc. Am. Math. Soc. 135(3), 777–785 (2007)

    Article  Google Scholar 

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

  20. 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)

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

  22. Weierstrass, K.: über die analytische darstellbarkeit sogenannter willkürlicher functionen einer reellen Veränderlichen, Sitzungsber. Akad. Berlin, pp. 633–639 (1885)

  23. 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)

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Acknowledgements

The authors are grateful to the referees for the useful comments and suggestions in improving the presentation of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences Aïn Chock, University Hassan II, Casablanca, Morocco

    Othman Tyr, Radouan Daher & Omar El Fourchi

  2. Laboratory MC, Polydisciplinary Faculty of Safi, University of Cadi Ayyad, Marrakech, Morocco

    Salah El Ouadih

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  1. Othman Tyr
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  2. Radouan Daher
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  3. Salah El Ouadih
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  4. Omar El Fourchi
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Correspondence to Othman Tyr.

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Tyr, O., Daher, R., Ouadih, S.E. et al. 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 (2021). https://doi.org/10.1007/s40590-021-00358-8

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