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Properties of a Family of Operators

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Abstract

In this paper we define a family of new nonsymmetric operators of generalized translation and describe their properties. For each of these operators, we introduce a generalized modulus of smoothness, for which the direct and inverse theorems of approximation theory are given.

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REFERENCES

  1. Z. Ditzian and V. Totik, Moduli of Smoothness, Springer, New York, 1987.

    Google Scholar 

  2. J. Löfstrëm and J. Peetre, “Approximation theorems connected with generalized translations,” Math. Ann., 181 (1969), 255–268.

    Google Scholar 

  3. S. Pawelke, “Ein Satz von Jackenschen Typ für algebraische Polynome,” Acta Sci. Math., 33 (1972), no. 3-4, 323–336.

    Google Scholar 

  4. P. Butzer, R. Stens, and M. Wehrens, “Higher order moduli of continuity based on the Jacobi translation operator and best approximation,” C. R. Math. Rend. Acad. Sci. Canada (1980), no. 2, 83–87.

    Google Scholar 

  5. M. K. Potapov, “On the structural and constructive characteristics of some classes of functions,” Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.], 131 (1974), 211–231.

    Google Scholar 

  6. M. K. Potapov, “On the structural characteristics of classes of functions with a given order of best Approximation,” Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.], 134 (1975), 260–277.

    Google Scholar 

  7. M. K. Potapov, “On the approximation by algebraic polynomials in the integral metric with Jacobi weight,” Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.] (1983), no. 3, 43–52.

    Google Scholar 

  8. M. K. Potapov and V. M. Fedorov, “On the Jackson theorems for the generalized modulus of smoothness,” Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.], 172 (1985), 291–298.

    Google Scholar 

  9. M. K. Potapov, “The operators of generalized translation in approximation theory,” in: Proceedings of the Second Mathematical Conference in Pristina, Pristina, 1997, pp. 27–36.

  10. M. K. Potapov, “On the approximation of functions characterized by a single nonsymmetric operator of generalized translation” Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.], 227 (1999), 243–259.

    Google Scholar 

  11. N. Ya. Vilenkin, Special functions and representation theory for groups [in Russian], Nauka, Moscow, 1965.

    Google Scholar 

  12. H. Bateman and A. Erdélyi, Higher Transcendental Functions, vol. 2, McGraw-Hill, New York-Toronto-London, 1953.

    Google Scholar 

  13. B. A. Khalilova, “On certain estimates for polynomials,” Izv. Akad. Nauk Azerbaidzhan. SSR Ser. Fiz.-Tekhn. (1974), no. 2, 46–55

    Google Scholar 

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Potapov, M.K. Properties of a Family of Operators. Mathematical Notes 69, 373–386 (2001). https://doi.org/10.1023/A:1010287509486

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