Advertisement

Approximation Theorems for Positive Linear Operators Associated with Hermite and Laguerre Polynomials

  • Grażyna KrechEmail author
Chapter

Abstract

We present some results regarding positive linear operators associated with Hermite and Laguerre expansions. We consider Poisson type integrals for orthogonal expansions and discuss their approximation properties in the \(L^p\) space. We also investigate operators of Szász–Mirakjan type defined via Hermite polynomials. We give the rates of convergence by means of the modulus of continuity and moduli of smoothness. We present Voronovskaya type theorems for these operators and discuss boundary value problems for Poisson integrals. We also consider some combinations of the operators presented here, study their approximation errors and prove the Voronovskaya type formula.

Keywords

Poisson integrals Linear operators Hermite and Laguerre expansions Approximation order Voronovskaya type theorem 

AMS 2010

Primary 41A25 Secondary 41A36 

References

  1. 1.
    P. Appell, J. Kampé de Fériet, Fonctions Hypergéométriques et Hypersphériques; Polynômes d’Hermite (Gauthier-Villars, Paris, 1926)zbMATHGoogle Scholar
  2. 2.
    J. Gosselin, K. Stempak, Conjugate expansions for Hermite functions. Illinois J. Math. 38, 177–197 (1994)Google Scholar
  3. 3.
    H. Johnen, K. Scherer, On equivalence of \(K\) functional and moduli of continuity and some applications, Constructive Theory of Functions of Several Variables, Proceedings of Conference (Oberwolfach, 1976), vol. 571, Lecture Notes in Mathematics (Springer, Berlin), pp. 119–140CrossRefGoogle Scholar
  4. 4.
    G. Krech, On some approximation theorems for the Poisson integral for Hermite expansions. Anal. Math. 40, 133–145 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    G. Krech, A note on some positive linear operators associated with the Hermite polynomials. Carpathian J. Math. 32, 71–77 (2016)MathSciNetzbMATHGoogle Scholar
  6. 6.
    G.M. Mirakjan, Approximation of continuous functions with the aid of polynomials of the form \(e^{-nx} \sum _{k=0}^{m_{n}}c_{k, n}x^{k}\). Comptes rendus de l’Académie des sciences de l’URSS 31, 201–205 (1941)Google Scholar
  7. 7.
    B. Muckenhoupt, Poisson integrals for Hermite and Laguerre expansions. Trans. Am. Math. Soc. 139, 231–242 (1969)MathSciNetCrossRefGoogle Scholar
  8. 8.
    M.A. Özarslan, O. Duman, Approximation properties of poisson integrals for orthogonal expansions. Taiwan. J. Math 12(5), 1147–1163 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    O. Szász, Generalization of S. Bernstein’s polynomials to the infinite interval. J. Res. Natl. Bur. Stand. 45(3), 239–245 (1950)MathSciNetCrossRefGoogle Scholar
  10. 10.
    G. Toczek, E. Wachnicki, On the rate of convergence and the Voronovskaya theorem for the poisson integrals for Hermite and Laguerre expansions. J. Approx. Theory 116, 113–125 (2002)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Faculty of Applied MathematicsAGH University of Science and TechnologyKrakówPoland

Personalised recommendations