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Some examples in approximation on the unit disk by reciprocals of polynomials

Part of the Lecture Notes in Mathematics book series (LNM,volume 1287)

Abstract

This paper is a continuation of the authors’ study of approximation by reciprocals of polynomials. A Jackson-type theorem for such approximants is established for a certain class of functions f analytic and nonzero in the disk |z|<1 and continuous on |z|≤1. Furthermore, we obtain the sharp degree of convergence for reciprocal polynomial approximation on |z|≤1 to functions f that are analytic on |z|≤1, nonzero in |z|<1, and vanish somewhere on |z|=1.

AMS subject classification 41A20

  • 41A17

The research of this author was conducted while visiting the Institute for Constructive Mathematics at the University of South Florida.

The research of this author was supported, in part, by the National Science Foundation.

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References

  1. Levin A.L. and Saff E.B., Degree of approximation of real functions by reciprocals of real and complex polynomials, SIAM J. Math. Analysis (to appear).

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  2. Levin A.L. and Saff E.B., Jackson type theorems in approximation by reciprocals of polynomials, Rocky Mountain Journal of Mathematics (to appear).

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  3. Lorentz G.G., Approximation of functions, Holt, Rinehart and Winston, New York, 1966.

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  4. Walsh J.L., On approximation to an analytic function by rational functions of best approximation, Math. Z., 38(1934), 163–176.

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© 1987 Springer-Verlag

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Levin, A.L., Saff, E.B. (1987). Some examples in approximation on the unit disk by reciprocals of polynomials. In: Saff, E.B. (eds) Approximation Theory, Tampa. Lecture Notes in Mathematics, vol 1287. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0078898

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  • DOI: https://doi.org/10.1007/BFb0078898

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-18500-0

  • Online ISBN: 978-3-540-47991-8

  • eBook Packages: Springer Book Archive