The relationship between the zeros of best approximations and differentiability
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- by Peter B. Borwein PDF
- Proc. Amer. Math. Soc. 92 (1984), 528-532 Request permission
Abstract:
We examine the relationship between the analytic properties of continuous functions on $[ - 1,1]$ and the location of the roots of the sequence of best polynomial approximations. We show that if the approximants have no zeros in a certain ellipse then the function being approximated must be analytic in this ellipse. We also show that the rate at which the zeros of the $n$th approximant tend to the interval $[ - 1,1]$ determines the global differentiability of the function under consideration.References
- E. W. Cheney, Introduction to approximation theory, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0222517 A. A. Gončar, On uniform convergence of Padé approximants, Math. USSR Sb. 44 (1983), 539 559.
- G. G. Lorentz, Approximation of functions, Holt, Rinehart and Winston, New York-Chicago, Ill.-Toronto, Ont., 1966. MR 0213785
- Maynard Thompson, Approximation by polynomials whose zeros lie on a curve, Duke Math. J. 31 (1964), 255–265. MR 160916 J. L. Walsh, The analogue for maximally convergent polynomials of Jentzsch’s theorem, Duke Math. J. 26 (1959), 605-616.
Additional Information
- © Copyright 1984 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 92 (1984), 528-532
- MSC: Primary 41A50; Secondary 41A10
- DOI: https://doi.org/10.1090/S0002-9939-1984-0760939-5
- MathSciNet review: 760939