The relationship between the zeros of best approximations and differentiability

Author:
Peter B. Borwein

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

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Abstract: We examine the relationship between the analytic properties of continuous functions on 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 th approximant tend to the interval determines the global differentiability of the function under consideration.

**[1]**E. W. Cheney,*Introduction to approximation theory*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0222517****[2]**A. A. Gončar,*On uniform convergence of Padé approximants*, Math. USSR Sb.**44**(1983), 539 559.**[3]**G. G. Lorentz,*Approximation of functions*, Holt, Rinehart and Winston, New York-Chicago, Ill.-Toronto, Ont., 1966. MR**0213785****[4]**Maynard Thompson,*Approximation by polynomials whose zeros lie on a curve*, Duke Math. J.**31**(1964), 255–265. MR**0160916****[5]**J. L. Walsh,*The analogue for maximally convergent polynomials of Jentzsch's theorem*, Duke Math. J.**26**(1959), 605-616.

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DOI:
https://doi.org/10.1090/S0002-9939-1984-0760939-5

Article copyright:
© Copyright 1984
American Mathematical Society