Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Interpolation polynomials which give best order of approximation among continuously differentiable functions of arbitrary fixed order on $ [-1,\,+1]$


Author: A. K. Varma
Journal: Trans. Amer. Math. Soc. 200 (1974), 419-426
MSC: Primary 41A05
DOI: https://doi.org/10.1090/S0002-9947-1974-0369999-5
MathSciNet review: 0369999
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The object of this paper is to show that there exists a polynomial $ {P_n}(x)$ of degree $ \leqslant 2n - 1$ which interpolates a given function exactly at the zeros of $ n$th Tchebycheff polynomial and for which $ \vert\vert f - {P_n}\vert\vert \leqslant {C_k}{w_k}(1/n,f)$ where $ {w_k}(1/n,f)$ is the modulus of continuity of $ f$ of $ k$th order.


References [Enhancements On Off] (What's this?)

  • [1] P. L. Butzer, On approximation theory (Proc. Conf. Oberwolfach, 1963), Birkhäuser, Basel, 1964. MR 31 #2537.
  • [2] G. Freud, Über ein Jacksonsches Interpolationsverfahren, On Approximation Theory (Proc. Conf. Oberwolfach, 1963), Birkhäuser, Basel, 1964, pp. 227-232. MR 32 #308. MR 0182826 (32:308)
  • [3] S. B. Stečkin, On the order of the best approximations of continuous functions, Izv. Akad. Nauk SSSR Ser. Mat. 15 (1951), 219-242. (Russian) MR 13, 29. MR 0041959 (13:29d)
  • [4] -, The approximation of periodic functions by Fejér sums, Trudy Mat. Inst. Steklov. 62 (1961), 48-60; English transl., Amer. Math. Soc. Transl. (2) 28 (1963), 269-282. MR 28 #5287a, b. MR 0162086 (28:5287b)
  • [5] A. K. Varma, The approximation of functions by certain trigonometric interpolation polynomials, Proc. Internat. Sympos. Austin, Texas, edited by G. G. Lorentz, 1973, pp. 511-515. MR 0333550 (48:11875)
  • [6] -, Trigonometric interpolation polynomials which gives best order of approximation among continuously differentiable functions, Proc. Internat. Sympos. Approximation Theory (Poznań, Poland, 1972) (to appear).
  • [7] A. Zygmund, The approximation of functions by typical means of their Fourier series, Duke Math. J. 12 (1945), 695-704. MR 7, 435. MR 0015539 (7:435b)
  • [8] -, Trigonometric series. Vols. 1, 2, 2nd rev. ed., Cambridge Univ. Press, New York, 1959. MR 21, #6498.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 41A05

Retrieve articles in all journals with MSC: 41A05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0369999-5
Keywords: Polynomial approximation, modulus of continuity, Hermite interpolation, typical means, Tchebycheff nodes
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society