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)

 
 

 

On the distribution of extremal points of general Chebyshev polynomials


Authors: András Kroó and Franz Peherstorfer
Journal: Trans. Amer. Math. Soc. 329 (1992), 117-130
MSC: Primary 41A50; Secondary 41A30
DOI: https://doi.org/10.1090/S0002-9947-1992-1012514-4
MathSciNet review: 1012514
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a linear subspace $ {\mathcal{U}_n} = {\operatorname{span}}[{\varphi _1}, \ldots ,{\varphi _n}]$ in $ C[a,b]$ we introduce general Chebyshev polynomials as solutions of the minimization problem $ {\operatorname{min}_{{a_i}}}{\left\Vert {{\varphi _n} - \sum\nolimits_{i = 1}^{n - 1} {{a_i}{\varphi _i}} } \right\Vert _C}$. For such a Chebyshev polynomial we study the distribution of its extremal points (maximum and minimum points) in terms of structural and approximative properties of $ {\mathcal{U}_n}$.


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

  • [1] N. I. Achieser, Vorlesungen über Approximations theorie, Akademie Verlag, Berlin, 1953. MR 0061692 (15:867c)
  • [2] P. B. Borwein, R. Grothmann, A. Kroó and E. B. Saff, The density of alternation points in rational approximation Proc. Amer. Math. Soc. 105 (1989), 881-888. MR 948147 (89h:41035)
  • [3] M. I. Kadec, On the distribution of points of maximal deviation in the approximation of continuous functions by polynomials, Uspekhi Mat. Nauk 15 (1960), 199-202. MR 0113079 (22:3920)
  • [4] S. Karlin and W. J. Studden, Tchebysheff systems: with applications in analysis and statistics, Interscience, New York, 1966. MR 0204922 (34:4757)
  • [5] R. C. Jones and L. A. Karlovitz, Equioscillation under nonuniqueness in the approximation of continuous functions, J. Approximation Theory 3 (1970), 138-145. MR 0264303 (41:8899)
  • [6] A. Kroó, On the unicity of best Chebyshev approximation of differentiable functions, Acta Sci. Math. (Szeged) 47 (1984), 377-389. MR 783312 (86g:41046)
  • [7] A. Kroó and F. Peherstorfer, Interpolatory properties of best $ {L_1}$-approximations, Math. Z. 196 (1987), 249-257. MR 910830 (88i:41042)
  • [8] -, Interpolatory properties of best rational $ {L_1}$-approximations, Constr. Approx. 4 (1988), 97-106. MR 916092 (89c:41013)
  • [9] -, On the zeros of polynomial of minimal $ {L_p}$-norm, Proc. Amer. Math. Soc. 101 (1987), 652-656. MR 911027 (88i:41043)
  • [10] G. G. Lorentz, Distribution of alternation points in uniform polynomial approximation, Proc. Amer. Math. Soc. 92 (1984), 401-403. MR 759662 (86e:41047)
  • [11] R. Feinerman and D. Newman, Polynomial approximation, Williams-Willins, Baltimore, Md., 1974. MR 0499910 (58:17657)
  • [12] E. Passow, Alternating parity of Tchebycheff systems, J. Approximation Theory 9 (1973), 295-298. MR 0355433 (50:7907)
  • [13] A. Pinkus and Z. Ziegler, Interlacing properties of the zeros of the error functions in best $ {L^p}$-approximations, J. Approximation Theory 27 (1979), 1-18. MR 554112 (81c:41063)
  • [14] E. B. Saff, Incomplete and orthogonal polynomials, Approximation Theory IV (C. K. Chui, L. L. Schumaker, and J. D. Ward., eds.), Academic Press, New York, 1983, pp. 219-256. MR 754347 (86b:41029)
  • [15] I. Singer, Best approximation in normed linear spaces, Springer, Berlin, Heidelberg, and New York, 1970. MR 0270044 (42:4937)
  • [16] Sp. Tashev, On the distribution of points of maximal deviation for the polynomials of best Chebyshev and Hausdorff approximations, Approximation and Function Spaces (Z. Ciesielski, ed.), North-Holland, Amsterdam, 1981, pp. 791-799. MR 649477 (83c:41029)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 41A50, 41A30

Retrieve articles in all journals with MSC: 41A50, 41A30


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1012514-4
Keywords: Chebyshev polynomials, density of extremal points
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society