On the distribution of extremal points of general Chebyshev polynomials
HTML articles powered by AMS MathViewer
- by András Kroó and Franz Peherstorfer PDF
- Trans. Amer. Math. Soc. 329 (1992), 117-130 Request permission
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 ${\operatorname {min}_{{a_i}}}{\left \| {{\varphi _n} - \sum \nolimits _{i = 1}^{n - 1} {{a_i}{\varphi _i}} } \right \|_C}$. 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
- N. I. Achieser, Vorlesungen über Approximationstheorie, Akademie-Verlag, Berlin, 1953 (German). MR 0061692
- P. B. Borwein, A. Kroó, R. Grothmann, and E. B. Saff, The density of alternation points in rational approximation, Proc. Amer. Math. Soc. 105 (1989), no. 4, 881–888. MR 948147, DOI 10.1090/S0002-9939-1989-0948147-0
- M. Ĭ. Kadec′, On the distribution of points of maximum deviation in the approximation of continuous functions by polynomials, Uspehi Mat. Nauk 15 (1960), no. 1 (91), 199–202 (Russian). MR 0113079
- Samuel Karlin and William J. Studden, Tchebycheff systems: With applications in analysis and statistics, Pure and Applied Mathematics, Vol. XV, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1966. MR 0204922
- R. C. Jones and L. A. Karlovitz, Equioscillation under nonuniqueness in the approximation of continuous functions, J. Approximation Theory 3 (1970), 138–145. MR 264303, DOI 10.1016/0021-9045(70)90021-3
- András Kroó, On the unicity of best Chebyshev approximation of differentiable functions, Acta Sci. Math. (Szeged) 47 (1984), no. 3-4, 377–389. MR 783312
- András Kroó and Franz Peherstorfer, Interpolatory properties of best $L_1$-approximants, Math. Z. 196 (1987), no. 2, 249–257. MR 910830, DOI 10.1007/BF01163659
- András Kroó and Franz Peherstorfer, Interpolatory properties of best rational $L_1$-approximations, Constr. Approx. 4 (1988), no. 1, 97–106. MR 916092, DOI 10.1007/BF02075450
- András Kroó and Franz Peherstorfer, On the zeros of polynomials of minimal $L_p$-norm, Proc. Amer. Math. Soc. 101 (1987), no. 4, 652–656. MR 911027, DOI 10.1090/S0002-9939-1987-0911027-9
- G. G. Lorentz, Distribution of alternation points in uniform polynomial approximation, Proc. Amer. Math. Soc. 92 (1984), no. 3, 401–403. MR 759662, DOI 10.1090/S0002-9939-1984-0759662-2
- Robert P. Feinerman and Donald J. Newman, Polynomial approximation, Williams & Wilkins Co., Baltimore, Md., 1974. MR 0499910
- Eli Passow, Alternating parity of Tchebycheff systems, J. Approximation Theory 9 (1973), 295–298. MR 355433, DOI 10.1016/0021-9045(73)90096-8
- Allan Pinkus and Zvi Ziegler, Interlacing properties of the zeros of the error functions in best $L^{p}$-approximations, J. Approx. Theory 27 (1979), no. 1, 1–18. MR 554112, DOI 10.1016/0021-9045(79)90093-5
- E. B. Saff, Incomplete and orthogonal polynomials, Approximation theory, IV (College Station, Tex., 1983) Academic Press, New York, 1983, pp. 219–256. MR 754347
- Ivan Singer, Best approximation in normed linear spaces by elements of linear subspaces, Die Grundlehren der mathematischen Wissenschaften, Band 171, Publishing House of the Academy of the Socialist Republic of Romania, Bucharest; Springer-Verlag, New York-Berlin, 1970. Translated from the Romanian by Radu Georgescu. MR 0270044
- Sp. Tashev, On the distribution of the points of maximal deviation for the polynomials of best Chebyshev and Hausdorff approximations, Approximation and function spaces (Gdańsk, 1979) North-Holland, Amsterdam-New York, 1981, pp. 791–799. MR 649477
Additional Information
- © Copyright 1992 American Mathematical Society
- 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