Müntz spaces and Remez inequalities
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- by Peter Borwein and Tamás Erdélyi PDF
- Bull. Amer. Math. Soc. 32 (1995), 38-42 Request permission
Abstract:
Two relatively long-standing conjectures concerning Müntz polynomials are resolved. The central tool is a bounded Remez type inequality for non-dense Müntz spaces.References
- J. M. Anderson, Müntz-Szasz type approximation and the angular growth of lacunary integral functions, Trans. Amer. Math. Soc. 169 (1972), 237–248. MR 310259, DOI 10.1090/S0002-9947-1972-0310259-4
- Joseph Bak and Donald J. Newman, Rational combinations of $x^{\lambda k}$, $\lambda _{k}\geq 0$ are always dense in $C[0, 1]$, J. Approximation Theory 23 (1978), no. 2, 155–157. MR 487180, DOI 10.1016/0021-9045(78)90101-6 S. N. Bernstein, Collected works: Vol 1. Constructive theory of functions (1905-1930), English translation, Atomic Energy Commission, Springfield, Virginia, 1958.
- Ralph Philip Boas Jr., Entire functions, Academic Press, Inc., New York, 1954. MR 0068627
- Peter Borwein, Zeros of Chebyshev polynomials in Markov systems, J. Approx. Theory 63 (1990), no. 1, 56–64. MR 1074081, DOI 10.1016/0021-9045(90)90113-5
- Peter B. Borwein, Variations on Müntz’s theme, Canad. Math. Bull. 34 (1991), no. 3, 305–310. MR 1127751, DOI 10.4153/CMB-1991-050-8
- Peter Borwein and Tamás Erdélyi, Notes on lacunary Müntz polynomials, Israel J. Math. 76 (1991), no. 1-2, 183–192. MR 1177339, DOI 10.1007/BF02782851
- Peter Borwein and Tamás Erdélyi, Lacunary Müntz systems, Proc. Edinburgh Math. Soc. (2) 36 (1993), no. 3, 361–374. MR 1242750, DOI 10.1017/S0013091500018472
- Peter Borwein and Tamás Erdélyi, Polynomials and polynomial inequalities, Graduate Texts in Mathematics, vol. 161, Springer-Verlag, New York, 1995. MR 1367960, DOI 10.1007/978-1-4612-0793-1
- Peter Borwein, Tamás Erdélyi, and John Zhang, Müntz systems and orthogonal Müntz-Legendre polynomials, Trans. Amer. Math. Soc. 342 (1994), no. 2, 523–542. MR 1227091, DOI 10.1090/S0002-9947-1994-1227091-4
- E. W. Cheney, Introduction to approximation theory, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0222517
- J. A. Clarkson and P. Erdös, Approximation by polynomials, Duke Math. J. 10 (1943), 5–11. MR 7813, DOI 10.1215/S0012-7094-43-01002-6
- Tamás Erdélyi, Remez-type inequalities on the size of generalized polynomials, J. London Math. Soc. (2) 45 (1992), no. 2, 255–264. MR 1171553, DOI 10.1112/jlms/s2-45.2.255
- Tamás Erdélyi, Remez-type inequalities and their applications, J. Comput. Appl. Math. 47 (1993), no. 2, 167–209. MR 1237312, DOI 10.1016/0377-0427(93)90003-T G. Freud, Orthogonal polynomials, Pergamon Press, Oxford, 1971.
- W. A. J. Luxemburg and J. Korevaar, Entire functions and Müntz-Szász type approximation, Trans. Amer. Math. Soc. 157 (1971), 23–37. MR 281929, DOI 10.1090/S0002-9947-1971-0281929-0 C. Müntz, Über den Approximationsatz von Weierstrass, H. A. Schwartz Festschrift, Berlin, 1914.
- D. J. Newman, Derivative bounds for Müntz polynomials, J. Approximation Theory 18 (1976), no. 4, 360–362. MR 430604, DOI 10.1016/0021-9045(76)90007-1 —, Approximation with rational functions, CBMS Regional Conf. Ser. in Math., vol. 41, Amer. Math. Soc., Providence, RI, 1978. E. J. Remez, Sur une propriété des polynômes de Tchebyscheff, Comm. Inst. Sci. Kharkow 13 (1936), 93-95.
- Theodore J. Rivlin, Chebyshev polynomials, 2nd ed., Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1990. From approximation theory to algebra and number theory. MR 1060735
- Laurent Schwartz, Étude des sommes d’exponentielles. 2ième éd, Publications de l’Institut de Mathématique de l’Université de Strasbourg, V. Actualités Sci. Ind., Hermann, Paris, 1959 (French). MR 0106383
- G. Somorjai, A Müntz-type problem for rational approximation, Acta Math. Acad. Sci. Hungar. 27 (1976), no. 1-2, 197–199. MR 430617, DOI 10.1007/BF01896775
- Otto Szász, Über die Approximation stetiger Funktionen durch lineare Aggregate von Potenzen, Math. Ann. 77 (1916), no. 4, 482–496 (German). MR 1511875, DOI 10.1007/BF01456964
- G. Somorjai, On the density of quotients of lacunary polynomials, Acta Math. Acad. Sci. Hungar. 30 (1977), no. 1-2, 149–154. MR 454462, DOI 10.1007/BF01895659
- A. K. Taslakyan, Some properties of Legendre quasipolynomials with respect to a Müntz system, Mathematics, No. 2 (Russian), Erevan. Univ., Erevan, 1984, pp. 179–189 (Russian, with Armenian summary). MR 875260
- M. von Golitschek, A short proof of Müntz’s theorem, J. Approx. Theory 39 (1983), no. 4, 394–395. MR 723231, DOI 10.1016/0021-9045(83)90083-7
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 32 (1995), 38-42
- MSC: Primary 41A17
- DOI: https://doi.org/10.1090/S0273-0979-1995-00553-7
- MathSciNet review: 1273395