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Available in print format 		Bulletin of the American Mathematical Society
Bulletin of the American Mathematical Society
ISSN 1088-9485(e) ISSN 0273-0979(p)

     

Müntz spaces and Remez inequalities

Author(s): Peter Borwein; Tamás Erdélyi
Journal: Bull. Amer. Math. Soc. 32 (1995), 38-42.
MathSciNet review: 1273395
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Abstract | References | Additional information

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:

Bibliography

[1]
J. M. Anderson, Müntz-Szász type approximation and the angular growth of lacunary integral functions, Trans. Amer. Math. Soc. 169 (1972), 237-248. MR 0310259 (46:9360)

[2]
J. Bak and D. J. Newman, Rational combinations of $ {x^{{\lambda _k}}}$, $ {\lambda _k} \geq 0$, are always dense in $ C[0,1]$, J. Approx. Theory 23 (1978), 155-157. MR 0487180 (58:6840)

[3]
S. N. Bernstein, Collected works: Vol 1. Constructive theory of functions (1905-1930), English translation, Atomic Energy Commission, Springfield, Virginia, 1958.

[4]
R. P. Boas, Entire functions, Academic Press, New York, 1954. MR 0068627 (16:914f)

[5]
P. B. Borwein, Zeros of Chebyshev polynomials in Markov systems, J. Approx. Theory 63 (1990), 56-64. MR 1074081 (92a:41002)

[6]
-, Variations on Müntz's theme, Canad. Math. Bull. 34 (1991), 305-310. MR 1127751 (92i:41010)

[7]
P. B. Borwein and T. Erdélyi, Notes on lacunary Müntz polynomials, Israel J. Math. 76 (1991), 183-192. MR 1177339 (93h:41014)

[8]
-, Lacunary Müntz systems, Proc. Edinburgh Math. Soc. (2) 36 (1993), 361-374. MR 1242750 (94i:41023)

[9]
-, Polynomials and polynomials inequalities, Springer-Verlag, New York (to appear). MR 1367960 (97e:41001)

[10]
P. B. Borwein, T. Erdélyi, and J. Zhang, Müntz systems and orthogonal Müntz-Legendre polynomials, Trans. Amer. Math. Soc. 342 (1994), 523-542. MR 1227091 (94f:42026)

[11]
E. W. Cheney, Introduction to approximation theory, McGraw-Hill, New York, 1966. MR 0222517 (36:5568)

[12]
J. A. Clarkson and P. Erdős, Approximation by polynomials, Duke Math. J. 10 (1943), 5-11. MR 0007813 (4:196e)

[13]
T. Erdélyi, Remez-type inequalities on the size of generalized polynomials, J. London Math. Soc. (2) 45 (1992), 255-264. MR 1171553 (93e:41022)

[14]
-, Remez type inequalities and their applications, J. Comput. Appl. Math. 47 (1993), 167-210. MR 1237312 (94m:26003)

[15]
G. Freud, Orthogonal polynomials, Pergamon Press, Oxford, 1971.

[16]
W. A. J. Luxemburg and J. Korevaar, Entire functions and Müntz-Szász type approximation, Trans. Amer. Math. Soc. 157 (1971), 23-37. MR 0281929 (43:7643)

[17]
C. Müntz, Über den Approximationsatz von Weierstrass, H. A. Schwartz Festschrift, Berlin, 1914.

[18]
D. J. Newman, Derivative bounds for Müntz polynomials, J. Approx. Theory 18 (1976), 360-362. MR 0430604 (55:3609)

[19]
-, Approximation with rational functions, CBMS Regional Conf. Ser. in Math., vol. 41, Amer. Math. Soc., Providence, RI, 1978.

[20]
E. J. Remez, Sur une propriété des polynômes de Tchebyscheff, Comm. Inst. Sci. Kharkow 13 (1936), 93-95.

[21]
T. J. Rivlin, Chebyshev polynomials, 2nd ed., Wiley, New York, 1990. MR 1060735 (92a:41016)

[22]
L. Schwartz, Etude des sommes d'exponentielles, Hermann, Paris, 1959. MR 0106383 (21:5116)

[23]
G. Somorjai, A Müntz-type problem for rational approximation, Acta. Math. Hungar. 27 (1976), 197-199. MR 0430617 (55:3622)

[24]
O. Szász, Über die Approximation steliger Funktionen durch lineare Aggregate von Potenzen, Math. Ann. 77 (1916), 482-496. MR 1511875

[25]
G. Szegő, On the density of quotients of lacunary polynomials, Acta Math. Hungar. 30 (1922), 149-154. MR 0454462 (56:12713)

[26]
A. K. Taslakyan, Some properties of Legendre quasi-polynomials with respect to a Müntz system, Mathematics 2 (1984), 179-189. (Russian, Armenian summary) MR 875260 (88e:33008)

[27]
M. von Golitschek, A short proof of Müntz Theorem, J. Approx. Theory 39 (1983), 394-395. MR 723231 (85b:41005)

Additional Information:

DOI: 10.1090/S0273-0979-1995-00553-7
PII: S 0273-0979(1995)00553-7
Keywords: Remez inequality, Müntz's Theorem, Müntz spaces, Dirichlet sums, density
Copyright of article: Copyright 1995, American Mathematical Society




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