Uniqueness of the optimal nodes of quadrature formulae
Author:
Borislav D. Bojanov
Journal:
Math. Comp. 36 (1981), 525546
MSC:
Primary 65D30; Secondary 41A55
MathSciNet review:
606511
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Abstract: We prove the uniqueness of the quadrature formula with minimal error in the space , of periodic differentiable functions among all quadratures with n free nodes , , of fixed multiplicities , respectively. As a corollary, we get that the equidistant nodes are optimal in for if .
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 N. I. Ahiezer, Lectures on Approximation Theory, "Nauka", Moscow, 1965. MR 0188672 (32:6108)
 [2]
 R. B. Barrar & H. L. Loeb, "On monosplines with odd multiplicities of least norm," J. Analyse Math., v. 33, 1978, pp. 1238. MR 516041 (80j:41013)
 [3]
 D. Barrow, "On multiple node Gaussian quadrature formulae," Math. Comp., v. 32, 1978, pp. 431439. MR 482257 (80j:41045)
 [4]
 R. Bellman, "On the positivity of determinants with dominant main diagonal," J. Math. Anal. Appl., v. 59, 1977, p. 210. MR 0441991 (56:380)
 [5]
 B. D. Bojanov, "Existence and characterization of monosplines of least deviation," Constructive Function Theory '77, Sofia, 1980, pp. 249268. BAN.
 [6]
 B. D. Bojanov, "Uniqueness of the monosplines of least deviation," Numerische Integration, ISNM 45, BirkhäuserVerlag, Basel, 1979, pp. 6797. MR 561282 (81f:65009)
 [7]
 L. Čakalov, "On a representation of Newton's quotients in the interpolation theory and its applications," Annuaire Univ. Sofia Fac. Math. Méc., v. 34, 1938, pp. 353405.
 [8]
 K. Jetter & G. Lange, "Die Eindeutigkeit optimaler polynomialer Monosplines," Math. Z., v. 158, 1978, pp. 2334. MR 0467094 (57:6961)
 [9]
 R. S. Johnson, "On monosplines of least deviation," Trans. Amer. Math. Soc., v. 96, 1960, pp. 458477. MR 0122938 (23:A270)
 [10]
 A. A. Ligun, "Exact inequalities for spline functions and best quadrature formulae for certain classes of functions," Mat. Zametki, v. 19, 1979, pp. 913926. MR 0427907 (55:937)
 [11]
 N. E. Lušpai, "Best quadrature formulae for classes of differentiable periodic functions," Mat. Zametki, v. 6, 1969, pp. 475482.
 [12]
 N. E. Lušpai, "Optimal quadrature formulae for classes of functions with an integrable rth derivative," Anal. Math., v. 5, 1979, pp. 6788. MR 535497 (80j:41048)
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 C. Micchelli, "The fundamental theorem of algebra for monosplines with multiplicities," Linear Operators and Approximation, ISNM v. 20, BirkhäuserVerlag, Basel, 1972, pp. 419430. MR 0393951 (52:14758)
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 V. P. Motorniĭ, "On the best quadrature formula of the form for certain classes of periodic differentiable functions," Izv. Akad. Nauk SSSR Ser. Mat., v. 38, 1974, pp. 583614. MR 0390610 (52:11435)
 [15]
 J. M. Ortega & W. C. Rhetnboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. MR 0273810 (42:8686)
 [16]
 L. L. Schumaker, "Zeros of spline functions and applications," J. Approximation Theory, v. 18, 1976, pp. 152168. MR 0430611 (55:3616)
 [17]
 J. T. Schwartz, Nonlinear Functional Analysis, Gordon and Breach, New York, 1969. MR 0433481 (55:6457)
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 A. A. Žensykbaev, "On the best quadrature formula on the class ," Dokl. Akad. Nauk SSSR, v. 227, 1976, pp. 277279. MR 0405816 (53:9608)
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 A. A. Žensykbaev, "Best quadrature formula for the class ," Anal. Math., v. 3, 1977, pp. 8393. MR 0447924 (56:6234)
 [20]
 A. A. Žensykbaev, "Best quadrature formula for certain classes of periodic functions," Izv. Akad. Nauk SSSR Ser. Mat., v. 41, 1977, pp. 11101124. MR 0471271 (57:11008)
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 A. A. Zensykbaev, "Characteristic properties of the best quadrature formulae," Sibirsk. Mat. Ž., v. 20, 1979, pp. 4968. MR 523136 (80f:41022)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198106065114
PII:
S 00255718(1981)06065114
Article copyright:
© Copyright 1981
American Mathematical Society
