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Uniqueness of the optimal nodes of quadrature formulae


Author: Borislav D. Bojanov
Journal: Math. Comp. 36 (1981), 525-546
MSC: Primary 65D30; Secondary 41A55
DOI: https://doi.org/10.1090/S0025-5718-1981-0606511-4
MathSciNet review: 606511
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Abstract: We prove the uniqueness of the quadrature formula with minimal error in the space $ \tilde W_q^r[a,b],1 < q < \infty $, of $ (b - a)$-periodic differentiable functions among all quadratures with n free nodes $ \{ {x_k}\} _1^n$, $ a = {x_1} < \cdots < {x_n} < b$, of fixed multiplicities $ \{ {v_k}\} _1^n$, respectively. As a corollary, we get that the equidistant nodes are optimal in $ \tilde W_q^r[a,b]$ for $ 1 \leqslant q \leqslant \infty$ if $ {v_1} = \cdots = {v_n}$.


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  • [1] 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. 12-38. MR 516041 (80j:41013)
  • [3] D. Barrow, "On multiple node Gaussian quadrature formulae," Math. Comp., v. 32, 1978, pp. 431-439. 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 $ {L_p}$ deviation," Constructive Function Theory '77, Sofia, 1980, pp. 249-268. BAN.
  • [6] B. D. Bojanov, "Uniqueness of the monosplines of least deviation," Numerische Integration, ISNM 45, Birkhäuser-Verlag, Basel, 1979, pp. 67-97. 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. 353-405.
  • [8] K. Jetter & G. Lange, "Die Eindeutigkeit $ {L_2}$-optimaler polynomialer Monosplines," Math. Z., v. 158, 1978, pp. 23-34. MR 0467094 (57:6961)
  • [9] R. S. Johnson, "On monosplines of least deviation," Trans. Amer. Math. Soc., v. 96, 1960, pp. 458-477. 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. 913-926. MR 0427907 (55:937)
  • [11] N. E. Lušpai, "Best quadrature formulae for classes of differentiable periodic functions," Mat. Zametki, v. 6, 1969, pp. 475-482.
  • [12] N. E. Lušpai, "Optimal quadrature formulae for classes of functions with an $ {L_p}$-integrable r-th derivative," Anal. Math., v. 5, 1979, pp. 67-88. MR 535497 (80j:41048)
  • [13] C. Micchelli, "The fundamental theorem of algebra for monosplines with multiplicities," Linear Operators and Approximation, ISNM v. 20, Birkhäuser-Verlag, Basel, 1972, pp. 419-430. MR 0393951 (52:14758)
  • [14] V. P. Motorniĭ, "On the best quadrature formula of the form $ \Sigma _{k = 1}^n\;{P_k}f({x_k})$ for certain classes of periodic differentiable functions," Izv. Akad. Nauk SSSR Ser. Mat., v. 38, 1974, pp. 583-614. 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. 152-168. MR 0430611 (55:3616)
  • [17] J. T. Schwartz, Nonlinear Functional Analysis, Gordon and Breach, New York, 1969. MR 0433481 (55:6457)
  • [18] A. A. Žensykbaev, "On the best quadrature formula on the class $ {W^r}{L_p}$," Dokl. Akad. Nauk SSSR, v. 227, 1976, pp. 277-279. MR 0405816 (53:9608)
  • [19] A. A. Žensykbaev, "Best quadrature formula for the class $ {W^r}{L_2}$," Anal. Math., v. 3, 1977, pp. 83-93. 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. 1110-1124. MR 0471271 (57:11008)
  • [21] A. A. Zensykbaev, "Characteristic properties of the best quadrature formulae," Sibirsk. Mat. Ž., v. 20, 1979, pp. 49-68. MR 523136 (80f:41022)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1981-0606511-4
Article copyright: © Copyright 1981 American Mathematical Society

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