Uniqueness of the optimal nodes of quadrature formulae

Bojanov, Borislav D.

doi:10.1090/S0025-5718-1981-0606511-4

Uniqueness of the optimal nodes of quadrature formulae
HTML articles powered by AMS MathViewer

by Borislav D. Bojanov PDF

Math. Comp. 36 (1981), 525-546 Request permission

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 ${v_1} = \cdots = {v_n}$.

References

N. I. Ahiezer, Lektsii po teorii approksimatsii, Second, revised and enlarged edition, Izdat. “Nauka”, Moscow, 1965 (Russian). MR 0188672
R. B. Barrar and H. L. Loeb, On monosplines with odd multiplicity of least norm, J. Analyse Math. 33 (1978), 12–38. MR 516041, DOI 10.1007/BF02790167
David L. Barrow, On multiple node Gaussian quadrature formulae, Math. Comp. 32 (1978), no. 142, 431–439. MR 482257, DOI 10.1090/S0025-5718-1978-0482257-0
Richard Bellman, On the positivity of determinants with dominant main diagonal, J. Math. Anal. Appl. 59 (1977), no. 1, 210. MR 441991, DOI 10.1016/0022-247X(77)90101-9

Constructive Function Theory

Borislav D. Bojanov, Uniqueness of the monosplines of least deviation, Numerische Integration (Tagung, Math. Forschungsinst., Oberwolfach, 1978) Internat. Ser. Numer. Math., vol. 45, Birkhäuser, Basel-Boston, Mass., 1979, pp. 67–97. MR 561282

Annuaire Univ. Sofia Fac. Math. Méc.

Kurt Jetter and Gerald Lange, Die Eindeutigkeit $L_{2}$-optimaler polynomialer Monosplines, Math. Z. 158 (1978), no. 1, 23–34 (German, with English summary). MR 467094, DOI 10.1007/BF01214562
R. S. Johnson, On monosplines of least deviation, Trans. Amer. Math. Soc. 96 (1960), 458–477. MR 122938, DOI 10.1090/S0002-9947-1960-0122938-4
A. A. Ligun, Sharp inequalities for spline functions, and best quadrature formulas for certain classes of functions, Mat. Zametki 19 (1976), no. 6, 913–926 (Russian). MR 427907

Mat. Zametki

N. E. Lušpaĭ, Optimal quadrature formulas for classes of functions with $L_{p}$ integrable $r$th derivative in $L_{p}$, Anal. Math. 5 (1979), no. 1, 67–88 (Russian, with English summary). MR 535497, DOI 10.1007/BF02079350
Charles Micchelli, The fundamental theorem of algebra for monosplines with multiplicities, Linear operators and approximation (Proc. Conf., Math. Res. Inst., Oberwolfach, 1971) Internat. Ser. Numer. Math., Vol. 20, Birkhäuser, Basel, 1972, pp. 419–430. MR 0393951
V. P. Motornyĭ, The best quadrature formula of the form $\sum _{k=1}^{n}p_{k}f(x_{k})$ for certain classes of periodic differentiable functions, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 583–614 (Russian). MR 0390610
J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York-London, 1970. MR 0273810
Larry L. Schumaker, Zeros of spline functions and applications, J. Approximation Theory 18 (1976), no. 2, 152–168. MR 430611, DOI 10.1016/0021-9045(76)90103-9
J. T. Schwartz, Nonlinear functional analysis, Notes on Mathematics and its Applications, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Notes by H. Fattorini, R. Nirenberg and H. Porta, with an additional chapter by Hermann Karcher. MR 0433481
A. A. Žensykbaev, The best quadrature formula on the class $W^{r}L_{p}$, Dokl. Akad. Nauk SSSR 227 (1976), no. 2, 277–279 (Russian). MR 0405816
A. A. Žensykbaev, Best quadrature formula for the class $W^{r}L_{2}$, Anal. Math. 3 (1977), no. 1, 83–93 (English, with Russian summary). MR 447924, DOI 10.1007/BF02333255
A. A. Žensykbaev, The best quadrature formula for some classes of periodic differentiable functions, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 5, 1110–1124, 1200 (Russian). MR 0471271
A. A. Žensykbaev, Characteristic properties of best quadrature formulas, Sibirsk. Mat. Zh. 20 (1979), no. 1, 49–68, 204 (Russian). MR 523136