×

A duality proof of Tchakaloff’s theorem. (English) Zbl 1001.41014

The Tchakaloff theorem is a classical result in the theory of cubature formulas. It was proved relatively late: V. Tchakaloff, Bull. Sci. Math., II. Ser. 81, 123-134 (1957; Zbl 0079.13908)]. The theorem asserts that for a given positive measure \(\mu\) on \(\mathbb{R}^n\), of compact support, and a given degree \(d\), there exists a finite atomic positive measure \(\nu\), sharing with \(\nu\) the same moment up to degree \(d\). The existence of the “cubature” measure \(\nu\) is now an excercise in convexity theory. However, its construction in dimensions \(n\) greater than 1 is rather involved and not algorithmic. In dimension \(n=1\) the theory of orthogonal polynomials, and their zeros, offers a satisfactory construction. The note under review offers a novel and very interesting duality proof of Tchakaloff’s theorem. The authors obtain optimal bounds for the number of points in the support of \(\nu\), plus a careful analysis, based on their previous work, of the uniqueness of this minimal representing measures.

MSC:

41A55 Approximate quadratures
41A63 Multidimensional problems

Citations:

Zbl 0079.13908
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Tchakaloff, V., Formules de cubatures mécaniques à coefficients non négatifs, Bull. Sci. Math. (2), 81, 123-134 (1957) · Zbl 0079.13908
[2] Mysovskikh, I. P., On Chakalov’s theorem, USSR Comput. Math. Math. Phys.. USSR Comput. Math. Math. Phys., Zh. Vychisl. Mat. i Mat. Fiz., 15, 1589-1593 (1975), 1627 · Zbl 0346.41027
[3] Putinar, M., On Tchakaloff’s theorem, Proc. Amer. Math. Soc., 125, 2409-2414 (1997) · Zbl 0886.41025
[4] J. Stochel, Solving the truncated moment problem solves the full moment problem, Glasgow Math. J., to appear; J. Stochel, Solving the truncated moment problem solves the full moment problem, Glasgow Math. J., to appear · Zbl 0995.44004
[5] Curto, R.; Fialkow, L., Solution of the truncated complex moment problem for flat data, Mem. Amer. Math. Soc., 119 (1996), no. 568 · Zbl 0876.30033
[6] Curto, R.; Fialkow, L., Flat extensions of positive moment matrices: Relations in analytic or conjugate terms, Oper. Theory Adv. Appl., 104, 59-82 (1998) · Zbl 0904.30020
[7] Curto, R.; Fialkow, L., Flat extensions of positive moment matrices: Recursively generated relations, Mem. Amer. Math. Soc., 136 (1998), no. 648 · Zbl 0913.47016
[8] Möller, H. M., Kubaturformeln mit minimaler Knotenzahl, Numer. Math., 25, 185-200 (1976) · Zbl 0319.65019
[9] Ralston, A., A First Course in Numerical Analysis (1965), McGraw-Hill: McGraw-Hill New York · Zbl 0139.31603
[10] Stroud, A. H., Approximate Calculation of Multiple Integrals (1971), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0379.65013
[11] Xu, Y., Common Zeros of Polynomials in Several Variables and Higher-Dimensional Quadrature. Common Zeros of Polynomials in Several Variables and Higher-Dimensional Quadrature, Pitman Res. Notes Math. Ser., 312 (1994), Longman Sci. Tech.: Longman Sci. Tech. Harlow · Zbl 0898.26004
[12] L. Fialkow, Multivariable quadrature and extensions of moment matrices, preprint (1996); L. Fialkow, Multivariable quadrature and extensions of moment matrices, preprint (1996)
[13] J.P. Gabardo, Truncated trigonometric moment problems and determinate measures, preprint (1999); J.P. Gabardo, Truncated trigonometric moment problems and determinate measures, preprint (1999) · Zbl 0947.42015
[14] Akhiezer, N. I., The Classical Moment Problem (1965), Hafner: Hafner New York · Zbl 0135.33803
[15] Fialkow, L., Positivity, extensions, and the truncated complex moment problem, (Multivariable Operator Theory (Seattle, WA, 1993). Multivariable Operator Theory (Seattle, WA, 1993), Contemporary Mathematics, 185 (1995), American Mathematical Society: American Mathematical Society Providence, RI), 133-150 · Zbl 0830.44007
[16] Curto, R.; Fialkow, L., The truncated complex \(K\)-moment problem, Trans. Amer. Math. Soc., 352, 2825-2855 (2000) · Zbl 0955.47011
[17] Conway, J., A Course in Functional Analysis. A Course in Functional Analysis, Graduate Texts in Mathematics (1990), Springer-Verlag: Springer-Verlag New York · Zbl 0706.46003
[18] Paulsen, V., Completely Bounded Maps and Dilations. Completely Bounded Maps and Dilations, Pitman Res. Notes Math. Ser., 146 (1986), Longman Sci. Tech.: Longman Sci. Tech. London · Zbl 0614.47006
[19] Conway, J., A Course in Operator Theory. A Course in Operator Theory, Graduate Studies in Mathematics, 21 (2000), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0936.47001
[20] Fialkow, L., Minimal representing measures arising from rank-increasing moment matrix extensions, J. Operator Theory, 42, 425-436 (1999) · Zbl 0992.47003
[21] A. Wilmshurst, Complex Harmonic Mappings and the Valence of Harmonic Polynomials, Ph.D. Dissertation, University of York (1994); A. Wilmshurst, Complex Harmonic Mappings and the Valence of Harmonic Polynomials, Ph.D. Dissertation, University of York (1994) · Zbl 0891.30009
[22] Reznick, B., Sums of even powers of real linear forms, Mem. Amer. Math. Soc., 96 (1992), no. 463 · Zbl 0762.11019
[23] R. Curto, L. Fialkow, Solution of the singular quartic moment problem, J. Operator Theory, to appear; R. Curto, L. Fialkow, Solution of the singular quartic moment problem, J. Operator Theory, to appear · Zbl 1019.47018
[24] Stochel, J.; Szafraniec, F. H., The complex moment problem and subnormality: A polar decomposition approach, J. Funct. Anal., 159, 432-491 (1998) · Zbl 1048.47500
[25] Berg, C.; Christensen, J. P.R.; Ressel, P., Harmonic Analysis on Semigroups (1984), Springer-Verlag: Springer-Verlag New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.