Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Tchakaloff's theorem and $ K$-integral polynomials in Banach spaces


Authors: Damián Pinasco and Ignacio Zalduendo
Journal: Proc. Amer. Math. Soc. 145 (2017), 3395-3408
MSC (2010): Primary 46E50; Secondary 28C20, 46G12, 46G20
DOI: https://doi.org/10.1090/proc/13520
Published electronically: January 25, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Tchakaloff's theorem gives a quadrature formula for polynomials of a given degree with respect to a compactly supported positive measure which is absolutely continuous with respect to Lebesgue measure. We study the validity of two possible analogues of Tchakaloff's theorem in an infinite-dimensional Banach space $ E$: a weak form valid when $ E$ has a Schauder basis, and a stronger form requiring conditions on the support of the measure as well as on the space $ E$.


References [Enhancements On Off] (What's this?)

  • [1] Raymundo Alencar, Multilinear mappings of nuclear and integral type, Proc. Amer. Math. Soc. 94 (1985), no. 1, 33-38. MR 781051, https://doi.org/10.2307/2044946
  • [2] C. Boyd and R. A. Ryan, Geometric theory of spaces of integral polynomials and symmetric tensor products, J. Funct. Anal. 179 (2001), no. 1, 18-42. MR 1807251, https://doi.org/10.1006/jfan.2000.3666
  • [3] Daniel Carando and Verónica Dimant, Duality in spaces of nuclear and integral polynomials, J. Math. Anal. Appl. 241 (2000), no. 1, 107-121. MR 1738337, https://doi.org/10.1006/jmaa.1999.6626
  • [4] Raúl E. Curto and Lawrence A. Fialkow, A duality proof of Tchakaloff's theorem, J. Math. Anal. Appl. 269 (2002), no. 2, 519-532. MR 1907129, https://doi.org/10.1016/S0022-247X(02)00034-3
  • [5] Raffaella Cilia and Joaquín M. Gutiérrez, Polynomial characterization of Asplund spaces, Bull. Belg. Math. Soc. Simon Stevin 12 (2005), no. 3, 393-400. MR 2173701
  • [6] D. Carando, V. Dimant, B. Duarte, and S. Lassalle, $ K$-bounded polynomials, Math. Proc. R. Ir. Acad. 98A (1998), no. 2, 159-171. MR 1759429
  • [7] Seán Dineen, Complex analysis on infinite-dimensional spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 1999. MR 1705327
  • [8] S. Mazur, Une remarque sur l'homéomorphie des champs fonctionnels, Studia Math. 1 (1929), 83-85.
  • [9] S. Mazur and S. Ulam, Sur les transformationes isométriques d'espaces vectoriels normés, C. R. Acad. Sci. Paris 194 (1932), 946-948.
  • [10] I. P. Mysovskikh, On Chakalov's theorem, U.S.S.R. Comput. Math. and Math. Phys. 15 (6) (1975), 221-227.
  • [11] Mihai Putinar, A note on Tchakaloff's theorem, Proc. Amer. Math. Soc. 125 (1997), no. 8, 2409-2414. MR 1389533, https://doi.org/10.1090/S0002-9939-97-03862-8
  • [12] Bruce Reznick, Sums of even powers of real linear forms, Mem. Amer. Math. Soc. 96 (1992), no. 463, viii+155. MR 1096187, https://doi.org/10.1090/memo/0463
  • [13] Hans Joachim Schmid, Interpolatorische Kubaturformeln, Dissertationes Math. (Rozprawy Mat.) 220 (1983), 122 (German). MR 735919
  • [14] Vladimir Tchakaloff, Formules de cubatures mécaniques à coefficients non négatifs, Bull. Sci. Math. (2) 81 (1957), 123-134 (French). MR 0094632
  • [15] D. Yost, Asplund spaces for beginners, Selected papers from the 21st Winter School on Abstract Analysis (Poděbrady, 1993), Acta Univ. Carolin. Math. Phys. 34 (1993), no. 2, 159-177. MR 1282979

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 46E50, 28C20, 46G12, 46G20

Retrieve articles in all journals with MSC (2010): 46E50, 28C20, 46G12, 46G20


Additional Information

Damián Pinasco
Affiliation: Departamento de Matemática, Universidad Torcuato Di Tella, Av. Figueroa Alcorta 7350 (C1428BCW), Buenos Aires, Argentina – and – CONICET
Email: dpinasco@utdt.edu

Ignacio Zalduendo
Affiliation: Departamento de Matemática, Universidad Torcuato Di Tella, Av. Figueroa Alcorta 7350 (C1428BCW), Buenos Aires, Argentina – and – CONICET
Email: nacho@utdt.edu

DOI: https://doi.org/10.1090/proc/13520
Keywords: Polynomials on Banach spaces, integral polynomial, nuclear polynomial, Tchakaloff's theorem.
Received by editor(s): October 29, 2015
Received by editor(s) in revised form: July 4, 2016, and August 31, 2016
Published electronically: January 25, 2017
Additional Notes: The authors were partially supported by CONICET (PIP 11220090100624).
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society