The Minlos lemma for positive-definite functions on additive subgroups of ℝ n

W. Banaszczyk

Studia Mathematica (1997)

  • Volume: 126, Issue: 1, page 13-25
  • ISSN: 0039-3223

Abstract

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Let H be a real Hilbert space. It is well known that a positive-definite function φ on H is the Fourier transform of a Radon measure on the dual space if (and only if) φ is continuous in the Sazonov topology (resp. the Gross topology) on H. Let G be an additive subgroup of H and let G p c ∧ (resp. G b ∧ ) be the character group endowed with the topology of uniform convergence on precompact (resp. bounded) subsets of G. It is proved that if a positive-definite function φ on G is continuous in the Gross topology, then φ is the Fourier transform of a Radon measure μ on G p c ∧ ; if φ is continuous in the Sazonov topology, μ can be extended to a Radon measure on G b ∧ .

How to cite

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Banaszczyk, W.. "The Minlos lemma for positive-definite functions on additive subgroups of $ℝ^n$." Studia Mathematica 126.1 (1997): 13-25. <http://eudml.org/doc/216440>.

@article{Banaszczyk1997,
abstract = {Let H be a real Hilbert space. It is well known that a positive-definite function φ on H is the Fourier transform of a Radon measure on the dual space if (and only if) φ is continuous in the Sazonov topology (resp. the Gross topology) on H. Let G be an additive subgroup of H and let $G_\{pc\}^∧$ (resp. $G_b^∧$) be the character group endowed with the topology of uniform convergence on precompact (resp. bounded) subsets of G. It is proved that if a positive-definite function φ on G is continuous in the Gross topology, then φ is the Fourier transform of a Radon measure μ on $G_\{pc\}^∧$; if φ is continuous in the Sazonov topology, μ can be extended to a Radon measure on $G_b^∧$.},
author = {Banaszczyk, W.},
journal = {Studia Mathematica},
keywords = {Minlos lemma; positive definite functions; Sazonov topology; Gross topology; Bochner-type representation theorems; Fourier transform; Radon measure},
language = {eng},
number = {1},
pages = {13-25},
title = {The Minlos lemma for positive-definite functions on additive subgroups of $ℝ^n$},
url = {http://eudml.org/doc/216440},
volume = {126},
year = {1997},
}

TY - JOUR
AU - Banaszczyk, W.
TI - The Minlos lemma for positive-definite functions on additive subgroups of $ℝ^n$
JO - Studia Mathematica
PY - 1997
VL - 126
IS - 1
SP - 13
EP - 25
AB - Let H be a real Hilbert space. It is well known that a positive-definite function φ on H is the Fourier transform of a Radon measure on the dual space if (and only if) φ is continuous in the Sazonov topology (resp. the Gross topology) on H. Let G be an additive subgroup of H and let $G_{pc}^∧$ (resp. $G_b^∧$) be the character group endowed with the topology of uniform convergence on precompact (resp. bounded) subsets of G. It is proved that if a positive-definite function φ on G is continuous in the Gross topology, then φ is the Fourier transform of a Radon measure μ on $G_{pc}^∧$; if φ is continuous in the Sazonov topology, μ can be extended to a Radon measure on $G_b^∧$.
LA - eng
KW - Minlos lemma; positive definite functions; Sazonov topology; Gross topology; Bochner-type representation theorems; Fourier transform; Radon measure
UR - http://eudml.org/doc/216440
ER -

References

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  1. [1] W. Banaszczyk, Additive Subgroups of Topological Vector Spaces, Lecture Notes in Math. 1466, Springer, Berlin, 1991. Zbl0743.46002
  2. [2] W. Banaszczyk, New bounds in some transference theorems in the geometry of numbers, Math. Ann. 296 (1993), 625-635. Zbl0786.11035
  3. [3] W. Banaszczyk, Inequalities for convex bodies and polar reciprocal lattices in ℝ n , Discrete Comput. Geom. 13 (1995), 217-231. Zbl0824.52011
  4. [4] W. Banaszczyk, Inequalities for convex bodies and polar reciprocal lattices in ℝ n II. Application of K-convexity, Discrete Comput. Geom. 16 (1996), 305-311. Zbl0868.52002
  5. [5] J. Kisyński, On the generation of tight measures, Studia Math. 30 (1968), 141-151. Zbl0157.37301
  6. [6] J. Lindenstrauss and V. D. Milman, The local theory of normed spaces and its applications to convexity, in: Handbook of Convex Geometry, P. M. Gruber and J. M. Wills (eds.), North-Holland, Amsterdam, 1993, 739-763. Zbl0791.52003
  7. [7] R. A. Minlos, Generalized stochastic processes and their extension to the measure, Trudy Moskov. Mat. Obshch. 8 (1959), 497-518 (in Russian). 
  8. [8] A. Pietsch, Nuclear Locally Convex Spaces, Springer, Berlin, 1972. 
  9. [9] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Univ. Press, Cambridge, 1989. Zbl0698.46008
  10. [10] N. Tomczak-Jaegermann, Banach-Mazur Distances and Finite-Dimensional Operator Ideals, Longman Sci. & Tech., Harlow, 1989. Zbl0721.46004
  11. [11] N. N. Vakhaniya, V. I. Tarieladze and S. A. Chobanyan, Probability Distributions on Banach Spaces, Nauka, Moscow, 1985 (in Russian); English transl.: D. Reidel, Dordrecht, 1987. Zbl0572.60003

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