# The Minlos lemma for positive-definite functions on additive subgroups of ${\mathbb{R}}^{n}$

Studia Mathematica (1997)

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

## Access Full Article

top## Abstract

top## How to cite

topBanaszczyk, 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

top- [1] W. Banaszczyk, Additive Subgroups of Topological Vector Spaces, Lecture Notes in Math. 1466, Springer, Berlin, 1991. Zbl0743.46002
- [2] W. Banaszczyk, New bounds in some transference theorems in the geometry of numbers, Math. Ann. 296 (1993), 625-635. Zbl0786.11035
- [3] W. Banaszczyk, Inequalities for convex bodies and polar reciprocal lattices in ${\mathbb{R}}^{n}$, Discrete Comput. Geom. 13 (1995), 217-231. Zbl0824.52011
- [4] W. Banaszczyk, Inequalities for convex bodies and polar reciprocal lattices in ${\mathbb{R}}^{n}$ II. Application of K-convexity, Discrete Comput. Geom. 16 (1996), 305-311. Zbl0868.52002
- [5] J. Kisyński, On the generation of tight measures, Studia Math. 30 (1968), 141-151. Zbl0157.37301
- [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] R. A. Minlos, Generalized stochastic processes and their extension to the measure, Trudy Moskov. Mat. Obshch. 8 (1959), 497-518 (in Russian).
- [8] A. Pietsch, Nuclear Locally Convex Spaces, Springer, Berlin, 1972.
- [9] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Univ. Press, Cambridge, 1989. Zbl0698.46008
- [10] N. Tomczak-Jaegermann, Banach-Mazur Distances and Finite-Dimensional Operator Ideals, Longman Sci. & Tech., Harlow, 1989. Zbl0721.46004
- [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

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.