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

Banaszczyk, W.

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

ℝ^{n}

Banaszczyk, W.

Studia Mathematica, Tome 122 (1997), p. 13-25 / Harvested from The Polish Digital Mathematics Library

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Résumé

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 $G_{p c}^{\land}$ (resp. $G_{b}^{\land}$ ) 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 $G_{p c}^{\land}$ ; if φ is continuous in the Sazonov topology, μ can be extended to a Radon measure on $G_{b}^{\land}$ .

Publié le : 1997-01-01

Zbl 0894.43004

EUDML-ID : urn:eudml:doc:216440

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     title = {The Minlos lemma for positive-definite functions on additive subgroups of $$\mathbb{R}$^n$
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     journal = {Studia Mathematica},
     volume = {122},
     year = {1997},
     pages = {13-25},
     zbl = {0894.43004},
     language = {en},
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Banaszczyk, W. The Minlos lemma for positive-definite functions on additive subgroups of $ℝ^n$
            . Studia Mathematica, Tome 122 (1997) pp. 13-25. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv126i1p13bwm/

Bibliographie

[00000] [1] W. Banaszczyk, Additive Subgroups of Topological Vector Spaces, Lecture Notes in Math. 1466, Springer, Berlin, 1991. | Zbl 0743.46002

[00001] [2] W. Banaszczyk, New bounds in some transference theorems in the geometry of numbers, Math. Ann. 296 (1993), 625-635. | Zbl 0786.11035

[00002] [3] W. Banaszczyk, Inequalities for convex bodies and polar reciprocal lattices in $ℝ^{n}$ , Discrete Comput. Geom. 13 (1995), 217-231. | Zbl 0824.52011

[00003] [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. | Zbl 0868.52002

[00004] [5] J. Kisyński, On the generation of tight measures, Studia Math. 30 (1968), 141-151. | Zbl 0157.37301

[00005] [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. | Zbl 0791.52003

[00006] [7] R. A. Minlos, Generalized stochastic processes and their extension to the measure, Trudy Moskov. Mat. Obshch. 8 (1959), 497-518 (in Russian).

[00007] [8] A. Pietsch, Nuclear Locally Convex Spaces, Springer, Berlin, 1972.

[00008] [9] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Univ. Press, Cambridge, 1989. | Zbl 0698.46008

[00009] [10] N. Tomczak-Jaegermann, Banach-Mazur Distances and Finite-Dimensional Operator Ideals, Longman Sci. & Tech., Harlow, 1989. | Zbl 0721.46004

[00010] [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. | Zbl 0572.60003