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1997 | 126 | 1 | 13-25
Tytuł artykułu

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

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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 $G_{pc}^∧$ (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 $G_b^∧$.
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Twórcy
  • Faculty of Mathematics, Łódź University, Banacha 22, 90-238 Łódź, Poland
Bibliografia
  • [1] W. Banaszczyk, Additive Subgroups of Topological Vector Spaces, Lecture Notes in Math. 1466, Springer, Berlin, 1991.
  • [2] W. Banaszczyk, New bounds in some transference theorems in the geometry of numbers, Math. Ann. 296 (1993), 625-635.
  • [3] W. Banaszczyk, Inequalities for convex bodies and polar reciprocal lattices in $ℝ^n$, Discrete Comput. Geom. 13 (1995), 217-231.
  • [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.
  • [5] J. Kisyński, On the generation of tight measures, Studia Math. 30 (1968), 141-151.
  • [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.
  • [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.
  • [10] N. Tomczak-Jaegermann, Banach-Mazur Distances and Finite-Dimensional Operator Ideals, Longman Sci. & Tech., Harlow, 1989.
  • [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.
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bwmeta1.element.bwnjournal-article-smv126i1p13bwm
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