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1997 | 126 | 1 | 13-25
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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 $G_b^∧$.
Słowa kluczowe
  • Faculty of Mathematics, Łódź University, Banacha 22, 90-238 Łódź, Poland
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