Gennadiy Feldman's Functional Equations and Characterization Problems on PDF

By Gennadiy Feldman

ISBN-10: 3037190450

ISBN-13: 9783037190456

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Extra info for Functional Equations and Characterization Problems on Locally Compact Abelian Groups (Ems Tracts in Mathematics)

Example text

Ij /m i;j D1 is a symmetric positive semidefinite matrix, ˇ 2 Rm , b 0. Since . y/ D 0 if and only if y D 0. s; snC1 / ¤ 0. s; snC1 / 2 RmC1 . s1 ; : : : ; sm ; k/ D expf "k 2 g: Since . 1 the distribution also has non-Gaussian factors. 1 has non-Gaussian factors. 9 implies directly the following statement. 10. Each nondegenerate Gaussian distribution on a group X has nonGaussian factors if and only if X is topologically isomorphic to the circle group T . 11. Let X be a connected group. X/ such that .

Let be a symmetric Gaussian distribution on a group X . y/ D 1g D f0g. This implies that the group X is connected, and . / D X. Thus studying Gaussian distributions on a group X we can restrict ourselves to the case when X is a connected group. We will prove now that an arbitrary symmetric Gaussian distribution on a connected group X is a continuous homomorphic image of a Gaussian distribution in a linear space. This space is either finite-dimensional or infinite-dimensional depending on the dimension of the group X .

X by the formula Œt  D pt . We note that because L is a complete separable metric group and the kernel Ker p is a closed subgroup of L, then G is also a complete separable metric group. L/ is a Borel set. L/ can not coincide with the group X . L/, then the group X is the union of its one-parameter subgroups, and hence X must be arcwise connected. 3, X Š Rm T n , which contradicts the condition of the proposition. 17)). B/ > 0, then B D X. L// D 0. Hence D s. 15. Assume now that a group X is connected and locally connected.

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Functional Equations and Characterization Problems on Locally Compact Abelian Groups (Ems Tracts in Mathematics) by Gennadiy Feldman


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