Some properties of the normal image of convex functions

N. V. Krylov

doi:10.1070/SM1978v034n02ABEH001154

Some properties of the normal image of convex functions

N. V. Krylov

School of Mathematics

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Abstract

Let z be a convex function defined in a convex domain D of a finite-dimensional Euclidean space. Denote by z ⁽ⁿ⁾ the convolutions of z with elements of a d-type sequence of test functions and let ? _z and ? _{z ⁽ⁿ⁾} be the measures of normal images corresponding to z and z ⁽ⁿ⁾. One of the main results of this work is that ? _{z ⁽ⁿ⁾}?? _z in variation on a compact K ? D if and only if ? _z is absolutely continuous on K with respect to Lebesgue measure. Bibliography: 9 titles.

Original language	English (US)
Pages (from-to)	161-171
Number of pages	11
Journal	Mathematics of the USSR - Sbornik
Volume	34
Issue number	2
DOIs	https://doi.org/10.1070/SM1978v034n02ABEH001154
State	Published - Feb 28 1978

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abstract = "Let z be a convex function defined in a convex domain D of a finite-dimensional Euclidean space. Denote by z (n) the convolutions of z with elements of a d-type sequence of test functions and let ? z and ? z (n) be the measures of normal images corresponding to z and z (n). One of the main results of this work is that ? z (n)?? z in variation on a compact K ? D if and only if ? z is absolutely continuous on K with respect to Lebesgue measure. Bibliography: 9 titles.",

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N2 - Let z be a convex function defined in a convex domain D of a finite-dimensional Euclidean space. Denote by z (n) the convolutions of z with elements of a d-type sequence of test functions and let ? z and ? z (n) be the measures of normal images corresponding to z and z (n). One of the main results of this work is that ? z (n)?? z in variation on a compact K ? D if and only if ? z is absolutely continuous on K with respect to Lebesgue measure. Bibliography: 9 titles.

AB - Let z be a convex function defined in a convex domain D of a finite-dimensional Euclidean space. Denote by z (n) the convolutions of z with elements of a d-type sequence of test functions and let ? z and ? z (n) be the measures of normal images corresponding to z and z (n). One of the main results of this work is that ? z (n)?? z in variation on a compact K ? D if and only if ? z is absolutely continuous on K with respect to Lebesgue measure. Bibliography: 9 titles.

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Some properties of the normal image of convex functions

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