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## Studia Mathematica

1993 | 106 | 2 | 121-128
Tytuł artykułu

### A multidimensional Lyapunov type theorem

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Given functions $f_1,...,f_ν ∈ ℒ^1(ℝ^n;ℝ^m)$, weights $p_1,...,p_ν: ℝ^n → [0,1]$ with $∑ p_i ≡ 1$, and any finite set of vectors $v_1,...,v_k ∈ ℝ^n ∖ {0}$, we prove the existence of a partition ${A_1,...,A_ν}$ of $ℝ^n$ such that the two functions $f_p = ∑_{i=1}^ν p_i f_i,$f_A = ∑_{i=1}^ν χ_{A_i}f_i$have the same integral not only over$ℝ^n$, but also over every single line$x' + ℝv_j$, for each j = 1,...,k and almost every x' in the orthogonal hyperplane$v_j^⊥$. Equivalently, the Fourier transforms of$f_p$,$f_A$satisfy$f̂_p(y) = f̂_A(y)$for every$y ∈ ⋃ v_j^⊥\$.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
121-128
Opis fizyczny
Daty
wydano
1993
otrzymano
1992-02-26
poprawiono
1993-02-01
Twórcy
autor
• S.I.S.S.A., Via Beirut 4, Trieste 34014, Italy
Bibliografia
• [1] Z. Artstein, Yet another proof of the Lyapunov convexity theorem, Proc. Amer. Math. Soc. 108 (1990), 89-91.
• [2] J. P. Aubin and A. Cellina, Differential Inclusions, Springer, New York 1984.
• [3] A. Bressan and F. Flores, Multivalued Aumann integrals and controlled wave equations, preprint S.I.S.S.A., Trieste 1992.
• [4] L. Cesari, Optimization. Theory and Applications, Springer, New York 1983.
• [5] P. R. Halmos, The range of a vector measure, Bull. Amer. Math. Soc. 54 (1948), 416-421.
• [6] J. Lindenstrauss, A short proof of Liapounoff's convexity theorem, J. Math. Mech. 15 (1966), 971-972.
• [7] A. A. Lyapunov, On completely additive vector-valued functions, Izv. Akad. Nauk SSSR Ser. Mat. 8 (1940), 465-478 (in Russian; French summary).
• [8] C. Olech, The Lyapunov theorem: its extensions and applications, in: Methods of Nonconvex Analysis, A. Cellina (ed.), Lecture Notes in Math. 1446, Springer, 1989, 84-103.
• [9] J. A. Yorke, Another proof of the Liapunov convexity theorem, SIAM J. Control 9 (1971), 351-353.
Typ dokumentu
Bibliografia
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