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Izv. RAN. Ser. Mat., 2010, Volume 74, Issue 3, Pages 169–224 (Mi izv2689)  

Extremal problems for integrals of non-negative functions

A. I. Stepanets, A. L. Shidlich

Institute of Mathematics, Ukrainian National Academy of Sciences

Abstract: We study the numbers $e_\sigma(f)$ that characterize the best approximation of the integrals of functions in $L_p(A,d\mu)$, $p>0$, by integrals of rank $\sigma$. We find exact values and orders as $\sigma\to\infty$ for the least upper bounds of these numbers on the classes of functions representable as products of a fixed non-negative function and functions in the unit ball $U_p(A)$ of $L_p(A,d\mu)$. The numbers $e_\sigma( \cdot )$ are used to obtain necessary and sufficient conditions for an arbitrary function in $L_p(A,d\mu)$ to lie in $L_s(A,d\mu)$, $0<p,s<\infty$. We discuss applications of the results obtained to the approximation of measurable functions (given by convolutions with summable kernels) by entire functions of exponential type.

Keywords: best approximations of integrals by integrals of finite rank, absolute convergence of integrals.

DOI: https://doi.org/10.4213/im2689

Full text: PDF file (865 kB)
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English version:
Izvestiya: Mathematics, 2010, 74:3, 607–660

Bibliographic databases:

UDC: 517.5
MSC: 41A50
Received: 28.06.2007
Revised: 23.03.2009

Citation: A. I. Stepanets, A. L. Shidlich, “Extremal problems for integrals of non-negative functions”, Izv. RAN. Ser. Mat., 74:3 (2010), 169–224; Izv. Math., 74:3 (2010), 607–660

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