Extremal problems for integrals of non-negative functions
Izvestiya. Mathematics , Tome 74 (2010) no. 3, pp. 607-660
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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 $L_s(A,d\mu)$, $0$.
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.
@article{IM2_2010_74_3_a7,
author = {A. I. Stepanets and A. L. Shidlich},
title = {Extremal problems for integrals of non-negative functions},
journal = {Izvestiya. Mathematics },
pages = {607--660},
publisher = {mathdoc},
volume = {74},
number = {3},
year = {2010},
language = {en},
url = {http://geodesic.mathdoc.fr/item/IM2_2010_74_3_a7/}
}
A. I. Stepanets; A. L. Shidlich. Extremal problems for integrals of non-negative functions. Izvestiya. Mathematics , Tome 74 (2010) no. 3, pp. 607-660. http://geodesic.mathdoc.fr/item/IM2_2010_74_3_a7/