Scientific articles
On the best approximation of some classes of periodic functions in the space $L_{2}$
M. R. Langarshoeva,
S. S. Khorazmshoevb a Moscow University of Humanities and Technology – Moscow Institute of Architecture and Construction
b Tajik Technical University named after academician M. S. Osimi
Abstract:
We consider the set
$L_{2}^{(r)}$ of
$2\pi$-periodic functions
$f\in L_{2}$ whose
$(r-1)$-th order derivative is absolutely continuous, and the
$r$-th order derivative
$f^{(r)}\in L_{2}.$ We solve the extremal problem of finding an exact Jackson–Stechkin type constant that connects the best polynomial approximation of functions from
$L_{2}^{(r)}$ with the average value of the generalized
$m$-th order modulus of continuity of their derivative
$f^{(r)}$ in the space
$L_{2}.$ We also consider the classes
$W_{m}^{(r)}(u)$ and
$W_{m}^{(r)}(u,\Phi)$ of functions from
$L_{2}^{(r)}$ such that the average value of the generalized
$m$-th order modulus of continuity of their derivative
$f^{(r)}$ is bounded from above by unity and, accordingly, by the value of some function
$\Phi(u).$ We calculate the exact values of the known
$n$-widths (according to Bernstein, to Gelfand, to Kolmogorov, linear, and projection) of the class
$W_{m}^{(r)}(u).$ Then we solve the extremal problem of finding the exact value of the best approximation for the class
$W_{m}^{(r)}(u,\Phi).$ The obtained results develop and complement some known results on the best approximation of various classes of functions in
$L_{2}.$ In the paper, we use methods for solving extremal problems in normed spaces, as well as the method developed by V. M. Tikhomirov
for estimating from below the
$n$-widths of functional classes in Banach spaces.
Keywords:
best polynomial approximation in $L_{2},$ extremal characteristic — generalized modulus of continuity, $n$-widths
UDC:
517.5,
519.65
MSC: 42A10,
41A17,
41A44 Received: 20.12.2024
Accepted: 13.03.2025
DOI:
10.20310/2686-9667-2025-30-149-56-65
Fulltext:
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