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2 papers
Sharp estimates for the convergence rate of “hyperbolic” partial sums of double fourier series in orthogonal polynomials
V. A. Abilova,
M. K. Kerimovb a Dagestan State University, ul. Gadzhieva 43a, Makhachkala, 367025, Russia
b Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119333, Russia
Abstract:
Two-variable functions
$f(x,y)$ from the class
$L_2=L_2((a,b)\times(c,d);p(x)q(y))$ with the weight
$p(x)q(y)$ and the norm
$$
||f||=\sqrt{\int_a^b\int_c^dp(x)q(x)f^2(x,y)dx\,dy}
$$
are approximated by an orthonormal system of orthogonal
$P_n(x)Q_n(y)$,
$n, m=0, 1,\dots$, with weights
$p(x)$ and
$q(y)$. Let
$$
E_N(f)=\inf_{P_N}||f-P_N||
$$
denote the best approximation of
$f\in L_2$ algebraic polynomials of the form
\begin{gather*}
P_N(x,y)=\sum_{0<n,m<N}a_{m,n}x^ny^m,\\
P_1(x,y)=\mathrm{const}.
\end{gather*}
Consider a double Fourier series of
$f\in L_2$ in the polynomials
$P_n(x)Q_m(y)$,
$n, m=0, 1,\dots$, and its “hyperbolic” partial sums
\begin{gather*}
S_1(f; x,y)=c_{0,0}(f)P_0(x)Q_0(y),\\
S_N(f; x,y)=\sum_{0<n,m<N}c_{n,m}(f)P_n(x)Q_m(y),\qquad N=2,3,\dots.
\end{gather*}
A generalized shift operator
$F_h$ and a
$k$th-order generalized modulus of continuity
$\Omega_k(A,h)$ of a function
$f\in L_2$
are used to prove the following sharp estimate for the convergence rate of the approximation:
\begin{gather*}
E_N(f)\leqslant(1-(1-h)^{2\sqrt{N}})^{-k}\,\Omega_k(f; h), \qquad h\in(0,1),\\
N=4,5,\dots;\qquad k=1,2,\dots.
\end{gather*}
Moreover, for every fixed
$N=4,9,16,\dots$, the constant on the right-hand side of this inequality is cannot be reduced.
Key words:
double Fourier series, “hyperbolic” partial sum of Fourier series, best approximation of functions by algebraic polynomials in two variables, generalized shift operator, generalized modulus of continuity.
UDC:
519.651 Received: 15.06.2012
Fulltext:
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