Abstract:
In this paper the sharp upper
bounds of approximation of functions of two variables with generalized Fourier–Chebyshev
polynomials for the class of functions $L_{2,\rho}^{(r)} (D)$, $r\in N,$ are calculated in
$L_{2,\rho}:=L_{2,\rho}(Q)$, where $\rho:=\rho(x,y)=1/\sqrt{(1-x^{2})(1-y^{2})}$,
$Q:=\{(x,y):-1\leq x,y\leq1\}$, and $D$ is a second order Chebyshev–Hermite operator.
The sharp estimates for the best polynomial approximation are obtained by means of
weighted average of module of continuity of $m$-th order with $D^r f$$(r\in Z_+)$ in
$L_{2,\rho}$. The sharp estimates for the best approximation of double Fourier
series in Fourier–Chebyshev orthogonal system in the classes of functions of several
variables which are characterized by generalized module of continuity are given. We first
form some classes of functions and then the corresponding methods of approximations,
“circular” by partial sum of Fourier–Chebyshev double series, since, unlike the
one-dimensional case, there is no natural way of expressing the partial sums of double
series. The shift operator plays a crucial role in the problems related to expansion of
functions in Fourier series in trigonometric system and estimating their best
approximation properties. Based on some previous known research we construct the
shift operator, which enables one to determine some classes of functions which
characterized by module of continuity. And for these classes of functions the upper bound
for the best mean squared approximation by “circular” partial sum of Fourier–Chebyshev
double series is calculated.
Key words:mean-squared
approximation, generalized module of continuity, Fourier–Tchebychev double series,
Kolmogorov type inequality, shift operator.
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