Weighted Approximation by Rational Operators
Copyright policy )Weighted Approximation by Rational Operators
In this interesting paper, the author considers Shephard-type rational operators \[ S_n(f; x)= \Biggl[\sum^{n-1}_{k=1}| x- x_k|^{-s} f(x_k)\Biggr]\Biggl/\Biggl[\sum^{n-1}_{k=1}| x- x_k|^{-s}\Biggr] \] for appropriately chosen nodes \(\{x_k\}\) in \([-1,1]\). Let \(\alpha> 0\) and \[ w(x)= (1- x^2)^\alpha,\quad x\in [-1,1]. \] The author establishes the esimates \[ \| w(f- S_n(f))\|_{L_\infty[-1, 1]}\leq C\omega_\varphi\Biggl(f; {1\over n}\Biggr)_w \] and a \(K\)-functional type estimate \[ \omega_\varphi\Biggl(f;{1\over n}\Biggr)\sim\| w(f- S_n(f))\|_{L_\infty[-1, 1]}+{1\over n}\| w\varphi S_n'(f)\|_{L_\infty[- 1,1]}, \] under appropriate assumptions on \(\alpha\), and the points \(\{x_k\}\).
\(K\)-functionals, Approximation by positive operators, Rate of convergence, degree of approximation, degree of approximation, Shephard operators
\(K\)-functionals, Approximation by positive operators, Rate of convergence, degree of approximation, degree of approximation, Shephard operators
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