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Approximating periodic functions by interpolation sums of Jackson type. (English. Russian original) Zbl 1195.42009
J. Math. Sci., New York 157, No. 4, 592-606 (2009); translation from Zap. Nauchn. Semin. POMI 357, 90-114 (2008).
Summary: Let
\[ \Phi_n(t)= \frac{1}{2\pi(n+1)} \bigg( \frac{\sin\frac{(n+1)t}{2}} {\sin\frac t2}\bigg)^2 \]
be the Fejér kernel, \(C\) be the space of continuous \(2\pi\)-periodic functions \(f\) with the norm \(\|f\|= \max_{x\in\mathbb R}|f(x)|\), let
\[ J_n(f,x)= \frac{2\pi}{n+1} \sum_{k=0}^n f(t_k)\Phi_n(x-t_k), \quad\text{where}\quad t_k= \frac{2\pi k}{n+1}, \]
be the Jackson polynomials of the function \(f\), and let
\[ \sigma_n(f,x)= \int_{-\pi}^\pi f(x+t)\Phi_n(t)\,dt \]
be the Fejér sums of \(f\). The paper presents upper bounds for certain quantities like
\[ |f(x)-J_n(f,x)|, \quad |J_n(f,x)- \sigma_n(f,x), \quad \|f-J_n(t)\|, \quad \|J_n(t)-\sigma n(f)\|, \]
which are exact in order for every function \(f\in C\). Special attention is paid to the constants occurring in the inequalities obtained.
42A10 Trigonometric approximation
41A05 Interpolation in approximation theory
Full Text: DOI
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