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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.
MSC:
42A10 Trigonometric approximation
41A05 Interpolation in approximation theory
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References:
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