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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
Full Text:
References:
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