Abstract
Приближение непрерывных 2π-периодических кусочно-гладких функций дискретными суммами Фурье
Highlights
In the present article we consider the problem of approximation of functions f ∈ CΩ2 by the polynomials Ln,N (f, x)
We show that instead of the estimate |f (x) − Ln,N (f, x)| c ln n/n, which follows from the well-known Lebesgue inequality, we found an exact order estimate |f (x) − Ln,N (f, x)| c/n (x ∈ R) which is uniform with respect to n (1 n N/2)
The proofs of these estimations are based on comparing of approximating properties of discrete and continuous finite Fourier series
Summary
Let N be a natural number greater than 1. Select N uniformly distributed points tk = 2πk/N + u (0 k N − 1), and denote by Ln,N (f ) = Ln,N (f, x) (1 n N/2) the trigonometric polynomial of order n possessing the least quadratic deviation from f with respect to the system {tk}kN=−01. In the present article we consider the problem of approximation of functions f ∈ CΩ2 by the polynomials Ln,N (f, x). We found a local estimate |f (x) − Ln,N (f, x)| c(ε)/n2 (|x − ai| ε) which is uniform with respect to n (1 n N/2) The proofs of these estimations are based on comparing of approximating properties of discrete and continuous finite Fourier series.
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