Approximation of continuous functions by trigonometric polynomials almost everywhere |
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Authors: | T V Radoslavova |
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Institution: | 1. V. A. Steklov Mathematical Institute, Academy of Sciences of the USSR, USSR
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Abstract: | We consider the problem of the rate of approximation of continuous 2π-periodic functions of class WrHω]C by trigonometric polynomials of order n on sets of total measure. We prove that when r≥0,ω(δ)δ ?1 → ∞ (δ → 0) there exists a function f ε WrHω]C such thatf ε WrHω]C and for any sequence {tn n=1 ∞ we have almost everywhere on 0, 2π] $\begin{array}{l} \overline {\mathop {\lim }\limits_{n \to \infty } } \left| {f(x) - t_n (x)} \right|n^r \omega ^{ - 1} (1/n) > C_x > 0, \\ \overline {\mathop {\lim }\limits_{n \to \infty } } \left| {\tilde f(x) - t_n (x)} \right|n^r \omega ^{ - 1} (1/n) > C_x > 0. \\ \end{array}$ |
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