Кусочно-монотонная и рациональная аппрок симации и равномерная сходим ость рядов Фурье

Abstract

Let

$$R_n [f] = \inf \left\{ {\mathop {\max }\limits_{ - \pi \leqq x \leqq \pi } \left| {f(x) - \frac{{P(x)}}{{Q(x)}}} \right|} \right\}$$

, whereP andQ denote polynomials (algebraic or trigonometric) of degree ≦n. Theorem 2a. If for a continuous 2π-periodic function f the condition

$$\sum\limits_{n = 1}^\infty {\frac{1}{n}R_n [f]< \infty } $$

holds, then the Fourier series of f converges to f(x) uniformly. Theorem 2b. Let {β _n } be a non-increasing sequence of positive numbers such that

$$\sum\limits_{n = 1}^\infty {\frac{1}{n}\beta _n = \infty } $$

Then there exists a continuous 2π-periodic function f₀ for which R_n[f₀]≦β_n for all n=1 and yet the Fourier series of f₀ diverges at x=0.R _n [f]may be replaced in these theorems byM _n [f], whereM _n [f] is the minimal uniform deviation off(x) from piecewise monotonie functionsМ _n (х) of order ≦n.