ISSN 1786-0091 APPROXIMATION PROPERTIES OF PARTIAL SUMS OF FOURIER SERIES

Abstract. In this paper we find class of functions for which the Lebesgue estimate can be improved. Let C ([0, 2pi]) denote the space of continuous function f with period 2pi. If f ∈ C ([0, 2pi]) then the function ωp (δ, f) = sup x sup |h|≤δ |∆p (x;h, f) | , ω1 (δ, f) = ω (δ, f) is called the modulus of continuity of the function f, where ∆1 (x;h, f) = f (x+ h) − f (x), ∆p+1 (x;h, f) = ∆p (x+ h;h, f)−∆p (x;h, f). Denote by Lipα the class of function f ∈ C ([0, 2pi]) for which ω (δ, f) ≤ c (f) δα and let Sn (f, x) be the n-th partial sum of the trigonometric Fourier series of the function f. The estimation of Lebesgue (see [Dz, p. 116], or [Ba, Ch. 1])is well known ‖f − Sn (f)‖C ≤ c