Local properties of functions and approximation by trigonometric polynomials

Abstract

Suppose Φ_{p, E} (p>0 an integer, E ⊂[0, 2π]) is a family of positive nondecreasing functionsϕ _x(t) (t>0, x∈ E) such thatϕ _x(nt)≤n^P ϕ _x(t) (n=0,1,...), t_n is a trigonometric polynomial of order at most n, and Δ ^l_h (f, x) (l>0 an integer) is the finite difference of orderl with step h of the functionf.THEOREM. Supposef (x) is a function which is measurable, finite almost everywhere on [0, 2π], and integrable in some neighborhood of each point xε E,ϕ _X εΦ_p,E and

$$\overline {\mathop {\lim }\limits_{\delta \to \infty } } |(2\delta )^{ - 1} \smallint _{ - \delta }^\delta \Delta _u^l (f,x)du|\varphi _x^{ - 1} (\delta ) \leqslant C(x)< \infty (x \in E).$$

. Then there exists a sequence {t _n } ^∞ _n=1 which converges tof (x) almost everywhere, such that for x ε E

$$\overline {\mathop {\lim }\limits_{n \to \infty } } |f(x) - l_n (x)|\varphi _x^{ - 1} (l/n) \leqslant AC(x),$$

where A depends on p andl.