Local properties of functions and approximation by trigonometric polynomials

Abstract

Suppose Φp, E (p>0 an integer, E ⊂[0, 2π]) is a family of positive nondecreasing functionsϕx(t) (t>0, x∈ E) such thatϕx(nt)≤nPϕx(t) (n=0,1,...), tn is a trigonometric polynomial of order at most n, and Δhl(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 xe E,ϕXeΦp,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 {tn}n=1∞ which converges tof (x) almost everywhere, such that for x e 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.