Approximation by entire functions on a countable union of segments on the real axis. 3. Further generalization

Abstract

In the present paper we consider the approximation of smooth functions from various classes with the help of entire functions of exponential type. We assume that our smooth functions f are defined on an union of countable set of segments {[an; bn]} lying on the real axis such that all of those segments are commensurable and all complementary intervals are commensurable too. Let ! be a modulus of continuity satisfying a classical Dini condition, namely x Z 0 !(t) t dt + x ∞ Z x !(t) t2 dt c · !(x). We consider classes of functions f such that f is bounded on S n∈Z [an; bn] and for all r 0, x1, x2 2 In one has a property |f(r)(x2) − f(r)(x1)| cf!(|x2 − x1|), f(0) def = f. We denote through T a set of entire functions of exponential type bounded on the real axis. The main result of our paper is following. Theorem. Let a function f and a modulus of continuity ! satisfy conditions mentioned above. Then for any 1 there exists a function F 2 T such that for x 2 S n∈Z [an; bn] one has an estimate |f(x) − F (x)| cf dr 1+ 1

(x, [n∈Z [an; bn])!(d1+ 1

(x, [n∈Z [an; bn])), where characteristic d (x, . . .) was introduced in our paper (Vestnik St.Petersburg Univ.: Math. 49, issue 4, 373–378 (2016)).