On Approximation of Continuous Functions by Entire Functions on Subsets of the Real Line

Abstract

We generalize the classical Bernstein theorem concerning the constructive description of classes of functions uniformly continuous on the real line. The approximation of continuous bounded functions by entire functions of exponential type on unbounded closed proper subsets of the real line is studded.

On approximation of continuous functions by trigonometric polynomials

“…Later, Levin [16] constructed the general theory of such mappings and used them to solve several extremal problems in classes of subharmonic functions. We are going to use the basic results of this theory adopted in [7] to investigate problems concerning approximation of continuous functions by entire functions of exponential type on subsets of the real line. Following [7, p. 92], we consider the case where E possesses the property…”

“…The inclusion A ω (E) ⊂ C ⋆ ω (E) follows from [7,Theorem 3]. In order to prove the converse inclusion, we adapt the reasoning from [1, pp.…”

“…Therefore, some metric properties of Φ and Ψ can be derived from the properties of ϕ proved in [7,Section 4]. For the convenience of the reader we repeat the relevant material from [4,6,7] without proofs, thus making our exposition self-contained.…”

“…Let f ∈ C ⋆ ω (E). According to [7,Theorem 2], for any n ∈ N there exists an entire function g n of exponential type at most n such that…”

“…The solution to Problem 1 follows directly relatively simply from the recent results given in [7] on the approximation of bounded continuous functions by entire functions of exponential type on subsets of the real line. The main idea is to introduce a new modulus of continuity by using a distance between the points on E that is not Euclidean (cf.…”