Approximations by entire functions on countable unions of segments of the real axis

Аннотация

The problem to approximate functions continuous on subsets of the real line by entire functions has a long history that started from the Jackson–Bernstein theorem on the approximation of 2- periodic functions by trigonometric polynomials naturally treated as exponential-type entire functions. In this paper, we deal with the problem referring to the concept of this theorem describing classes of functional spaces via the rate of their possible approximation by entire functions. A key example is the Bernstein theorem describing the class of bounded functions from Holder classes over the whole axis by exponential-type entire functions. The key point is that the approximation rate at a neighborhood of the segment edge exceeds the one that originally appeared in the theory of approximation functions from Holder classes on segments (this allows us to coordinate the direct and inverse theorems for that case, i.e., to recover the holder smoothness from the approximation rate in the said scale). In the present paper, we present a direct theorem on the possibility of a prescribed-rate approximation of functions from Holder classes on countable unions of segments by entire functions. Earlier, such approximations were not considered. Also, we provide general definitions and important lemmas used for further constructing approximating functions. In the second part of the work, we provide a proof of the direct theorem. In our further papers, to obtain a constructive description of the smoothness class by means of the approximation rate, we will prove the corresponding inverse theorem. Usually, to deduce such assertions, one needs a fact similar to the Bernstein theorem on the estimate of the norm of an entire function via the norm of the function itself. In our case, we need an assertion similar to the Akhiezer–Levin theorem estimating an entire function on the axis via its values on a subset of the axis.