TY - JOUR

T1 - Approximation by entire functions on countable unions of segments of the real axis: 2. proof of the main theorem

AU - Silvanovich, O. V.

AU - Shirokov, N. A.

N1 - Silvanovich, O.V., Shirokov, N.A. Approximation by entire functions on countable unions of segments of the real axis: 2. proof of the main theorem. Vestnik St.Petersb. Univ.Math. 50, 35–43 (2017). https://doi.org/10.3103/S1063454117010125

PY - 2017/1

Y1 - 2017/1

N2 - In this study, we consider an approximation of entire functions of Hölder classes on a countable union of segments by entire functions of exponential type. It is essential that the approximation rate in the neighborhood of segment ends turns out to be higher in the scale that had first appeared in the theory of polynomial approximation by functions of Hölder classes on a segment and made it possible to harmonize the so-called “direct” and “inverse” theorems for that case, i.e., restore the Hölder smoothness by the rate of polynomial approximation in this scale. Approximations by entire functions on a countable union of segments have not been considered earlier. The first section of this paper presents several lemmas and formulates the main theorem. In this study, we prove this theorem on the basis of earlier given lemmas.

AB - In this study, we consider an approximation of entire functions of Hölder classes on a countable union of segments by entire functions of exponential type. It is essential that the approximation rate in the neighborhood of segment ends turns out to be higher in the scale that had first appeared in the theory of polynomial approximation by functions of Hölder classes on a segment and made it possible to harmonize the so-called “direct” and “inverse” theorems for that case, i.e., restore the Hölder smoothness by the rate of polynomial approximation in this scale. Approximations by entire functions on a countable union of segments have not been considered earlier. The first section of this paper presents several lemmas and formulates the main theorem. In this study, we prove this theorem on the basis of earlier given lemmas.

KW - Hölder classes

KW - entire functions of exponential type

KW - approximation on subsets of the real axis

UR - https://link.springer.com/article/10.3103/S1063454117010125

M3 - Article

VL - 50

SP - 35

EP - 43

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 1

ER -