TY - JOUR

T1 - Approximation by entire functions on a countable union of real-axis segments 4. Inverse theorem

AU - Sil’vanovich, O. V.

AU - Shirokov, N. A.

N1 - Sil’vanovich, O.V., Shirokov, N.A. Approximation by Entire Functions on a Countable Union of Real-Axis Segments. 4. Inverse Theorem. Vestnik St.Petersb. Univ.Math. 51, 267–275 (2018). https://doi.org/10.3103/S1063454118030081

PY - 2018

Y1 - 2018

N2 - For more than a century, the constructive description of functional classes in terms of the possible rate of approximation of its functions by means of functions chosen from a certain set remains among the most important problems of approximation theory. It turns out that the nonuniformity of the approximation rate due between the points of the domain of the approximated function is substantial. For instance, it was only in the mid-1950s that it was possible to constructively describe Holder classes on the segment [–1; 1] in terms of the approximation by algebraic polynomials. For that particular case, the constructive description requires the approximation at neighborhoods of the segment endpoints to be essentially better than the one in a neighborhood of its midpoint. A possible approximation quality test is to find out whether the approximation rate provides a possibility to reconstruct the smoothness of the approximated function. Earlier, we investigated the approximation of classes of smooth functions on a countable union of segments on the real axis. In the present paper, we prove that the rate of the approximation by the entire exponential-type functions provides the possibility to reconstruct the smoothness of the approximated function, i.e., a constructive description of classes of smooth functions is possible in terms of the specified approximation method. In an earlier paper, that result is announced for Holder classes, but the construction of a certain function needed for the proof is omitted. In the present paper, we use another proof; it does not apply the specified function.

AB - For more than a century, the constructive description of functional classes in terms of the possible rate of approximation of its functions by means of functions chosen from a certain set remains among the most important problems of approximation theory. It turns out that the nonuniformity of the approximation rate due between the points of the domain of the approximated function is substantial. For instance, it was only in the mid-1950s that it was possible to constructively describe Holder classes on the segment [–1; 1] in terms of the approximation by algebraic polynomials. For that particular case, the constructive description requires the approximation at neighborhoods of the segment endpoints to be essentially better than the one in a neighborhood of its midpoint. A possible approximation quality test is to find out whether the approximation rate provides a possibility to reconstruct the smoothness of the approximated function. Earlier, we investigated the approximation of classes of smooth functions on a countable union of segments on the real axis. In the present paper, we prove that the rate of the approximation by the entire exponential-type functions provides the possibility to reconstruct the smoothness of the approximated function, i.e., a constructive description of classes of smooth functions is possible in terms of the specified approximation method. In an earlier paper, that result is announced for Holder classes, but the construction of a certain function needed for the proof is omitted. In the present paper, we use another proof; it does not apply the specified function.

KW - smooth functions

KW - entire functions

KW - approximation

UR - http://www.scopus.com/inward/record.url?scp=85052677210&partnerID=8YFLogxK

U2 - 10.3103/S1063454118030081

DO - 10.3103/S1063454118030081

M3 - Article

VL - 51

SP - 267

EP - 275

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 3

ER -