Research output: Contribution to journal › Article › peer-review
Approximation by entire functions on a countable union of segments on the real axis:3. Further generalization. / Silvanovich, O.V.; Shirokov , N. A. .
In: Vestnik St. Petersburg University: Mathematics, Vol. 51, No. 2, 2018, p. 164-168.Research output: Contribution to journal › Article › peer-review
}
TY - JOUR
T1 - Approximation by entire functions on a countable union of segments on the real axis:3. Further generalization
AU - Silvanovich, O.V.
AU - Shirokov , N. A.
N1 - Silvanovich, O.V., Shirokov, N.A. Approximation by Entire Functions on a Countable Union of Segments on the Real Axis: 3. Further Generalization. Vestnik St.Petersb. Univ.Math. 51, 164–168 (2018). https://doi.org/10.3103/S1063454118020085
PY - 2018
Y1 - 2018
N2 - In this paper, an approximation of functions of extensive classes set on a countable unit of segments of a real axis using the entire functions of exponential type is considered. The higher the type of the approximating function is, the higher the rate of approximation near segment ends can be made, compared with their inner points. The general approximation scale, which is nonuniform over its segments, depending on the type of the entire function, is similar to the scale set out for the first time in the study of the approximation of the function by polynomials. For cases with one segment and its approximation by polynomials, this scale has allowed us to connect the so-called direct theorems, which state a possible rate of smooth function approximation by polynomials, and the inverse theorems, which give the smoothness of a function approximated by polynomials at a given rate. The approximations by entire functions on a countable unit of segments for the case of Hölder spaces have been studied by the authors in two preceding papers. This paper significantly expands the class of spaces for the functions, which are used to plot an approximation that engages the entire functions with the required properties.
AB - In this paper, an approximation of functions of extensive classes set on a countable unit of segments of a real axis using the entire functions of exponential type is considered. The higher the type of the approximating function is, the higher the rate of approximation near segment ends can be made, compared with their inner points. The general approximation scale, which is nonuniform over its segments, depending on the type of the entire function, is similar to the scale set out for the first time in the study of the approximation of the function by polynomials. For cases with one segment and its approximation by polynomials, this scale has allowed us to connect the so-called direct theorems, which state a possible rate of smooth function approximation by polynomials, and the inverse theorems, which give the smoothness of a function approximated by polynomials at a given rate. The approximations by entire functions on a countable unit of segments for the case of Hölder spaces have been studied by the authors in two preceding papers. This paper significantly expands the class of spaces for the functions, which are used to plot an approximation that engages the entire functions with the required properties.
KW - smooth functions
KW - entire functions of exponential type
KW - approximation on the real axis subset
UR - http://www.scopus.com/inward/record.url?scp=85048661742&partnerID=8YFLogxK
U2 - 10.3103/S1063454118020085
DO - 10.3103/S1063454118020085
M3 - Article
VL - 51
SP - 164
EP - 168
JO - Vestnik St. Petersburg University: Mathematics
JF - Vestnik St. Petersburg University: Mathematics
SN - 1063-4541
IS - 2
ER -
ID: 28190754