TY - JOUR

T1 - Examples of the Best Piecewise Linear Approximation with Free Nodes

AU - Malozemov, V. N.

AU - Tamasyan, G. Sh.

N1 - Malozemov, V.N., Tamasyan, G.S. Examples of the Best Piecewise Linear Approximation with Free Nodes. Vestnik St.Petersb. Univ.Math. 51, 380–385 (2018). https://doi.org/10.3103/S1063454118040118

PY - 2018

Y1 - 2018

N2 - The monograph published by E.Ya. Remez in 1957 addressed numerical methods with Chebyshev approximations. Particularly, the problem of the best uniform approximation of a function that is convex on an interval with continuous piecewise linear functions with free nodes was considered. In 1975, A.M. Vershik, V.N. Malozemov, and A.B. Pevnyi developed a general approach for constructing the best piecewise polynomial approximations with free nodes. The notion of partition with equal deviations was introduced, and it was found that such partition exists and generates the best piecewise polynomial approximation. In addition, a numerical method for constructing a partition with equal deviations was proposed. This paper gives three examples to describe the general approach to solving the problem of the best piecewise linear approximation with free nodes. In the case of an arbitrary continuous function, its best piecewise linear approximation in general is not continuous. It is continuous when approximating strictly convex and strictly concave functions.

AB - The monograph published by E.Ya. Remez in 1957 addressed numerical methods with Chebyshev approximations. Particularly, the problem of the best uniform approximation of a function that is convex on an interval with continuous piecewise linear functions with free nodes was considered. In 1975, A.M. Vershik, V.N. Malozemov, and A.B. Pevnyi developed a general approach for constructing the best piecewise polynomial approximations with free nodes. The notion of partition with equal deviations was introduced, and it was found that such partition exists and generates the best piecewise polynomial approximation. In addition, a numerical method for constructing a partition with equal deviations was proposed. This paper gives three examples to describe the general approach to solving the problem of the best piecewise linear approximation with free nodes. In the case of an arbitrary continuous function, its best piecewise linear approximation in general is not continuous. It is continuous when approximating strictly convex and strictly concave functions.

KW - Chebyshev approximations

KW - piecewise linear function

KW - partition with equal deviations

U2 - 10.3103/S1063454118040118

DO - 10.3103/S1063454118040118

M3 - Article

VL - 51

SP - 380

EP - 385

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 4

ER -