TY - JOUR

T1 - Degree of Simultaneous Coconvex Polynomial Approximation

AU - Kopotun, K.

AU - Leviatan, D.

N1 - Publisher Copyright:
© 1998, Birkhäuser Verlag, Basel.

PY - 1998/8/1

Y1 - 1998/8/1

N2 - Let f ɛ C1[-1, 1] change its convexity finitely many times in the interval, say s times, at Ys : -1 < Ys < ... < Y1 < 1. We estimate the degree of simultaneous approximation of ƒ and its derivative by polynomials of degree n, which change convexity exactly at the points Ys, and their derivatives. We show that provided n is sufficiently large, depending on the location of the points Ys, the rate of approximation can be estimated by C(s)/n times the second Ditzian-Totik modulus of smoothness of ƒ′. This should be compared to a recent paper by the authors together with I. A. Shevchuk where ƒ is merely assumed to be continuous and estimates of coconvex approximation are given by means of the third Ditzian-Totik modulus of smoothness. However, no simultaneous approximation is given there.

AB - Let f ɛ C1[-1, 1] change its convexity finitely many times in the interval, say s times, at Ys : -1 < Ys < ... < Y1 < 1. We estimate the degree of simultaneous approximation of ƒ and its derivative by polynomials of degree n, which change convexity exactly at the points Ys, and their derivatives. We show that provided n is sufficiently large, depending on the location of the points Ys, the rate of approximation can be estimated by C(s)/n times the second Ditzian-Totik modulus of smoothness of ƒ′. This should be compared to a recent paper by the authors together with I. A. Shevchuk where ƒ is merely assumed to be continuous and estimates of coconvex approximation are given by means of the third Ditzian-Totik modulus of smoothness. However, no simultaneous approximation is given there.

KW - Coconvex polynomial approximation

KW - Jackson estimates

UR - http://www.scopus.com/inward/record.url?scp=85054013549&partnerID=8YFLogxK

U2 - 10.1007/BF03322045

DO - 10.1007/BF03322045

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AN - SCOPUS:85054013549

SN - 1422-6383

VL - 34

SP - 150

EP - 155

JO - Results in Mathematics

JF - Results in Mathematics

IS - 1-2

ER -