TY - JOUR

T1 - Interpolatory pointwise estimates for monotone polynomial approximation

AU - Kopotun, K. A.

AU - Leviatan, D.

AU - Shevchuk, I. A.

N1 - Publisher Copyright:
© 2017 Elsevier Inc.

PY - 2018/3/15

Y1 - 2018/3/15

N2 - Given a nondecreasing function f on [−1,1], we investigate how well it can be approximated by nondecreasing algebraic polynomials that interpolate it at ±1. We establish pointwise estimates of the approximation error by such polynomials that yield interpolation at the endpoints (i.e., the estimates become zero at ±1). We call such estimates “interpolatory estimates”. In 1985, DeVore and Yu were the first to obtain this kind of results for monotone polynomial approximation. Their estimates involved the second modulus of smoothness ω2(f,⋅) of f evaluated at 1−x2/n and were valid for all n≥1. The current paper is devoted to proving that if f∈Cr[−1,1], r≥1, then the interpolatory estimates are valid for the second modulus of smoothness of f(r), however, only for n≥N with N=N(f,r), since it is known that such estimates are in general invalid with N independent of f. Given a number α>0, we write α=r+β where r is a nonnegative integer and 0<β≤1, and denote by Lip⁎α the class of all functions f on [−1,1] such that ω2(f(r),t)=O(tβ). Then, one important corollary of the main theorem in this paper is the following result that has been an open problem for α≥2 since 1985 If α>0, then a function f is nondecreasing and in Lip⁎α if and only if, there exists a constant C such that, for all sufficiently large n, there are nondecreasing polynomials Pn, of degree n, such that |f(x)−Pn(x)|≤C([Formula presented])α,x∈[−1,1].

AB - Given a nondecreasing function f on [−1,1], we investigate how well it can be approximated by nondecreasing algebraic polynomials that interpolate it at ±1. We establish pointwise estimates of the approximation error by such polynomials that yield interpolation at the endpoints (i.e., the estimates become zero at ±1). We call such estimates “interpolatory estimates”. In 1985, DeVore and Yu were the first to obtain this kind of results for monotone polynomial approximation. Their estimates involved the second modulus of smoothness ω2(f,⋅) of f evaluated at 1−x2/n and were valid for all n≥1. The current paper is devoted to proving that if f∈Cr[−1,1], r≥1, then the interpolatory estimates are valid for the second modulus of smoothness of f(r), however, only for n≥N with N=N(f,r), since it is known that such estimates are in general invalid with N independent of f. Given a number α>0, we write α=r+β where r is a nonnegative integer and 0<β≤1, and denote by Lip⁎α the class of all functions f on [−1,1] such that ω2(f(r),t)=O(tβ). Then, one important corollary of the main theorem in this paper is the following result that has been an open problem for α≥2 since 1985 If α>0, then a function f is nondecreasing and in Lip⁎α if and only if, there exists a constant C such that, for all sufficiently large n, there are nondecreasing polynomials Pn, of degree n, such that |f(x)−Pn(x)|≤C([Formula presented])α,x∈[−1,1].

KW - Degree of approximation

KW - Jackson-type interpolatory estimates

KW - Monotone polynomial approximation

UR - http://www.scopus.com/inward/record.url?scp=85035236027&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2017.11.038

DO - 10.1016/j.jmaa.2017.11.038

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AN - SCOPUS:85035236027

SN - 0022-247X

VL - 459

SP - 1260

EP - 1295

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 2

ER -