TY - JOUR

T1 - No Jackson-Type Estimates for Piecewise q-Monotone, q≥3, Trigonometric Approximation

AU - Leviatan, D.

AU - Motorna, O. V.

AU - Shevchuk, I. A.

N1 - Publisher Copyright:
© 2022, Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2022/10

Y1 - 2022/10

N2 - We say that a function f ∈ C[a, b] is q-monotone, q ≥ 2, if f ∈ Cq-2(a, b), i.e., belongs to the space of functions with (q -2)th continuous derivative in (a, b), and f(q-2) is convex in this space. Let f be continuous and 2-periodic. Assume that it changes its q-monotonicity finitely many times in. We are interested in estimating the degree of approximation of f by trigonometric polynomials, which are co-q-monotone with this function, namely, trigonometric polynomials that change their q-monotonicity exactly at the points where f does. These Jackson-type estimates are valid for piecewise monotone (q = 1) and piecewise convex (q = 2) approximations. However, we prove, that no estimates of this kind are valid, in general, for the co-q-monotone approximation with q ≥ 3.

AB - We say that a function f ∈ C[a, b] is q-monotone, q ≥ 2, if f ∈ Cq-2(a, b), i.e., belongs to the space of functions with (q -2)th continuous derivative in (a, b), and f(q-2) is convex in this space. Let f be continuous and 2-periodic. Assume that it changes its q-monotonicity finitely many times in. We are interested in estimating the degree of approximation of f by trigonometric polynomials, which are co-q-monotone with this function, namely, trigonometric polynomials that change their q-monotonicity exactly at the points where f does. These Jackson-type estimates are valid for piecewise monotone (q = 1) and piecewise convex (q = 2) approximations. However, we prove, that no estimates of this kind are valid, in general, for the co-q-monotone approximation with q ≥ 3.

UR - http://www.scopus.com/inward/record.url?scp=85140852570&partnerID=8YFLogxK

U2 - 10.1007/s11253-022-02099-x

DO - 10.1007/s11253-022-02099-x

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:85140852570

SN - 0041-5995

VL - 74

SP - 757

EP - 772

JO - Ukrainian Mathematical Journal

JF - Ukrainian Mathematical Journal

IS - 5

ER -