TY - JOUR

T1 - Comonotone approximation by hybrid polynomials

AU - Leviatan, D.

AU - Shcheglov, M. V.

AU - Shevchuk, I. A.

N1 - Publisher Copyright:
© 2023 Elsevier Inc.

PY - 2024/1/15

Y1 - 2024/1/15

N2 - Let f(x):=g(x)+ax, where g∈C˜, the space of continuous 2π-periodic functions and a∈R. Denote ‖g‖:=maxx∈R|g(x)|, and let ω(f,t), denote the modulus of continuity of f. Let Tn be the set of trigonometric polynomials Tn of degree n(x):=Tn(x)+ax, a∈R, a hybrid polynomial. If f is monotone, then g was called, by Salem and Zygmund, of monotone type. If f has an even number of extremal points in (−π,π], then we estimate inf{‖f−Qn‖:Qns.t.f′(x)Qn′(x)≥0, a.e. in R}, the error of its best comonotone approximation, in the uniform norm, by hybrid polynomials. We obtain Jackson-type estimates for the approximation of f by hybrid polynomials for a wide class of functions f. We also show cases where such estimates are invalid.

AB - Let f(x):=g(x)+ax, where g∈C˜, the space of continuous 2π-periodic functions and a∈R. Denote ‖g‖:=maxx∈R|g(x)|, and let ω(f,t), denote the modulus of continuity of f. Let Tn be the set of trigonometric polynomials Tn of degree n(x):=Tn(x)+ax, a∈R, a hybrid polynomial. If f is monotone, then g was called, by Salem and Zygmund, of monotone type. If f has an even number of extremal points in (−π,π], then we estimate inf{‖f−Qn‖:Qns.t.f′(x)Qn′(x)≥0, a.e. in R}, the error of its best comonotone approximation, in the uniform norm, by hybrid polynomials. We obtain Jackson-type estimates for the approximation of f by hybrid polynomials for a wide class of functions f. We also show cases where such estimates are invalid.

KW - Comonotone approximation by hybrid polynomials

KW - Degree of approximation

KW - Jackson-type estimates

UR - http://www.scopus.com/inward/record.url?scp=85152266015&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2023.127286

DO - 10.1016/j.jmaa.2023.127286

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AN - SCOPUS:85152266015

SN - 0022-247X

VL - 529

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 2

M1 - 127286

ER -