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