Comonotone approximation of twice differentiable periodic functions

In the case where a 2π-periodic function f is twice continuously differentiable on the real axis ℝ and changes its monotonicity at different fixed points y _i ∈ [− π, π), i = 1,…, 2s, s ∈ ℕ (i.e., on ℝ, there exists a set Y := {y _i } _i∈ℤ of points y _i = y _i+2s + 2π such that the function f does not decrease on [y _i , y _i−1] if i is odd and does not increase if i is even), for any natural k and n, n ≥ N(Y, k) = const, we construct a trigonometric polynomial T _n of order ≤n that changes its monotonicity at the same points y _i ∈ Y as f and is such that

*20c || f - Tn || ￡ fracc( k,s )n2upomega k( f",1 mathordvphantom 1 n n ) ( || f - Tn || ￡ fracc( r + k,s )nrupomega k( f(r),1 mathord	/	vphantom 1 n n ),    f ? C(r),    r 3 2 ), begin{array}{*{20}{c}} {left| {f - {T_n}} right| leq frac{{cleft( {k,s} right)}}{{{n^2}}}{{{upomega }}_k}left( {f',{1 mathord{left/{vphantom {1 n}} right.} n}} right)} {left( {left| {f - {T_n}} right| leq frac{{cleft( {r + k,s} right)}}{{{n^r}}}{{{upomega }}_k}left( {{f^{(r)}},{1 mathord{left/{vphantom {1 n}} right.} n}} right),quad f in {C^{(r)}},quad r geq 2} right),} end{array}
Keywords:
本文献已被 SpringerLink 等数据库收录！