Comonotone approximation of twice differentiable periodic functions

Negative results in coconvex approximation of periodic functions

“…Remark 1.4. We do not discuss the comonotone approximation in the Introduction, we only note, that in the comonotone approximation of periodic functions more positive, but less negative results are known, see, [7], [8], [1] for more details. However in the last Section we formulate the analogs of Theorems 1.2 and 1.3 for the comonotone (co-1-monotone) approximation.…”

“…Let F be a function, guaranteed by Theorem 5.2. Since F is a 2π-periodic function and F ∈ ∆ (1) (Y s ) ∩ C (1) , the function…”

“…is also 2π-periodic and f ∈ ∆ (2) (Y s ) ∩ C (1) . For each polynomial T n ∈ ∆ (2) (Y s ) ∩ T n (5.1) implies 2) n (f, Y s ) ≥ cE (1) n (F, Y s ) − ω 3 (F, 1/n). Therefore…”

“…We say that a 2π-periodic function f ∈ C(R) is piecewise monotone with respect to Y s ∈ Y s , if it is a nondecreasing function on [y 1 , y 0 ] and it changes its monotonicity at the points Y s , that is, if (−1) i−1 f is nondecreasing on [y i , y i−1 ], 1 ≤ i ≤ 2s. We denote by ∆ (1) (Y s ) the collection of all such piecewise monotone functions. For a function g ∈ ∆ (1) (Y s ) denote by E (1) n (g, Y s ) := inf…”

“…We denote by ∆ (1) (Y s ) the collection of all such piecewise monotone functions. For a function g ∈ ∆ (1) (Y s ) denote by E (1) n (g, Y s ) := inf…”