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1.
Suppose that a continuous 2π-periodic function f on the real axis ? changes its monotonicity at different ordered fixed points y i ∈ [?π,π), i = 1, …, 2s, s ∈ ?. In other words, there is a set Y: = {y i } i∈? of points y i = y i+2s + 2π on ? such that f is nondecreasing on [y i ,y i?1] if i is odd and not increasing if i is even. For each n ≥ N(Y), we construct a trigonometric polynomial P n of order ≤ n changing its monotonicity at the same points y i ∈ Y as f and such that $$ \parallel f - P_n \parallel \leqslant c(s) \omega _2 \left( {f,\frac{\pi } {n}} \right), $$ where N(Y) is a constant depending only on Y, c(s) is a constant depending only on s, ω2(f,·) is the modulus of continuity of second order of the function f, and ∥ · ∥ is the max-norm. 相似文献
2.
H. A. Dzyubenko 《Ukrainian Mathematical Journal》2009,61(4):519-540
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 )n2\upomega k( f",1 \mathord\vphantom 1 n n ) ( || f - Tn || £ \fracc( r + k,s )nr\upomega 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{{c\left( {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{{c\left( {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} 相似文献
3.
In Part I of the paper, we have proved that, for every α > 0 and a continuous function f, which is either convex (s = 0) or changes convexity at a finite collection Y
s
= {y
i
}
s
i=1 of points y
i
∈ (-1, 1),
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