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On Sikkema-Kantorovič polynomials of order K

Research paper by Cao Jiading

Indexed on: 01 Jun '89Published on: 01 Jun '89Published in: Approximation Theory and its Applications



Abstract

Let an≥0 and F(u)∈C [0,1], Sikkema constructed polynomials:\(B_n (\alpha _n ,F,x) = \sum\limits_{i = 0}^n F \left( {\frac{i}{{n + \alpha _n }}} \right)\left( {\begin{array}{*{20}c} n \\ i \\ \end{array} } \right)x^i (1 - x)^{n - 1} \), ifαn≡0, then Bn (0, F, x) are Bernstein polynomials.Let\(f(t) \in L[0,{\mathbf{ }}1],{\mathbf{ }}F{\mathbf{ }}{}_\kappa (u) = \frac{1}{{(k - 1)!}}\int_0^u {(u - v)^{k - 1} f(v)dv} \), we constructe new polynomials in this paper:Qn(k)(αn,f(t))=dk/dxkBn+k (αn,Fk(u),x), which are called Sikkema-Kantorovic polynomials of order k. Ifαn≡0, k=1, then Qn(1) (0, f(t), x) are Kantorovič polynomials Pn(f). Ifαn=0, k=2, then Qn(2), (0, f(t), x) are Kantorovič polynomials of second order (see Nagel). The main result is:Theorem 2. Let 1≤p≤∞, in order that for every f∈LP [0, 1],\(\mathop {\lim }\limits_{n \to \infty } \left\| {Q_n^{(k)} (\alpha _n ,f,x) - f(x)} \right\|_p = 0\), it is sufficient and necessary that\(\mathop {\lim }\limits_{n \to \infty } \frac{{\alpha _n }}{n} = 0\),§ 1. Let f(t) de a continuous function on [a, b], i. e., f∈C [a, b], we define[1–2],[8–10]:\(L^\infty [a,{\mathbf{ }}b] = C[a,{\mathbf{ }}b],{\mathbf{ }}\left\| f \right\|_\infty = \left\| f \right\|_c = \begin{array}{*{20}c} {max} \\ {a \leqslant 1 \leqslant b} \\ \end{array} \left| {f(t)} \right|.\).As usual, for the space Lp [a,b](1≤p<∞), we have\(\left\| f \right\|_P = \left\{ {\smallint _a^b \left| {f(t)} \right|{\mathbf{ }}{}^Pdt} \right\}^{\frac{1}{P}} \)and L[a, b]=l1[a, b].Letαn⩾0and F(u)∈C[0,1],Sikkema-Bernstein polynomials[3] [4].