SOME APPROXIMATION THEOREMS FOR MODIFIED BERNSTEIN-DURRMEYER OPERATORS

The modified Bernstein-Durrmeyer operators discussed in this paper are given byM_nf≡M_n(f,x)=(n+2)P_(n,k)∫_0~1p_n+1.k(t)f(t)dt,whereWe will show,for 0<α<1 and 1≤p≤∞

Some approximation theorems for modified Bernstein-Durrmeyer operators

The modified Bernstein-Durrmeyer operators discussed in this paper are given by $$M_n f \equiv M_n (f,x) = (n + 2)\sum\limits_{k = 0}^n {P_{n,k} (x)\int_0^1 {P_{n + 1,k} (t)f(t)dt} } $$ where $$P_{nk} (x) = \left( \begin{gathered} n \hfill \\ k \hfill \\ \end{gathered} \right)x^k (1 - x)^{n - k} $$ We will show, for 0<α<1 and 1≤p≤∞ $$\begin{gathered} \left\| {M_n f - f} \right\|_p = o(n^{ - \alpha } ) \Leftrightarrow \omega _\varphi ^2 (f,t)_p = o(t^{2\alpha } ), \hfill \\ \left| {M_n f - f(x)} \right| \leqslant M\left( {\tfrac{{x(1 - x)}}{n} + \tfrac{1}{{n^2 }}} \right)^{\tfrac{\alpha }{2}} \Leftrightarrow \omega (f,t) = O(t^\alpha ) \hfill \\ \end{gathered} $$ where $$\begin{gathered} \omega _\varphi ^2 (f,t)_p = \mathop {\sup }\limits_{0 \leqslant k \leqslant t} \left\| {\Delta _{h\varphi }^2 f(x)} \right\|_{L_p 0,1]} ,\varphi ^2 (x) = x(1 - x), \hfill \\ \Delta _{h\varphi }^2 f(x) = \sum\limits_{k = 0}^2 {( - 1)^k } \left( \begin{gathered} 2 \hfill \\ k \hfill \\ \end{gathered} \right)f(x + (1 - k)h\varphi ), if \left {x - h\varphi (x),x + h\varphi (x)} \right] \subset 0,1] \hfill \\ \end{gathered} $$ Δ _h? ² f(x)=0, otherwise.