Generalized Baskakov-beta operators (Q1047547)

Denote \[ C_{\gamma}[0,\infty):=\left\{ f\in C[0,\infty): f(t)=O(t^{\gamma}) \text{ as } t\to\infty \text{ for some } \gamma>0\right\}. \] For \(f\in C_{\gamma}[0,\infty)\) and \(\alpha>0\) consider the modified Baskakov-beta operators, introduced by Wang in 2005: \[ B_{n,\alpha}(f,x)= \sum_{k=0}^\infty p_{n,k,\alpha}(x)\int_0^\infty b_{n,k,\alpha}(t)f(t)\,dt= \int_0^\infty W_{n,\alpha}(x,t)f(t)\,dt, \] where \[ \begin{aligned} p_{n,k,\alpha}(x)&= \frac{\Gamma(n/\alpha+k)}{\Gamma(k+1)\Gamma(n/\alpha)}\cdot \frac{(\alpha x)^k}{(1+\alpha x)^{(n/\alpha)+k}},\\ b_{n,k,\alpha}(t)&= \frac{1}{B(n/\alpha,k+1)}\cdot \frac{\alpha(\alpha t)^k}{(1+\alpha t)^{n/\alpha+k+1}}\end{aligned} \] and \[ W_{n,\alpha}(x,t)= \sum_{k=0}^\infty p_{n,k,\alpha}(x)b_{n,k,\alpha}(t). \] The authors give an estimation in terms of higher order modulus of continuity for these modified Baskakov-beta operators: Theorem 9. Let \(f\in C_{\gamma}[0,\infty)\), and suppose \(0<a<a_{1}<b_{1}<b<\infty\). Then, for all \(n\) sufficiently large, we have \[ \left\| B_{n,\alpha}^{(r)}(f,\bullet)-f^{(r)}(\cdot)\right\|_{C[a_{1},b_{1}]}\leq M\left\{\omega_{2}(f^{(r)},n^{-1/2},a,b)+n^{-1}\left\|f\right\|_{\gamma}\right\}. \] The last section of this paper is devoted to the following inverse theorem in simultaneous approximation: Theorem 10. Let \(0<\beta<2\), \(0<a_1<a_2< b_2<b_1<\infty\), \(f\in C_0^r\) and \(f(t)=O(t^\gamma)\). Then in the following statements (i) \(\Rightarrow\) (ii) (i) \(\| B_{n,\alpha}^{(r)}(f,\bullet)-f^{(r)}(\cdot)\|_{C[a_{1},b_{1}]}=O(n^{-\beta/2})\), (ii) \(f^{(r)}\in \text{Lip}^{\ast}(\beta,a_{2},b_{2})\).