The pointwise approximation by left Szász-Mirakjan quasi-interpolants operators (Q1884519)

The Szász-Mirakyan operators are defined by \[ S_n(f,x):=e^{-nx}\sum_{k=0}^\infty f(\frac kn)\frac{(nx)^k}{k!},\quad f\in C_B[0,\infty), \] where \(C_B[0,\infty)\) is the set of bounded continuous functions on \([0,\infty)\). Since \(S_n\) may be viewed as a mapping of the space of polynomials of degree \(n\) onto itself, it possesses an inverse which has the representation \(S^{-1}_n=:\sum_{k=0}^n\alpha_{n,k}D^k\), where \(\alpha_{n,k}(x)\) is a polynomial of degree \(k\), and \(D:=\frac d{dx}\). For \(0\leq\nu\leq n\) one defines the left Szász-Mirakyan quasi-interpolant \[ S_n^{(\nu)}f:=\sum_{k=0}^\nu\alpha_{n,k}D^kS_nf,\quad f\in C_B[0,\infty). \] Finally, for \(0\leq\lambda\leq1\), \(\varphi(x):=\sqrt x\), and \(f\in C_B[0,\infty)\), denote the \(r\)th Ditzian-Totík modulus of smoothness \[ \omega^r_{\varphi^\lambda}(f,t):=\sup_{0<h\leq t}\| \Delta^r_{h\varphi^\lambda(\cdot)}f(\cdot)\| , \] where \(\Delta\) is the ordinary symmetric difference and the function inside the norm is taken to be zero where it is not defined. The authors prove that given \(f\) and \(r\geq1\), for \(n\geq2r-1\), \(1-\frac1r\leq\lambda\leq1\), and \(0<\alpha<2r\), we have \[ | S_n^{(2r-1)}(f,x)-f(x)| =O\biggl(\biggl(\frac{\varphi^{1-\lambda}(x)}{\sqrt n}\biggr)^\alpha\biggr)\Longleftrightarrow\omega^{2r}_{\varphi^\lambda}(f,t)=O(t^\alpha), \] while the implication \(\Leftarrow\) is invalid for \(0\leq\lambda<1-\frac1r\). However, with \(\delta_n(x):=\max\{\frac1{\sqrt n},\varphi(x)\}\), we have the following equivalence for all \(0\leq\lambda\leq1\), \[ | S_n^{(2r-1)}(f,x)-f(x)| =O\biggl(\biggl(\frac{\delta_n^{1-\lambda}(x)}{\sqrt n}\biggr)^\alpha\biggr)\Longleftrightarrow\omega^{2r}_{\varphi^\lambda}(f,t)=O(t^\alpha). \]