Some approximation results for operators of Szász-Mirakjan-Durrmeyer type (Q2836155)

In the papers by \textit{M. Herzog} [Matematiche 54, No. 1, 77--90 (1999; Zbl 0960.41015); Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 43, No. 1, 77--94 (2003; Zbl 1129.41308)], a certain modification of the well-known discrete operators was introduced with the aid of a modified Bessel function of the first kind \(I_{\nu}\). Now the author proposes an integral case of the operators in the following manner NEWLINE\[NEWLINE L_{n}^{\nu}(f;x)= \begin{cases} \frac {n}{I_{\nu}(nx)}\sum \limits_{k=0}^{\infty} p_{n,k,\nu}(x)\int \limits_0^{\infty} q_{n,k,\nu}(t)f(t)\,{\operatorname {d}}t, & \qquad x> 0; \\ f(0), & \qquad x=0, \end{cases} NEWLINE\]NEWLINE where NEWLINE\[NEWLINE p_{n,k,\nu}(x)=\frac {\bigl ((nx)/2\bigr)^{2k+\nu}}{\Gamma (k+1)\Gamma (k+\nu +1)}, NEWLINE\]NEWLINE NEWLINE\[NEWLINE q_{n,k,\nu}(t)=\frac {e^{-nt}(nt)^{2k+\nu}}{\Gamma (2k+\nu +1)} NEWLINE\]NEWLINE for \(n\in N\), \(\nu \geq 0\) and \(x\geq 0\).NEWLINENEWLINEDirect theorems on the degree of approximation for functions belonging to exponential weighted spaces \(E_q\) are proved. For example, in Corollary 3.3.1. for \(f\in E_q\), \(f\) differentiable in the neighbourhood of the point \(x\) and such that \(f''(x)\) exists, we have NEWLINE\[NEWLINE \bigl | L_{n}^{\nu}(f;x)-f(x)\bigr | =O(n^{-1}) \quad \text{as} \quad n\to \infty . NEWLINE\]NEWLINE The rate of convergence is estimated by using the norm of weighted spaces, the weighted modulus of continuity and the modulus of smoothness.