On the image of the limit q-Durrmeyer operator

Abstract

The focus of this work is on the properties of the $q$-Durrmeyer operators $M_{n,q},$ $n\in N,$ and $M_{\infty ,q}$ introduced, for $q\in (0,1),$ by V. Gupta and H. Wang. First, it is shown that, for each $f\in C[0,1],$ the sequence ${\left\{M_{n,q}f\right\}}_{n\in N}$ converges to $M_{\infty ,q}f$ uniformly on $[0,1]$ with a rate not slower than $C_{q,f}{q}^n$, which refines the previously available result by V. Gupta and H. Wang, and implies the possibility of an analytic continuation for $M_{\infty ,q}f$ into a neighbourhood of $[0,1].$ Further investigation shows that $M_{\infty ,q}f$ admits an analytic continuation as an entire function regardless of $f\in C[0,1]$. Finally, the growth estimates for these functions are received and applied to describe the point spectrum of $M_{\infty ,q}.$ The paper also addresses the significant differences between the properties of $M_{\infty ,q}$ and the previously well-known limit $q$-Bernstein operator $B_{\infty ,q}.$