LAUSR

AbstractLet $$\{\delta _{n} \}^{\infty }_{n=1}$${δn}n=1∞   and   $$\{\delta ^{\prime }_{n}\}^{\infty }_{n=1}$${δn′}n=1∞ be two sequences of positive numbers, and $$\begin{aligned} C_{n,x}= \{ k: k\in N\cup \{0\} \,\,\text{ and }\,\, n(x-\delta ^{\prime }_{n})\le k\le n(x+\delta _{n}) \}. \end{aligned}$$Cn,x={k:k∈N∪{0}andn(x-δn′)≤k≤n(x+δn)}.For any continuous function $$f: [0,\infty ) \rightarrow \mathbb {{\mathbb {R}}}$$f:[0,∞)→R, we define a new localized Szász–Mirakjan operator as follows: $$\begin{aligned} S_{n,\delta _{n},\delta ^{\prime }_{n}}(f,x)=e^{-nx}\sum _{k\in C_{n,x}}\frac{(nx)^{k}}{k!}f\left( \frac{k}{n}\right) , \,\, x\ge 0. \end{aligned}$$Sn,δn,δn′(f,x)=e-nx∑k∈Cn,x(nx)kk!fkn,x≥0.We call this bi-shift localized Szász–Mirakjan operators. Certain new convergence theorems are obtained for such operators when the limits both $$\lim _{n\rightarrow \infty }\delta _{n}\sqrt{n}$$limn→∞δnn and $$\lim _{n\rightarrow \infty }\delta ^{\prime }_{n}\sqrt{n}$$limn→∞δn′n exist.