Approximation Under Exponential Growth Conditions by Szász and Baskakov Type Operators in the Complex Plane

In this chapter, firstly for q > 1 the exact order near to the best approximation, 1 q n $$\frac{1} {q^{n}}$$ , is obtained in approximation by complex q-Favard–Szász–Mirakjan, q-Szász–Kantorovich operators and q-Baskakov operators attached to functions of exponential growth conditions, which are entire functions or analytic functions defined only in compact disks (without to require to be defined on the whole axis [ 0 , + ∞ ) $$[0,+\infty )$$ ). Quantitative Voronovskaja-type results of approximation order 1 q 2 n $$\frac{1} {q^{2n}}$$ are proved. For q-Szász–Kantorovich operators, the case q = 1 also is considered, when the exact order of approximation 1 n $$\frac{1} {n}$$ is obtained. Approximation results for a link operator between the Phillips and Favard–Szász–Mirakjan operators are also obtained. Then, by using a sequence b n a n : = λ n > 0 $$\frac{b_{n}} {a_{n}}:=\lambda _{n}> 0$$ , a n , b n > 0 $$a_{n},b_{n}> 0$$ , n ∈ ℕ $$n \in \mathbb{N}$$ with the property that λ n → 0 $$\lambda _{n} \rightarrow 0$$ as fast we want, we obtain the approximation order O ( λ n ) $$O(\lambda _{n})$$ for the generalized Szász–Faber operators and the generalized Baskakov–Faber operators attached to analytic functions of exponential growth in a continuum G ⊂ ℂ $$G \subset \mathbb{C}$$ . Several concrete examples of continuums G are given for which these generalized operators can explicitly be constructed. Finally, approximation results for complex Baskakov–Szász–Durrmeyer operators are presented.