caa a22 4500 606200487 CHVBK 20210128100944.0 cr unu---uuuuu 210128e20150601xx s 000 0 eng 10.1007/s00025-015-0443-5 doi (NATIONALLICENCE)springer-10.1007/s00025-015-0443-5 Approximation by the Complex form of a Link Operator Between the Phillips and the Szász-Mirakjan Operators [Elektronische Daten] [Sorin Gal, Vijay Gupta] The link operator $${P_{\alpha}^\rho(f,x)=\sum_{k=1}^\infty s_{\alpha,k}(x)\int_0^\infty \theta_{\alpha,k}^\rho(t)f(t)dt+e^{-\alpha x}f(0)}$$ P α ρ ( f , x ) = ∑ k = 1 ∞ s α , k ( x ) ∫ 0 ∞ θ α , k ρ ( t ) f ( t ) d t + e - α x f ( 0 ) , $${\alpha, \rho > 0}$$ α , ρ > 0 , $${x\in [0, +\infty)}$$ x ∈ [ 0 , + ∞ ) , $${s_{\alpha,k}(x)=e^{-\alpha x} \frac{(\alpha x)^k}{k!}, \theta_{\alpha,k}^\rho(t)=\frac{\alpha\rho}{\Gamma(k\rho)} e^{-\alpha\rho t}(\alpha\rho t)^{k\rho-1}}$$ s α , k ( x ) = e - α x ( α x ) k k ! , θ α , k ρ ( t ) = α ρ Γ ( k ρ ) e - α ρ t ( α ρ t ) k ρ - 1 , between the Phillips operator (obtained for $${\rho=1}$$ ρ = 1 ) and the Szász-Mirakjan operator (obtained for $${\rho\to +\infty}$$ ρ → + ∞ ), was introduced by Pǎltǎnea in (Carpathian J Math 24:378-385, 2008), for which he proved uniform convergence to f (as $${\alpha\to +\infty}$$ α → + ∞ ) in any compact subinterval $${[0, b]\subset [0, +\infty)}$$ [ 0 , b ] ⊂ [ 0 , + ∞ ) . In this paper, for entire functions f of some exponential growth in $${\mathbb{C}}$$ C , quantitative estimates in approximation and in Voronovskaya-type result in any closed disk $${\overline{\mathbb{D}_{r}} \subset \mathbb{C}}$$ D r ¯ ⊂ C are obtained for $${P_{\alpha}^\rho(f,z)}$$ P α ρ ( f , z ) . Springer Basel, 2015 Complex generalized Szász-Mirakjan operators nationallicence Voronovskaya type result nationallicence exact order nationallicence exponential growth nationallicence entire function nationallicence Gal Sorin Department of Mathematics and Computer Science, University of Oradea, Str. Universitatii No. 1, 410087, Oradea, Romania aut Gupta Vijay Department of Mathematics, Netaji Subhas Institute of Technology, Sector 3 Dwarka, 110078, New Delhi, India aut Results in Mathematics Springer Basel 67/3-4(2015-06-01), 381-393 1422-6383 67:3-4<381 2015 67 25 https://doi.org/10.1007/s00025-015-0443-5 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00025-015-0443-5 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Gal Sorin Department of Mathematics and Computer Science, University of Oradea, Str. Universitatii No. 1, 410087, Oradea, Romania aut NATIONALLICENCE 700 1- Gupta Vijay Department of Mathematics, Netaji Subhas Institute of Technology, Sector 3 Dwarka, 110078, New Delhi, India aut NATIONALLICENCE 773 0- Results in Mathematics Springer Basel 67/3-4(2015-06-01), 381-393 1422-6383 67:3-4<381 2015 67 25