By Sorin G. Gal

This monograph, as its first major objective, goals to review the overconvergence phenomenon of significant periods of Bernstein-type operators of 1 or numerous complicated variables, that's, to increase their quantitative convergence houses to bigger units within the complicated airplane instead of the genuine periods. The operators studied are of the subsequent varieties: Bernstein, Bernstein-Faber, Bernstein-Butzer, q-Bernstein, Bernstein-Stancu, Bernstein-Kantorovich, Favard-Szasz-Mirakjan, Baskakov and Balazs-Szabados. the second one major goal is to supply a examine of the approximation and geometric homes of different types of complicated convolutions: the de los angeles Vallee Poussin, Fejer, Riesz-Zygmund, Jackson, Rogosinski, Picard, Poisson-Cauchy, Gauss-Weierstrass, q-Picard, q-Gauss-Weierstrass, Post-Widder, rotation-invariant, Sikkema and nonlinear. a number of functions to partial differential equations (PDE) are also offered. some of the open difficulties encountered within the reviews are proposed on the finish of every bankruptcy. For extra examine, the monograph indicates and advocates related reviews for different complicated Bernstein-type operators, and for different linear and nonlinear convolutions.

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**Example text**

Butzer’s Linear Combination of Bernstein Polynomials [q] In the paper of Butzer [51], were inductively introduced the operators Ln (x) of [0] real variable x ∈ [0, 1] by setting Ln (f ) (x) := Bn (f )(x) and [2q−2] q (2q − 1) L[2q] n (f ) (x) = 2 L2n (f ) (x) − L[2q−2] (f ) (x), n for q ∈ N. For example, for q = 1 one obtains [0] [0] L[2] n (f ) (x) := 2L2n (f ) (x) − Ln (f ) (x) = 2B2n (f )(x) − Bn (f )(x). 1), he proved that |L[2q−2] (f )(x) − f (x)| = O(n−q ). n The first main result of this section is the extension of Butzer’s result to the case of complex variable and can be stated as follows.

0) → f (0), as n → ∞. We obviously have nf (1/n) = f (1/n)−f (1/n) In what follows, we will use the well-known pointwise estimate for Bernstein polynomials when f ∈ C 2 [0, 1], given by x(1 − x) , for all x ∈ [0, 1], n ∈ N, n denotes the uniform norm in C[0, 1]. |Bn (f )(x) − f (x)| ≤ C f where · We get Bn (f )(1/n) − Bn (f )(0) (1/n) Bn (f )(1/n) − f (1/n) f (1/n) − f (0) = + → f (0), (1/n) (1/n) nBn (f )(1/n) = as n → ∞, since by the above estimate we have Bn (f )(1/n) − f (1/n) C f ≤ (1/n) n → 0, for n → ∞.

Proof. According to the above considerations, there exists g analytic in D r such that f = T (g), that is g = T −1 (f ) (therefore F can be extended by continuity on ∂D1 . By hypothesis on f it follows that f cannot be of the form f (z) = c0 F0 (z) + c1 F1 (z) where F0 and F1 are the Faber polynomials of degree 0 and 1 respectively and c0 , c1 ∈ C. This immediately implies that g is not a polynomial of degree ≤ 1. First we have Bn (T −1 (f )) = T −1 [Bn (f ; G)]. Indeed, n Bn (T −1 (f ))(z) = p=0 n n n ∆p1/n T −1 (f )(0)z p = ∆p1/n F (0)z p , p p p=0 Bernstein-Type Operators of One Complex Variable since T −1 (f )(ξ) = 1 2πi |w|=1 f [Ψ(w)] w−ξ dw 1 2πi T −1 [Bn (f ; G)](z) = n = p=0 n = p=0 25 = F (ξ), and |w|=1 Bn (f ; G)[Ψ(w)] dw w−z 1 n ∆p1/n F (0) 2πi p |w|=1 Fp [Ψ(w)] dw w−z n ∆p1/n F (0)z p , p since according to Gaier [76], p.