By Sorin G Gal

This monograph, as its first major aim, goals to check the overconvergence phenomenon of significant periods of Bernstein-type operators of 1 or numerous advanced 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 next varieties: Bernstein, Bernstein-Faber, Bernstein-Butzer, q-Bernstein, Bernstein-Stancu, Bernstein-Kantorovich, Favard-Szasz-Mirakjan, Baskakov and Balazs-Szabados. the second one major target is to supply a examine of the approximation and geometric homes of different types of complicated convolutions: the de l. a. Vallee Poussin, Fejer, Riesz-Zygmund, Jackson, Rogosinski, Picard, Poisson-Cauchy, Gauss-Weierstrass, q-Picard, q-Gauss-Weierstrass, Post-Widder, rotation-invariant, Sikkema and nonlinear. numerous functions to partial differential equations (PDE) are also provided. a few of the open difficulties encountered within the reviews are proposed on the finish of every bankruptcy. For extra learn, the monograph indicates and advocates related reports for different complicated Bernstein-type operators, and for different linear and nonlinear convolutions.

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For any 1 < r < R the following estimate |Bn (f ; G)(z) − f (z)| ≤ C , z ∈ Gr , n ∈ N, n holds, where C > 0 depends on f , r and Gr but it is independent of n. ˜ \ G is Proof. First we note that since G is a continuum then it follows that C simply connected. By the proof of Theorem 2, p. 52 in Suetin [186], for any fixed ∞ 1 < β < R we have f (z) = k=0 ak (f )Fk (z) uniformly in Gβ , where ak (f ) are f [Ψ(u)] 1 the Faber coefficients and are given by ak (f ) = 2πi |u|=β uk+1 du. Note here that G ⊂ Gβ .

4 we can write T2i n,2q−1 (z) T2i n,2q (z) (2q) f (2q−1) (z) + f (z) (2i n)q−1 (2q − 1)! (2q)! aq−1 [z(1 − z)]q (2q) = (1 − 2z)[z(1 − z)]q−1 f (2q−1) (z) + f (z) (2q − 1)! 2q (q)! 1 1 + Fi (z)f (2q−1) (z) + Gi (z)f (2q) (z), n n where Fi (z) and Gi (z) are polynomials bounded in Dr by constants independent of n. Collecting all the above considerations in conclusion we obtain L[2q−2] (f )(z) − f (z) n = f (q+1) (z) 2q − 1 z(1 − z)Pq (z) + q−1 aq−1 (1 − 2z)[z(1 − z)]q−1 f (2q−1) (z) (q + 1)! 2 1 2q − 1 [z(1 − z)]q (2q) f (z) + Kq (f )(z) , + q−1 · q 2 2 q!

Therefore the theorem has been proved. Remarks. 1 suggests that for f ∈ AR , the (m ) limit of the iterates Bn n (f )(z) represents the semigroup of operators T (t)(f )(z) defined on the locally convex space (Fr´echet) AR . 1 extend some related results in the case of iterates of real Bernstein polynomials on [0, 1] (see Karlin-Ziegler [114], Kelisky-Rivlin [115]). 7 hold for their iterates too. 5in bernstein Bernstein-Type Operators of One Complex Variable 31 In the proofs of these properties we need the following two auxiliary lemmas.