By Sorin G Gal

This monograph, as its first major target, goals to review the overconvergence phenomenon of vital sessions of Bernstein-type operators of 1 or a number of complicated variables, that's, to increase their quantitative convergence homes to greater units within the complicated airplane instead of the genuine durations. The operators studied are of the next kinds: 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 learn of the approximation and geometric houses of various kinds 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 purposes to partial differential equations (PDE) are also offered. the various open difficulties encountered within the experiences are proposed on the finish of every bankruptcy. For additional study, the monograph indicates and advocates comparable experiences for different advanced Bernstein-type operators, and for different linear and nonlinear convolutions.

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**Example text**

Remark. 2, in a similar way we can prove that in the case when r = 1 the pointwise estimate 2p Bn (f )(z) − f (z) − j=1 f (j) (z) −j |z| · |1 − z|Cp,1 (f ) n Tn,j (z) ≤ , ∀|z| ≤ 1, j! (k − 2p + 1)(k − 2p) < ∞. 2 is exactly np+1 . More exactly, the second main result of this paper is the following. 4. (Gal [89]) Let R > 1 and let f : DR → C be an analytic ∞ k function, say f (z) = k=0 ck z . If f is not a polynomial of degree ≤ 2p then for any 1 ≤ r < R and any natural number p we have 2p Bn (f ) − f − j=1 f (j) −j n Tn,j j!

N Case 2. We have ∞ k=n+1 |ak (f )| · |Bn (Fk ; G)(z) − Fk (z)| ≤ ∞ k=n+1 ∞ |ak (f )| · |Bn (Fk ; G)(z)| + k=n+1 |ak (f )| · |Fk (z)|. 5in bernstein Bernstein-Type Operators of One Complex Variable 23 By the estimates mentioned in the case 1), we immediately get ∞ k=n+1 |ak (f )| · |Fk (z)| ≤ C(r, β, f ) ∞ k=n+1 dk , for all z ∈ Gr , with d = r/β. Also, ∞ k=n+1 |ak (f )| · |Bn (Fk ; G)(z)| = ≤ ∞ k=n+1 ∞ k=n+1 n |ak (f )| · |ak (f )| · p=0 n p=0 Dn,p,k · Fp (z) Dn,p,k · |Fp (z)|. But for p ≤ n < k and taking into account the estimates obtained in the Case 1) we get |ak (f )| · |Fp (z)| ≤ C(r, β, f ) rk rp ≤ C(r, β, f ) , for all z ∈ Gr , βk βk which implies ∞ k=n+1 |ak (f )| · |Bn (Fk ; G)(z) − Fk (z)| ≤ C(r, β, f ) = C(r, β, f ) ∞ Dn,p,k k=n+1 p=0 ∞ k k=n+1 n+1 = C(r, β, f ) n r β k r β d , 1−d with d = r/β.

5 and by the linearity and continuity of T −1 we get C ≤ Bn (g) − g n r = Bn (g) − T −1 (f ) ≤ |T −1 r = T −1 [Bn (f ; G)] − T −1 (f ) | · Bn (f ; G) − f Gr ≤ M Bn (f ; G) − f r Gr , which proves the lower estimate. On the other hand we have T [Bn (g)] = Bn (T (g); G). Indeed, n ∆p1/n g(0)Fp (z), T [Bn (g)](z) = p=0 and n Bn (T (g); G)(z) = p=0 n ∆p1/n H(0)Fp (z), p where according to Gaier [76], p. 49. 17’) we have H(w) = 1 2πi |u|=1 T (g)[Ψ(u)] du = g(w). 5 and by the linearity and continuity of T we obtain Bn (f ; G) − f Gr = Bn (T (g); G) − T (g) ≤ |T | · Bn (g) − g r = T [Bn (g)] − T (g) C ≤ , n Gr which proves the upper estimate and the theorem.