In this paper we find the exact orders of approximation of analytic functions by the complex versions of several potentials (including the Flett potential) generated by the Gamma function and by some singular integrals. | Turk J Math 35 (2011) , 443 – 456. ¨ ITAK ˙ c TUB doi: Approximation by complex potentials generated by the Gamma function Sorin G. Gal Abstract In this paper we find the exact orders of approximation of analytic functions by the complex versions of several potentials (including the Flett potential) generated by the Gamma function and by some singular integrals. Key words and phrases: Complex potentials, singular integrals, order of approximation 1. Introduction In the real case, the approximation properties of the potentials such as those of Riesz, Bessel, generalized Riesz, generalized Bessel and Flett have been studied by many authors, see . Kurokawa [5], Gadjiev-AralAliev [3], Uyhan-Gadjiev-Aliev [7], Sezer [6], Aliev-Gadjiev-Aral [1] and their references. Let us recall that in the real case, the classical Bessel type parabolic potential is defined for any f ∈ L (R2 ), 1 ≤ p 0 , Γ(α) is the Gamma function and W (y, τ ) = 2 √ 1 e−y /(4τ) 4πτ is the Gauss-Weierstrass kernel. It is known that formally we can write −α/2 ∂2 ∂ B (f)(x, t) = I − + f(x, t), ∂x2 ∂t α and the following convergence properties hold (see Uyhan-Gadjiev-Aliev [7]): (i) if f ∈ Lp (R2 ), 1 ≤ p 1 . For R > 0 let us denote DR = {z ∈ C; |z| 0 and that f : DR → C, with R > 1 , is analytic in DR , that is ∞ f(z) = k=0 ak z k , for all z ∈ DR . ∞ −iu ) α (i) For Ut (f)(z) = πt −∞ f(ze u2 +t2 du we have that FU (f)(z) given by (2) is analytic in DR and we can write FUα (f)(z) = ∞ ak · k=0 1 · z k , z ∈ DR . (k + 1)α Also, if f is not constant function for q = 0 , and not a polynomial of degree ≤ q − 1 for q ∈ N, then for all 1 ≤ r 0 is independent of z (and α ) but depends on f and r . Now, let q ∈ N ∪ {0} and 1 ≤ r 0 we get q! FUα (f)(z) − f(z) α (q) (q) dv |[FU (f)] (z) − f (z)| = 2π (v − z)q+1 γ ≤ Cr1 (f)α · 446 q 2πr1 , · 2π (r1 − r)q+1 (5) GAL which proves the upper estimate [FUα(f)](q) − f (q) r ≤ C ∗ α, (6) with C ∗ depending .
