ON THE CAUCHY TRANSFORM AND COMPLEX CUBIC SPLINES

. In this paper one approximates the Cauchy transform of a complex function on a simple closed curve, using an interpolation cubic spline function given by Iancu (1987).


INTRODUCTION
considered the Cauchy transform (1) T f where Γ is a simple closed curve, in the complex plane, with continuously differentiable parametrization.For the evaluation of T f (z) there are investigated numerical methods which are based on replacing f (z) by a uniformly convergent sequence {Φ n (z) | n ≥ 1} to f (z).Based on the fact that, if Φ n converge to f , then T Φ n converge to T f with the same speed, Atkinson (1972) has studied the cases in which the functions Φ n (z) are defined as piecewise linear and piecewise quadratic interpolation function to f (z) at a given set of node points on Γ.
In this paper we give an extension of Atkinson's results for the case when Φ n (z) are taken as interpolating complex cubic spline functions for f on the Γ curve, and, also, an extension of our previous results (Iancu, 1987;1989).
One denotes where Let f be a continuous function on Γ, about which we know that it takes the values: (4) Let us consider the interpolating function of the form (Iancu, 1987): uniquely determined by the conditions system:

THE ERROR ANALYSIS
Next, we prove the following: Lemma 1.Let be the complex spline function (5) determined by the conditions (6).For any function f ∈ C(Γ) that is interpolated by the spline function (5), we have: is the modulus of continuity of f on Γ and h = max k∈{1,2,...,n+1} Proof.On the base of uniform norm definition, we have: By using (5), one obtains: (6), having in view the definition of the modulus of continuity (8), the expression of the interpolating cubic spline (5), by using Eq. ( 9), we can give a simple formula for the difference f (z) − Φ n (z).So, we have From here, one obtains: If the node points of the partition (2) are taken so that |h k | = h, then, from (10), results: Remark 1.Because the system (6) determines uniquely the complex values M j and m j , where j = 1, 2, . . ., n, the formula ( 9) can be written in the form: Let the Banach space H µ (Γ) of the functions which satisfy the Hölder condition where A is the Hölder constant and µ ∈ (0, 1] is the Hölder exponent.
We have the following result: Theorem 1.Let be the functions f and Proof.Taking into account the results of Atkinson (1972), Chien-Ke Lu (1982), Muskhelishvilli (1953) we obtain: where By using (5), and taking into account the conditions of the Theorem 1 we have obtained the evaluation of M µ (f − Φ n ).So we have: Considering the case when the node points are equidistant |h k | = h, k = 1, 2, . . ., n + 1 and taking into account ( 6), we have: In conclusion, and, so, the theorem is proved.

THE CAUCHY TRANSFORM OF Φ n
We have the following proposition: Proposition 1.Let us consider a complex cubic spline function of the form (5) in the conditions (6).Then, for the Cauchy transform T Φ n (z), by using the formula (1), we obtain the following result: where