The approach methods to the defined functions on the closed curve

THE APPROACH METHODS TO THE DEFINED FUNCTIONS ON THE CLOSED CURVE Melek DEMİRHAN Mathematics Department, Higher Licence Thesis, 2002 The Thesis Advisor: Assoc.Prof. Dr. Nizami MUSTAFAYEV SUMMARY In this thesis, the approximation of the functions which are defined on a closed y curve is investigated, that can be written depending on the ability of the difference of two analytic functions (that functions are analytic inside and outside of the y) which belong to Holder class on the y. The necessary basic information has been given in the first part. In the second chapter, the best approximation is examined to the functions which are analytic in a simply connected domain and continuous on its closure with polynamials and the speed of approximation is shown. In the third chapter of the thesis the best approximation is given to the functions which are defined an a closed curve with the nth. degree rational polynomials. Besides this, for the density function in the singular integral with Cauchy kernel which satisfies the Holder Condition the best approximation is examined with the nth degree rational polynamials. Key Words: Cauchy Type integrals, The Best Approximation, The Polynamial, The Rational Polynomial.