Simultaneous approximation on Birkhoff systems and relation with interpolation

ABSTRACTSIMULTANEOUS APPROXIMATION ON BIRKHOFF SYSTEMSANDRELATION WITH INTERPOLATIONCOŞKUN Sıdıka TubaM.Sc.in Department of Mathematics.Supervisor: Asst.Prof.Dr.Mehmet AÇIKGÖZTemmuz 2005, 45 pagesBy simultaneous approximation we mean the approximation of a differentiablefunction f by polinomials with respect to the semi-norm . F given by= max f ( ki ) ,f F i = 0,1,..., ρwhere the ki are integers satisfying 0 = ko < k1 < ... < k ρ . . is the uniformnorm.We can approximate to f with algebraic and trigonometric polynomials.In this study, the extremal sets which have the same definations in algebraic andtrigonometric approximation are examined. Then the differences and the similaritiesare discovered. Moreover, the uniqueness of best simultaneous approximation isinvestigated and the question `under sufficient condition, it can be unique?? isanswered.In Haar space, an incidence matrice is existed by using the extremal sets and thebest approximation polynomials. The Birkhoff interpolation problem which isdecided for this incidence matrice, is solved by changing the matrice and using thePolya conditions which is known as the Birkhoff conditions.Key Words-Simultaneous Approximation, Extremal Sets, IncidenceMatrice, Birkhoff Interpolations.