Volume 61 Issue 2 (2011) - Годишник на ТУ - София - Технически ...

The exact definitions of points and uniform convergence of the SS and how to studythe limit curves for the class of smoothness are given in [1]. If for random data from0zero level f i | i Z, the limit curve f k (x)generated by the scheme has a class ofsmoothness C n [ a,b]we call that the scheme is of this class. Depending on whetherthe subscript i in (1) is even or odd Chaikin scheme is disintegrated in two relationsffk 12ik 12i13414ffkikiIf the vector a is a constant then the scheme is called stationary. The basic function ( ) of a stationary scheme is obtained if the scheme started by points of zero-level0 xiif 0 0 | i Z ,i 0for i 0where 0 is the symbol of Kroniker. For these initial data basis1for i 0function of Chaikin algorithm is a B-spline with support [-2,1] [2], i.e.0for x 2 2(x 2) / 2 for 2 x 12 0 ( x) 3 (2x 1) /4 for 1 x 0 (3) 2( x 2) / 2 for 0 x 10 for x 1Other basic functions are ( x) 0 ( x i)for i Z.(4)i3. Examples of "memorizing" of some elementary functionsusing the scheme of Chaikin"Memorizing" of a function is finding an element of best uniform approximation of afunction f (x), by applying modified algorithm of Remez. For this purpose it must beshown that an element of best uniform approximation exists. It should also be shownthat the algorithm of Remez is converging to an element of best uniform approximation.From the theorem 1.3 [5] follows that in considered by this item examplesthe element of best uniform approximations exists and it is unique. Moreover ifmodified Remez algorithm proposed by Nurnberg in [5] is used this element of thebest uniform approximation can be found.The proposed by Nurnberg modified single-point algorithm of Remez solves twomain problems arising from the application of this algorithm for splines. First, the algorithmavoids the cases in which at some step the main determinant of system (6) isequal to zero. Second, the modified algorithm provides convergence of the sequenceof elements p k (x)to the element of best uniform approximation p * ( x).1434ffki1ki1(2)41