On the best polynomials approximation of segment functions

An algorithm for finding the best approximation polynomial for a continuous multivalued segment function defined on a set of segments X is proposed, where X=(⋃n1j1=0[aj1,bj1])∪(⋃nk=0xk) with (⋃n1j1=0[aj1,bj1])∩(⋃nk=0xk)=∅. The disjoint segments [aj1,bj1] and points xk belong to a bounded segment [A,B]⊂R. We assume that the functions f1 and f2 are continuous on the set X, and everywhere on X the value of the function f1(x) does not exceed the value of the function f2(x). The operator assigning to each x∈X the segment [(x,f1(x)),(x,f2(x))] will be called the segments function F(x) defined on X. Since the functions f1 and f2 are continuous, the segments function F is an upper h-semicontinuous mapping. The polynomial Pm=∑mi=0aixi of the best approximation in the Hausdorff metric on the set X of a segment function F with a vector of coefficients a⃗ =(a0,a1,…,am)∈Rm+1 is a solution to the extremal problem mina⃗ ∈Rm+1maxx∈Xmax(Pm(x)-f1(x),f2(x)-Pm(x)). It is shown by methods of constructive function theory that, for any functions f1(x)≤f2(x) continuous on X, there exists some polynomial of best approximation in the Hausdorff metric as the segment function F(x) is upper h-semicontinuous on X. An algorithm for describing the set E of coefficients a⃗ of polynomials of the best approximation of a segment function is proposed. Necessary and sufficient conditions for the uniqueness of the polynomial of best approximation of the segment function are obtained. The results of numerical experiments carried out using the proposed algorithm are presented.