On the Best Polynomials Approximation of Segment Functions

Trynin, A. Yu.
Vladikavkaz Mathematical Journal 2023. Vol. 25. Issue 1.

Abstract:
An algorithm for finding the best approximation polynomial for a continuous multivalued segment function defined on a set of segments \(X\) is proposed, where \( X=\big(\bigcup_{j_{1}=0}^{n_1}[a_{j_1},b_{j_1}]\big)\cup\big(\bigcup_{k=0}^n x_k\big)\) with \(\big(\bigcup_{j_{1}=0}^{n_1}[a_{j_1},b_{j_1}]\big)\cap \big(\bigcup_{k=0}^n x_k\big)=\varnothing\). The disjoint segments \([a_{j_1},b_{j_1}]\) and points \(x_k\) belong to a bounded segment \([A,B]\subset\mathbb{R}\). We assume that the functions \(f_{1}\) and \(f_{2}\) are continuous on the set \(X\), and everywhere on \(X\) the value of the function \(f_{1}(x)\) does not exceed the value of the function \( f_{2} (x)\). The operator assigning to each \(x\in X\) the segment \([(x,f_{1}(x)),(x,f_{2}(x))]\) will be called the segments function \({\cal F } (x)\) defined on \(X\). Since the functions \(f_{1}\) and \(f_{2}\) are continuous, the segments function \({\cal F}\) is an upper \(h\)-semicontinuous mapping. The polynomial \(P_{m}=\sum_{i=0}^{m}a_{i}x^{i}\) of the best approximation in the Hausdorff metric on the set \(X\) of a segment function \({\cal F}\) with a vector of coefficients \(\vec{a}=(a_0,a_1,\dots,a_m)\in {\mathbb{R}^{m+1}}\) is a solution to the extremal problem \( \min_{\vec{a}\in {\mathbb{R}^{m+1}}} \max_{x\in X}\max(P_{m}(x)-f_{1}(x),f_{2}(x)-P_{m}(x)).\) It is shown by methods of constructive function theory that, for any functions \(f_{1}(x)\le f_{2}(x)\) continuous on \(X\), there exists some polynomial of best approximation in the Hausdorff metric as the segment function \({\cal F} (x)\) is upper \(h\)-semicontinuous on \(X\). An algorithm for describing the set \(E\) of coefficients \(\vec{a}\) 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.

Keywords: best approximation of functions, polynomial approximation, segment function

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