Chebyshev approximation by piecewise continuous functions
Chand, Donald Rajinder
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This paper discusses the following problem of approximation theory: a continuous function f(x) is to approximated over an interval alpha </= x </= B by N not necessarily connected polynomials of a given degree n, in such a way that the maximum error magnitude is a minimum. Each polynomial is associated with one subinterval [uj, uj+1]. If the end points Ui were specified in advance, the problem would reduce to N independent problems of the same type, namely, the fitting of a single polynomial in the Chebyshev sense. Here, however, the end points are taken as unknowns and the principal problem is to determine them. The paper presents a proof of the existence of the best approximation and examples showing the solution, in general, is not unique. However, in the special case of approximation of convex functions by line segments, the solution is shown to be unique. Further in this case a simple characterization of the solution is obtained and it is shown that the problem may be reduced analytically to a stage where in order to determine Ui computationally, it is only necessary to solve a system of equations rather than minimize a function. Results obtained by a dynamic programming method using a digital computer (IBM 7090) are used for illustration.
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