Allen A. Goldstein
ABSTRACT
There has been a renewed interest of late in the problem of approximation in the Tchebycheff sense, that is of minimizing the maximum residues. Several methods 1, 2, 3, have already been discussed and many examples of approximating functions of one variable by polynomials and rational functions, etc. have been published. On the other hand the important problem of solving an overdetermined system of linear equations in the Tchebycheff sense has not been given the attention it deserves. The possession of a practical method for the solution of this problem would make possible “best” approximations of functions of several variables by linear combinations of arbitrary functions.
It is not widely known that an algorithm for solving this problem exists and is due to de la Vallee Poussin.4 Unfortunately the algorithm is unwieldy but it is still useful in the computation of a minimum approximation. We shall therefore outline the results of de la Vallee Poussin before discussing some new methods with which we have been experimenting.
- 1.Hastings Cecil, Jr. "Approximations for Digital Computers" Princeton University Press 1955. Google Scholar
Digital Library
- 2.Selfridge, R. G. "Approximations With Least Maximum Error", Pacific Journal of Mathematics, Vol. 3, No. 1, March 1953, pp 247-255.Google ScholarCross Ref
- 3.Carlson, Bengt and Goldstein, Max. "Rational Approximations of Functions" U.S. Atomic Energy Commission, Los Alamos Scientific Lab. August 1955Google Scholar
- 4.Poussin, Ch. J. de la Vallee "Sur la Methode de I'approximation Minimum", Societe Scientifiaue de Bruxelles, Annales, Seconde Partie, Memires, Vol. 35 1911, pp 1-16. The above paper has been translated with a commentary by H. E. Salzer. Copies may be obtained by writing him.Google Scholar
Index Terms
- On the method of minimum (or "best") approximation and the method of least nth powers
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