Existence questions for the problem of Chebyshev approximation by interpolating rationals
Abstract: This paper considers a problem of Chebyshev approximation by interpolating rationals. Examples are given which show that best approximations may not exist. Sufficient conditions for existence are established, some of which can easily be checked in practice. Illustrative examples are also presented.
-  N. I. Achieser, Theory of approximation, Translated by Charles J. Hyman, Frederick Ungar Publishing Co., New York, 1956. MR 0095369
-  D. BRINK, Tchebycheff Approximation by Reciprocals of Polynomials on , Thesis, Michigan State University, East Lansing, Mich., 1972.
-  Claude Gilormini, Approximation rationnelle avec des nœuds, C. R. Acad. Sci. Paris Sér. A-B 263 (1966), A286–A287 (French). MR 0204933
-  D. C. Handscomb (ed.), Methods of numerical approximation, Pergamon Press, Oxford-New York-Toronto, Ont., 1966. Lectures delivered at a Summer School held at Oxford University, Oxford, September, 1965. MR 0455292
-  Henry L. Loeb, Un contre-exemple à un résultat de M. Claude Gilormini, C. R. Acad. Sci. Paris Sér. A-B 266 (1968), A237–A238 (French). MR 0230012
-  Günter Meinardus, Approximation of functions: Theory and numerical methods, Expanded translation of the German edition. Translated by Larry L. Schumaker. Springer Tracts in Natural Philosophy, Vol. 13, Springer-Verlag New York, Inc., New York, 1967. MR 0217482
-  A. L. Perrie, Uniform rational approximation with osculatory interpolation, J. Comput. System Sci. 4 (1970), 509–522. MR 0275018, https://doi.org/10.1016/S0022-0000(70)80026-5
-  J. L. Walsh, The existence of rational functions of best approximation, Trans. Amer. Math. Soc. 33 (1931), no. 3, 668–689. MR 1501609, https://doi.org/10.1090/S0002-9947-1931-1501609-5
-  Jack Williams, Numerical Chebyshev approximation by interpolating rationals, Math. Comp. 26 (1972), 199–206. MR 0373230, https://doi.org/10.1090/S0025-5718-1972-0373230-6
- N. I. AHIEZER, Lectures on the Theory of Approximation, Kharkov, 1940; English transl., Ungar, New York, 1956. MR 3, 234; 20 #1872. MR 0095369 (20:1872)
- D. BRINK, Tchebycheff Approximation by Reciprocals of Polynomials on , Thesis, Michigan State University, East Lansing, Mich., 1972.
- C. GILORMINI, "Approximation rationelle avec des noeuds," C. R. Acad. Sci. Paris Sér. A--B, v. 263, 1966, pp. A286-A287. MR 34 #4768. MR 0204933 (34:4768)
- D. C. HANDSCOMB, Methods of Numerical Approximation, Pergamon Press, London, 1966. MR 0455292 (56:13531)
- H. L. LOEB, "Un contre-exemple à un résultat de M. Claude Gilormini," C. R. Acad. Sci. Paris Sér. A--B, v. 266, 1968, pp. A237-A238. MR 37 #5578. MR 0230012 (37:5578)
- G. MEINARDUS, Approximation of Functions: Theory and Numerical Methods, Springer Tracts in Natural Philosophy, vol. 13, Springer-Verlag, New York, 1967. MR 36 #571. MR 0217482 (36:571)
- A. L. PERRIE, "Uniform rational approximation with osculatory interpolation," J. Comput. System Sci., v. 4, 1970, pp. 509-522. MR 43 #776. MR 0275018 (43:776)
- J. L. WALSH, "The existence of rational functions of best approximation," Trans. Amer. Math. Soc., v. 33, 1931, pp. 668-689. MR 1501609
- J. WILLIAMS, "Numerical Chebyshev approximation by interpolating rationals," Math. Comp., v. 26, 1972, pp. 199-206. MR 0373230 (51:9431)
Retrieve articles in Mathematics of Computation with MSC: 41A50
Retrieve articles in all journals with MSC: 41A50
Keywords: Existence of best rational interpolants, Chebyshev approximation, uniform rational approximation with interpolatory constraints
Article copyright: © Copyright 1974 American Mathematical Society