The degree of approximation by Chebyshevian splines

DeVore, R.; Richards, F.

doi:10.1090/S0002-9947-1973-0336160-9

The degree of approximation by Chebyshevian splines
HTML articles powered by AMS MathViewer

by R. DeVore and F. Richards PDF

Trans. Amer. Math. Soc. 181 (1973), 401-418 Request permission

Abstract:

This paper studies the connections between the smoothness of a function and its degree of approximation by Chebyshevian splines. This is accomplished by proving companion direct and inverse theorems which give a characterization of smoothness in terms of degree of approximation. A determination of the saturation properties is included.

References

G. J. Butler and F. B. Richards, An $L^{p}$ saturation theorem for splines, Canadian J. Math. 24 (1972), 957–966. MR 308658, DOI 10.4153/CJM-1972-096-1
G. Freud and V. Popov, On approximation by spline functions, Proceedings of the Conference on the Constructive Theory of Functions (Approximation Theory) (Budapest, 1969) Akadémiai Kiadó, Budapest, 1972, pp. 163–172. MR 0397246
Dieter Gaier, Saturation bei Spline-Approximation und Quadratur, Numer. Math. 16 (1970), 129–140 (German). MR 273816, DOI 10.1007/BF02308865
Samuel Karlin, Total positivity. Vol. I, Stanford University Press, Stanford, Calif., 1968. MR 0230102
Samuel Karlin and William J. Studden, Tchebycheff systems: With applications in analysis and statistics, Pure and Applied Mathematics, Vol. XV, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1966. MR 0204922
G. G. Lorentz, Approximation of functions, Holt, Rinehart and Winston, New York-Chicago, Ill.-Toronto, Ont., 1966. MR 0213785
Günter Meinardus, Approximation von Funktionen und ihre numerische Behandlung, Springer Tracts in Natural Philosophy, Vol. 4, Springer-Verlag, Berlin-New York, 1964 (German). MR 0176272, DOI 10.1007/978-3-642-85646-4
V. A. Popov and Bl. H. Sendov, The classes that are characterizable by the best approximation by spline functions, Mat. Zametki 8 (1970), 137–148 (Russian). MR 279502
Franklin Richards, On the saturation class for spline functions, Proc. Amer. Math. Soc. 33 (1972), 471–476. MR 294958, DOI 10.1090/S0002-9939-1972-0294958-4
Karl Scherer, On the best approximation of continuous functions by splines, SIAM J. Numer. Anal. 7 (1970), 418–423. MR 267732, DOI 10.1137/0707034
A. F. Timan, Teorij pribli+enij funkciĭ deĭstvitel’nogo peremennogo, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1960 (Russian). MR 0117478