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An Lp Saturation Theorem for Splines

Published online by Cambridge University Press:  20 November 2018

G. J. Butler  and
F. B. Richards
G. J. Butler
Affiliation:
University of Alberta, Edmonton, Alberta
F. B. Richards
Affiliation:
University of Alberta, Edmonton, Alberta
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Let 1 be a subdivision of [0, 1], and let denote the class of functions whose restriction to each sub-interval is a polynomial of degree at most k. Gaier [1] has shown that for uniform subdivisions △n (that is, subdivisions for which

if and only if f is a polynomial of degree at most k. Here, and subsequently, denotes the usual norm in Lp[0, 1], 1 ≦ p ≦ ∞, and we should emphasize that functions differing only on a set of Lebesgue measure zero are identified.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 5 , October 1972 , pp. 957 - 966
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Gaier, D., Saturation bei Spline-Approximation und Quadratur, Numer. Math. 16 (1970), 129–140.Google Scholar
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3. Popov, V. and Sendov, Bl., Classes characterized by best possible approximations by spline functions, Math. Notes No. 2, 18 (1970), 550–557.Google Scholar
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5. Riesz, F., Systeme integrierbarer Funktionen, Math. Ann. 69 (1910), 449–497.Google Scholar
6. Schoenberg, I. J., On interpolation by spline functions and its minimal properties, On Approximation Theory (Intern. Ser. Numerical Math. (ISNM) 5 (1964), 109–129, Birkhauser, Basel/Stuttgart).Google Scholar