Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-02T20:03:34.787Z Has data issue: false hasContentIssue false

Simultaneous monotone approximation in low-order mean

Published online by Cambridge University Press:  17 April 2009

Robert Huotari
Affiliation:
Department of MathematicsIdaho State University Pocatello, ID 83209United States of America
Salem Sahab
Affiliation:
Mathematics DepartmentKing Abdulaziz UniversityJeddah 21413Saudi Arabia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Suppose that f, gL[0,1] have discontinuities of the first kind only. Using the measure, max{‖fhp, ‖ghp}, of simultaneous Lp approximation, we show that the best simultaneous approximations, hp, to f and g by nondecreasing functions converge uniformly as p → 1. Part of the proof involves a discussion of discrete simultaneous approximation in a general context. We discuss the inheritance of properties of f and g by hp, and of hp by h1.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Bruckner, A.M., Differentiation of Real Functions: Lecture notes in mathematics 659 (Springer-Verlag, Berlin, Heidelberg, New York, 1978).CrossRefGoogle Scholar
[2]Darst, R.B. and Huotari, R., ‘Best L 1 -approximation of bounded approximately continuous functions on [0,1] by nondecreasing functions’, J. Approx. Theory 43 (1985), 178189.CrossRefGoogle Scholar
[3]Goel, D.S., Holland, A.S.B., Nasim, C. and Sahney, B.N., ‘On best simultaneous approximation in normed linear spaces’, Canad. Math. Bull. 17 (1974), 523527.CrossRefGoogle Scholar
[4]Goel, D.S., Holland, A.S.B., Nasim, C. and Sahney, B.N., ‘Characterization of an element of best l p-simultaneous approximation’, S. Ramanaujan Memorial Volume, Madras (1984), 1014.Google Scholar
[5]Huotari, R. and Legg, D., ‘Best monotone approximation in L 1 [0,1]’, Proc. Amer. Math. Soc. 94 (1985), 279282.Google Scholar
[6]Huotari, R., Legg, D., Meyerowitz, A. and Townsend, D., ‘The natural best L 1-approximation by nondecreasing functions’, J. Approx. Theory 52 (1988), 132140.CrossRefGoogle Scholar
[7]Huotari, R., Meyerowitz, A. and Sheard, M., ‘Best monotone approximants in L 1 [0,1]’, J. Approx. Theory 47 (1986), 8591.CrossRefGoogle Scholar
[8]Huotari, R. and Sahab, S., ‘Simultaneous monotone Lp approximation, p → ∞’, Canad. Math. Bull. (to appear).Google Scholar
[9]Legg, D. and Townsend, D., ‘Sets of best L 1 approximants’, J. Approx. Theory 59 (1989), 316320.CrossRefGoogle Scholar
[10]Landers, D. and Rogge, L., ‘Natural choice of L 1 approximants’, J. Approx. Theory 33 (1981), 268280.CrossRefGoogle Scholar
[11]Landers, D. and Rogge, L., ‘On projections and monotony in Lp-spaces’, Manuscripta Math. 26 (1979), 363369.CrossRefGoogle Scholar
[12]Natanson, I. P., Theory of functions of a real variable (Ungar, New York, 1955).Google Scholar
[13]Philips, G.M. and Sahney, B.N., ‘Best simultaneous approximation in the L 1 and L 2 norms’, in Theory of approximation with applications, Editors Law, A.G. and Sahney, B.N. (Academic Press, New York, 1976).Google Scholar
[14]Powell, M.J.D., Approximation theory and methods (Cambridge, 1981).CrossRefGoogle Scholar
[15]Sahab, S.On the monotone simultaneous approximation on [0,1], Bull. Austral. Math. Soc. 39 (1988), 401411.CrossRefGoogle Scholar
[16]Sahab, S., ‘Natural choice of best L 1-simultaneous approximants’, Tamkang J. Math. 20 (1989), 147157.Google Scholar
[17]Sahab, S., ‘Best simultaneous approximation of quasicontinuous functions by monotone functions’, J. Austral. Math. Soc. (Ser. A) (to appear).Google Scholar
[18]Singer, I., Best approximation in normed linear spaces (Springer Verlag, Berlin, Heidelberg, New York, 1970).CrossRefGoogle Scholar
[19]Van Rooij, A.C.M. and Schikhof, W.H., A second course on real functions (Cambridge, 1982).CrossRefGoogle Scholar