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Best Simultaneous approximation of quasi-continuous functions by monotone functions

Part of: Approximations and expansions

Published online by Cambridge University Press:  09 April 2009

Salem M. A. Sahab
Salem M. A. Sahab
Affiliation:
King Abdulaziz UniversityP.O. Box 9028 Jeddah 21413, Saudi Arabia
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Abstract

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Let Q denote the Banach space (under the sup norm) of quasi-continuous functions on the unit interval [0, 1]. Let ℳ denote the closed convex cone comprised of monotone nondecreasing functions on [0, 1]. For f and g in Q and 1 < p < ∞, let hp denote the best Lp-simultaneous approximant of f and g by elements of ℳ. It is shown that hp converges uniformly as p → ∞ to a best L∞-simultaneous approximant of f and g by elements of ℳ. However, this convergence is not true in general for any pair of bounded Lebesgue measurable functions. If f and g are continuous, then each hp is continuous; so is limp→∞ hp = h∞.

MSC classification

Secondary: 41A28: Simultaneous approximation 41A30: Approximation by other special function classes 41A65: Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 50 , Issue 3 , June 1991 , pp. 391 - 408
Copyright
Copyright © Australian Mathematical Society 1991

References

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