Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-25T13:40:28.593Z Has data issue: false hasContentIssue false
  • English
  • Français

Simultaneous Monotone Lp Approximation, p → ∞

Published online by Cambridge University Press:  20 November 2018

Robert Huotari  and
Salem Sahab
Robert Huotari
Affiliation:
Department of Mathematics, Idaho State University, Pocatello, Idaho, USA 83209 Internet: [email protected]
Salem Sahab
Affiliation:
Mathematics Department, King Abdulaziz University, Jeddah 214I3, Saudi 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, g € L∞ [0,1 ] have discontinuities of the first kind only. Using the measure, max{ ∥f — h∥p, ∥g — h∥p}, of simultaneous Lp approximation, we show that the best simultaneous approximations f and g by nondecreasing functions converge uniformly as p → ∞. Part of the proof involves a discussion of discrete simultaneous approximation in a general context. Assuming only that f and g are approximately continuous, we show that their simultaneous best monotone Lp approximation is continuous.

Keywords

41A2840A0540A3041A30
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 34 , Issue 3 , 01 September 1991 , pp. 343 - 350
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Bruckner, A. M., Differentiation of Real Functions. Lecture Notes in Mathematics, 659, Springer- Verlag, Berlin-Heidelberg-New York, 1978.Google Scholar
2. Cheney, E. W., Multivariate Approximation Theory: Selected Topics. CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1986.Google Scholar
3. Darst, R. B. and Huotari, R., Best L\ -approximation of bounded approximately continuous functions on [0,11] by nondecreasingfunctions, J. Approx. Theory 43 (1985), 178–189.Google Scholar
4. Darst, R. B. and Sahab, S., Approximation of continuous and quasi-continuous functions by monotone functions, J. Approx. Theory 38 (1983), 9–27.Google Scholar
5. Descloux, J., Approximation in LP and Chebyshevapproximation,]. Soc. Ind. Appl. Math. 11(1963), 1017— 1026.Google Scholar
6. Goel, D. S., Holland, A. S. B., C. Nasim and Sahney, B. N., On best simultaneous approximation in normed linear spaces, Canad. Math. Bull. 17 (1974), 523–527.Google Scholar
7. Huotari, R., Tubular sets in R”, with applications to multivariate approximation, to appear.Google Scholar
8. Huotari, R., The p-limit selection in uniform approximation, to appear.Google Scholar
9. Huotari, R., Legg, D. and Townsend, D., The Polya algorithm on cylindrical sets , J. Approx. Theory 53( 1988), 335–349.Google Scholar
10. Landers, D. and Rogge, L., On projections and monotony in Lp-spaces , Manuscripta Math. 26(1979), 363– 369.Google Scholar
11. Legg, D. andD. Townsend, Best monotone approximation in L∞[0,1], J. Approx. Theory 42 (1984), 30–35.Google Scholar
12. H, W. Ling and McLaughlin, H. W., Approximation of random functions, J. Approx. Theory 20 (1977), 10–22.Google Scholar
13. Robertson, T., Wright, F. T. and Dykstra, R. L., Order Restricted Statistical Inference. Wiley, New York, 1988.Google Scholar
14. Sahab, S., On the monotone simultaneous approximation on [0,1], Bull. Austral. Math. Soc. 39 (1988), 401–411.Google Scholar
15. Van, A. C. M. Rooij and Schikhof, W. H., A Second Course on Real Functions. Cambridge, 1982.Google Scholar