Best approximation by monotone functions. (English) Zbl 0626.41015
Denote by \(L_ p\) the space of pth power integrable functions on the unit interval with usual norm and let \(M_ p\) be the set of non- decreasing functions in \(L_ p\). The following statements are the main results of the paper: a) Let \(1\leq p<\infty\), \(f\in L_ p\) and g be a best approximation in \(M_ p\). If x is a Lebesgue point of f then g is continuous at x. b) If \(p=1\) and every point of (0,1) is a Legesgue point of f then its best approximation in \(M_ 1\) is unique.
Reviewer: A.Kroo
MSC:
| 41A50 | Best approximation, Chebyshev systems |
| 41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |
Keywords:
Lebesgue pointReferences:
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