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Concentration of the error between a function and its polynomial of best uniform approximation

Published online by Cambridge University Press:  20 January 2009

Maurice Hasson
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Abstract

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Let f be a continuous real valued function defined on [−1, 1] and let En(f) denote the degree of best uniform approximation to f by algebraic polynomial of degree at most n. The supremum norm on [a, b] is denoted by ∥.∥[a, b] and the polynomial of degree n of best uniform approximation is denoted by Pn. We find a class of functions f such that there exists a fixed a ∈(−1, 1) with the following property

for some positive constants C and N independent of n. Moreover the sequence is optimal in the sense that if is replaced by then the above inequality need not hold no matter how small C > 0 is chosen.

We also find another, more general class a functions f for which

infinitely often.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 41 , Issue 3 , October 1998 , pp. 447 - 463
Copyright
Copyright © Edinburgh Mathematical Society 1998

References

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