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Another Note on Polynomial vs Rational Approximation

✍ Scribed by Boris Shekhtman

Publisher: Elsevier Science
Year: 1996
Tongue: English
Weight: 221 KB
Volume: 85
Category: Article
ISSN: 0021-9045
DOI: 10.1006/jath.1996.0047

No coin nor oath required. For personal study only.

✦ Synopsis

Let E be a subspace of C(X) and let R(E)= gÂh: g, h # E ; h>0]. We make a simple, yet intriguing observation: if zero is a best approximation to f from E, then zero is a best approximation to f from R(E ).

We also prove that if

That extends the results of P. Borwein and S. Zhou who proved it for the case when E n is the space of algebraic or trigonometric polynomials of degree n.

1996 Academic Press, Inc. * equioscillates i.e. there are points ! 1 , ...,

then gÂh(! j ) 0 for j even and gÂh(! j ) 0 for j odd. Since h is strictly positive, the function g # P n should satisfy the same condition g(! j ) 0 for j even and g(! j ) 0 for j odd.

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