In No Hurry to Deviate : Extensions of Bernstein's Lethargy Theorem

Start: 02/21/2018 - 4:15pm
End  : 02/21/2018 - 5:15pm


Asuman Aksoy (CMC)


The formal beginnings of approximation theory date back to 1885, with Weierstrass' celebrated approximation theorem. The discovery that every continuous function defined on a closed interval $[a,b]$ can be uniformly approximated as closely as desired by a polynomial function immediately prompted many new questions. One such question concerned approximating functions with polynomials of limited degree. That is, if we limit ourselves to polynomials of degree at most $n$, what can be said of the best approximation? As it turns out, there is no unified answer to this question. In fact, S. N. Bernstein (1938) showed that there exists functions whose best approximation converges arbitrarily slowly as the degree of the polynomial rises. In this talk, we take this aptly-named ``Lethargy theorem" of Bernstein and present two extensions. We'll show one of these extensions shrinks the interval for best approximation by half while the other gives a surprising equivalence to reflexivity in Banach spaces. Put colloquially, this shows that if you are in no hurry to deviate, you might reach a nice spot!

(Joint work with Q. Peng and G. Lewicki).

Freeberg Forum, LC 62, Kravis Center, CMC

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