On a "Lethargic" Theorem

Start: 09/21/2015 - 4:00pm
End  : 09/21/2015 - 5:00pm

Applied Math Seminar

Asuman Aksoy (Claremont McKenna College)


One of the most important theorems used in constructive theory of functions is called Berstein's "lethary" theorem [3]. The theorem states that if $ \{d_i\}  $ is a sequence of nonnegative numbers with $ \lim d_i =0 $ then there exists a function $ f\in C[0,1] $ such that dist$ (f, P_i) = d_i $ for $ i=0,1,2,3,\dots $, where C[0,1] denotes the Banach space of all continuous, real-valued functions defined on the interval $ [0,1] $ with supremum norm, and $ P_i $ denotes the space of all polynomials of degree $  \leq i $. In this talk, after examining the developments in this theory [1], we present a lethargic theorem for Fr ́echet spaces [2]. This is joint work with G. Lewicki.


[1] J. Almira and A. G. Aksoy On Shapiro’s Lethargy Theorem and Some Appli- cations, Jean. J. Approx. 6(1), 87 − 116, 2014.

[2] A. G. Aksoy and G. Lewicki Bernstein’s Lethargy Theorem in Fr ́echet spaces, arXiv:1503.06190.

[3] S. N. Bernstein, Collected Works, II Moskow,: Akad Nauk SSR. 1954.

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