One of the most important theorems used in constructive theory of functions is called Berstein's "lethary" theorem [3]. The theorem states that if is a sequence of nonnegative numbers with
then there exists a function
such that dist
for
, where C[0,1] denotes the Banach space of all continuous, real-valued functions defined on the interval
with supremum norm, and
denotes the space of all polynomials of degree
. 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.
References
[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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