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On an Extremal Problem in Fourier Series

Published online by Cambridge University Press:  20 November 2018

Lee Lorch*
Affiliation:
University of Alberta
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Let f(x) be a bounded odd function, - π < x < π, |f(X)| ≤ 1, with non-negative Fourier coefficients bk, k = 1,2, ….

Otto Szász [l] proved anew the existence of a bounded set of numbers {βn}, n = 1,2,…, such that

where βn is the smallest constant satisfying the above inequality and added that 2/π ≤ βn ≤ 4/π. He pointed out [1, p. 170] that β1 = 4/π and raised the question of the value of βn for n > 1.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1960

References

1. Otto, Szász, Some extremum problems in the theory of Fourier series, Amer. J. of Math. 61 (1939), 165-177.Google Scholar