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Bounds for the integral of a non-negative function in terms of its Fourier coefficients

Published online by Cambridge University Press:  24 October 2008

M. S. Longuet-Higgins
M. S. Longuet-Higgins
Affiliation:
National Institute of OceanographyWormley
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Abstract

The first 2N + 1 Fourier coefficients of an unknown, non-negative function f(θ) are given, and it is required to find bounds for ∫Ef(θ) dθ, where E is some given region of integration. We also wish to find the interval E for which the bounds are most strict, when the width of E is specified. f(θ) may represent a distribution of energy in the interval 0 ≤ θ ≤ 2π; the object is to determine where the energy is chiefly located.

In the present paper we show that if the energy is located mainly in the neighbourhood of not more than M distinct points, significant lower bounds for ∫Ef(θ) dθ can be found in terms of the first 2M + 1 Fourier coefficients. The effectiveness of the method is illustrated by applying the inequalities to some known functions.

The results have application in determining the direction of propagation of ocean waves and other forms of energy.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 51 , Issue 4 , October 1955 , pp. 590 - 603
Copyright
Copyright © Cambridge Philosophical Society 1955

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References

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