Article contents
Bounds for the integral of a non-negative function in terms of its Fourier coefficients
Published online by Cambridge University Press: 24 October 2008
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
References
REFERENCES
- 3
- Cited by