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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
Affiliation:
National Institute of OceanographyWormley

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) , 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) 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
Copyright
Copyright © Cambridge Philosophical Society 1955

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References

REFERENCES

(1)Barber, N. F.Finding the direction of travel of sea waves. Nature, Lond., 174 (1954), 1048–50.CrossRefGoogle Scholar
(2)Carathéodory, C. Über den Variabilitätsbereich der Koeffizienten von Potenzreihen die gegebene Werte nicht annehmen. Math. Ann. 64 (1907), 95115.CrossRefGoogle Scholar
(3)Carathéodory, C.Über den Variabilitätsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen. R.C. Circ. mat. Palermo, 32 (1911), 193217.Google Scholar
(4)Carathéodoby, C. and Fejér, . Über den Zusammenhang der Extremen von harmonischen Funktionen mit ihren Koeffizienten und über den Picard-Landau'schen Satz. R.C. Circ. mat. Palermo, 32 (1911), 218–39.Google Scholar
(5)Dirac, P. A. M.The principles of quantum mechanics (Oxford, 1930).Google Scholar
(6)Fischer, E.Über das Carathéodory'sche Problem, Potenzreihen mit positivem reellen Teil betreffend. R.C. Circ. mat. Palermo, 32 (1911), 240–56.Google Scholar
(7)Frobenius, G.Ableitung eines Satzes von Carathéodory aus einer Formel von Kronecker. S. B. preuss. Akad. Wiss. (1912), pp. 1631.Google Scholar
(8)Riesz, F.Sur certains systèmes singuliers d'équations intégrates. Ann. sci. Éc. norm. sup., Paris, (3), 28 (1911), 3361.CrossRefGoogle Scholar
(9)Riesz, F.Les systémes d'équations linéaires à une infinité d'inconnues (Paris, 1913, reprinted 1952).Google Scholar
(10)Schur, I.Über einen Satz von C. Carathéodory. S.B. preuss. Akad. Wiss. (1912), pp. 415.Google Scholar
(11)Toeplitz, O.Über die Fourier'sche Entwickelung positiver Funktionen. R.C. Circ. mat. Palermo, 32 (1911), 191–2.Google Scholar
(12)Zygmund, A.Trigonometrical series, 2nd ed. (New York, 1952).Google Scholar