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Bohr-Favard Inequalities


If f has no spectrum in [-lambda,lambda], then

 ||f||_infty<=pi/(2lambda)||f^'||_infty
(1)

(Bohr 1935). A related inequality states that if A_k is the class of functions such that

 f(x)=f(x+2pi),f(x),f^'(x),...,f^((k-1))(x)
(2)

are absolutely continuous and int_0^(2pi)f(x)dx=0, then

 ||f||_infty<=4/pisum_(nu=0)^infty((-1)^(nu(k+1)))/((2nu+1)^(k+1))||f^((k))(x)||_infty
(3)

(Northcott 1939). Further, for each value of k, there is always a function f(x) belonging to A_k and not identically zero, for which the above inequality becomes an equality (Favard 1936). These inequalities are discussed in Mitrinovic et al. (1991).


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References

Bohr, H. "Ein allgemeiner Satz über die Integration eines trigonometrischen Polynoms." Prace Matem.-Fiz. 43, 1935.Favard, J. "Application de la formule sommatoire d'Euler à la démonstration de quelques propriétés extrémales des intégrale des fonctions périodiques ou presquepériodiques." Mat. Tidsskr. B, 81-94, 1936. Reviewed in Zentralblatt f. Math. 16, 58-59, 1939.Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Inequalities Involving Functions and Their Integrals and Derivatives. Dordrecht, Netherlands: Kluwer, pp. 71-72, 1991.Northcott, D. G. "Some Inequalities Between Periodic Functions and Their Derivatives." J. London Math. Soc. 14, 198-202, 1939.Tikhomirov, V. M. "Approximation Theory." In Analysis II. Convex Analysis and Approximation Theory (Ed. R. V. Gamkrelidze). New York: Springer-Verlag, pp. 93-255, 1990.

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Bohr-Favard Inequalities

Cite this as:

Weisstein, Eric W. "Bohr-Favard Inequalities." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Bohr-FavardInequalities.html

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