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Almost Periodic Function

A function representable as a generalized Fourier series. Let R be a metric space with metric rho(x,y). Following Bohr (1947), a continuous function x(t) for (-infty<t<infty) with values in R is called an almost periodic function if, for every epsilon>0, there exists l=l(epsilon)>0 such that every interval [t_0,t_0+l(epsilon)] contains at least one number tau for which

rho[x(t),x(t+tau)]<epsilon

for (-infty<t<infty). Another formal description can be found in Krasnosel'skii et al. (1973).

Every almost periodic function is bounded and uniformly continuous on the entire real line.

See also

Fourier Series, Periodic Function

This entry contributed by Ronald M. Aarts

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References

Bohr, H. Almost Periodic Functions. New York: Chelsea, 1947.Besicovitch, A. S. Almost Periodic Functions. New York: Dover, 1954.Corduneanu, C. Almost Periodic Functions. New York: Wiley Interscience, 1961.Krasnosel'skii, M. A.; Burd, V. Sh.; and Kolesov, Yu. S. Nonlinear Almost Periodic Oscillations. New York: Wiley, 1973.Levitan, B. M. Almost-Periodic Functions. Moscow, 1953.Montgomery, H. L. "Harmonic Analysis as Found in Analytic Number Theory." In Twentieth Century Harmonic Analysis--A Celebration. Proceedings of the NATO Advanced Study Institute Held in Il Ciocco, July 2-15, 2000 (Ed. J. S. Byrnes). Dordrecht, Netherlands: Kluwer, pp. 271-293, 2001.

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Almost Periodic Function

Cite this as:

Aarts, Ronald M. "Almost Periodic Function." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/AlmostPeriodicFunction.html

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