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 FunctionThis 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.Referenced on Wolfram|Alpha
Almost Periodic FunctionCite this as:
Aarts, Ronald M. "Almost Periodic Function." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/AlmostPeriodicFunction.html