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Almost periodic function
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an almost periodic function is, loosely speaking, a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> of a <a href="/facts/Real_number/R02gw5Pb">real</a> variable that is <a href="/facts/Periodic_function/Sm8lJQsF">periodic</a> to within any desired level of accuracy, given suitably long, well-distributed "almost-periods". The concept was first studied by <a href="/facts/Harald_Bohr/JaC3Dmqa">Harald Bohr</a> and later generalized by <a href="/facts/Vyacheslav_Stepanov/8oAYDYW5">Vyacheslav Stepanov</a>, <a href="/facts/Hermann_Weyl/4C5HhyGU">Hermann Weyl</a> and <a href="/facts/Abram_Samoilovitch_Besicovitch/u80WU3n2">Abram Samoilovitch Besicovitch</a>, amongst others. There is also a notion of almost periodic functions on <a href="/facts/Locally_compact_abelian_group/oNGQJjvE">locally compact abelian groups</a>, first studied by <a href="/facts/John_von_Neumann/aUk6urof">John von Neumann</a>. </p><p>Almost periodicity is a property of <a href="/facts/Dynamical_system/782gREl5">dynamical systems</a> that appear to retrace their paths through <a href="/facts/Phase_space/qPe4Fq57">phase space</a>, but not exactly. An example would be a <a href="/facts/Planetary_system/6mcVtrTC">planetary system</a>, with <a href="/facts/Planet/rbPjE6Bx">planets</a> in <a href="/facts/Orbit/nD2TPySG">orbits</a> moving with <a href="/facts/Orbital_period/94f56uh2">periods</a> that are not <a href="/facts/Commensurability_(mathematics)/yBFCLN1Q">commensurable</a> (i.e., with a period vector that is not <a href="/facts/Proportionality_(mathematics)/xDQmfh6t">proportional</a> to a vector of <a href="/facts/Integer/PUwwrawx">integers</a>). A <a href="/facts/Kronecker%27s_theorem_on_diophantine_approximation/Fbtxu3G8">theorem of Kronecker</a> from <a href="/facts/Diophantine_approximation/WGVMuWF4">diophantine approximation</a> can be used to show that any particular configuration that occurs once, will recur to within any specified accuracy: if we wait long enough we can observe the planets all return to within, say, a <a href="/facts/Second_of_arc/tybs6YGC">second of arc</a> to the positions they once were in. </p>