Almost periodic function

Summary

An almost periodic function is a function that nearly repeats itself. For any tolerance there are many shifts that change it by less than the tolerance. On the real line, the uniformly almost periodic functions are exactly the uniform limits of finite sums $\sum c_k e^{i\lambda_k x}$ with arbitrary real frequencies $\lambda_k$. They extend the theory of Fourier series from periodic functions to functions such as $\cos x + \cos(\sqrt{2} x)$. The notion was created by Harald Bohr, and it extends to abelian groups, where it is linked to the Bohr compactification.