Dini–Lipschitz criterion

Not to be confused with Dini criterion.

In mathematics, the Dini–Lipschitz criterion is a sufficient condition for the Fourier series of a periodic function to converge uniformly at all real numbers. It was introduced by Dini (1872), as a strengthening of a weaker criterion introduced by Lipschitz (1864). The criterion states that the Fourier series of a periodic function f converges uniformly on the real line if

where ω is the modulus of continuity of f with respect to δ.


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