Dini–Lipschitz criterion

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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 Ulisse Dini (1872), as a strengthening of a weaker criterion introduced by Rudolf Lipschitz (1864). The criterion states that the Fourier series of a periodic function f converges uniformly on the real line if

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

References[edit]

  • Dini, Ulisse (1872), Sopra la serie di Fourier, Pisa
  • Golubov, B.I. (2001) [1994], "Dini–Lipschitz_criterion", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4