Dini–Lipschitz 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 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

lim δ → 0 + ω ( δ , f ) log ⁡ ( δ ) = 0 {\displaystyle \lim _{\delta \rightarrow 0^{+}}\omega (\delta ,f)\log(\delta )=0}

where ω {\displaystyle \omega } is the modulus of continuity of f with respect to δ {\displaystyle \delta } .

References

  • Dini, Ulisse (1872), Sopra la serie di Fourier, Pisa
  • Golubov, B. I. (2001) [1994], "Dini-Lipschitz criterion", Encyclopedia of Mathematics, EMS Press