= Dini test for Fourier series
{c}
{title2=$\int_0^\delta|f(t+s)+f(t-s)-L_+-L_-|\,ds/s<\infty$}
For an integrable periodic function with one-sided limits $L_+,L_-$ at $t$, the displayed condition implies that its classical <Fourier partial sums> converge to $(L_++L_-)/2$. To prove it use the <Dirichlet kernel> $D_n(s)=\sin((n+1/2)s)/\sin(s/2)$ and its integral $2\pi$. Subtract the target value and combine the positive and negative halves of the convolution integral. The resulting difference is
$$
\frac1{2\pi}\int_0^\pi\frac{f(t+s)+f(t-s)-L_+-L_-}{\sin(s/2)}\sin((n+1/2)s)ds.
$$
The coefficient of the sine is integrable near zero by the Dini condition, and away from zero by integrability of $f$. The <Riemann-Lebesgue lemma> therefore makes this expression tend to zero. <Step functions> satisfy the condition because the numerator vanishes in a sufficiently small punctured neighborhood. This is a local Fourier convergence test, distinct from <Dini's theorem> on uniform convergence of monotone sequences of <continuous functions>.
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