 Dini test for Fourier series

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 isThe 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.