Dini's Test

A test for the convergence of Fourier Series. Let

$$\phi_x(t)\equiv f(x+t)+f(x-t)-2f(x),$$

then if

$$\int_0^\pi {\vert\phi_x(t)\vert\,dt\over t}$$

is Finite, the Fourier Series converges to $f(x)$ at $x$.

See also Fourier Series

References

Sansone, G. Orthogonal Functions, rev. English ed. New York: Dover, pp. 65-68, 1991.