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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


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

© 1996-9 Eric W. Weisstein