Fourier's Theorem/Integral Form

Theorem

Let $f: \R \to \R$ be a real function which satisfies the Dirichlet conditions on $\R$.

Then:

$\dfrac {\map f {t^+} + \map f {t^-} } 2 = \ds \int_{-\infty}^\infty e^{2 \pi i t s} \paren {\int_{-\infty}^\infty e^{-2 \pi i t s} \map f t \rd t} \rd s$

where:

$\map f {t^+}$ and $\map f {t^-}$ denote the limit from above and the limit from below of $f$ at $t$.

Continuous Point

Let $f$ be continuous at $t \in \R$.

Then:

$\ds \map f t = \int_{-\infty}^\infty e^{2 \pi i t s} \paren {\int_{-\infty}^\infty e^{-2 \pi i t s} \map f t \rd t} \rd s$

Proof

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Source of Name

This entry was named for Joseph Fourier.

Sources

• 1978: Ronald N. Bracewell: The Fourier Transform and its Applications (2nd ed.) ... (previous) ... (next): Chapter $2$: Groundwork: The Fourier transform and Fourier's integral theorem