Fourier's Theorem/Integral Form/Continuous Point

Theorem

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

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

At a point of continuity we have:

\(\ds \dfrac {\map f {t^+} + \map f {t^-} } 2\)

\(=\)

\(\ds \dfrac {\map f t + \map f t} 2\)

as $\map f t = \map f {t^+} = \map f {t^-}$ at a point of continuity

\(\ds \)

\(=\)

\(\ds \dfrac {2 \map f t} 2\)

\(\ds \)

\(=\)

\(\ds \map f t\)

The result follows from Fourier's Theorem: Integral Form.

$\blacksquare$

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