Multiplication formula for the Fourier transform

Remarks

Both sides equal the double integral $\int\int f(x) g(y) e^{-2\pi i x \cdot y}\, dx\, dy$. This double integral converges absolutely, with bound $\norm{f}_1 \norm{g}_1$, so Fubini's theorem allows either order of integration. Absolute convergence of each side also follows from $\abs{\hat f} \le \norm{f}_1$ and $g \in$ $L^1$.

Example: take $n = 1$, $f = \mathbf{1}_{[-1/2, 1/2]}$ and $g(y) = e^{-\pi y^2}$, so that $\hat g = g$. The right side is $\int_{-1/2}^{1/2} e^{-\pi y^2}\, dy$. The left side is $\int_\R \frac{\sin \pi y}{\pi y} e^{-\pi y^2}\, dy$, and the formula says the two numbers are equal.

The formula is the basis of the definition of the Fourier transform of a tempered distribution $T$, namely $\hat T(\varphi) = T(\hat \varphi)$. By the formula, this agrees with the transform of $f$ when $T$ is integration against $f$.

Going deeper

For a student

The formula is the main tool in proofs of the Fourier inversion theorem: one pairs $\hat f$ with a Gaussian of shrinking width and moves the transform onto the Gaussian. It is also what makes the extension of the transform to tempered distributions consistent with the integral definition. With complex conjugates it leads to the Parseval form $\int f \bar g = \int \hat f \overline{\hat g}$ on $L^1 \cap L^2$.