Terpujilah Kristus.
\section{Homomorfisme antara Grup Konvolusi Fungsi dan Grup Perkalian Fungsi}
Andaikan ada himpunan $C(\mathbb{R}, \mathbb{C})$ yang berisi semua fungsi dari $\mathbb{R}$ ke $\mathbb{C}$.
Transformasi Fourier $F \,:\, C(\mathbb{R}, \mathbb{C}) \to C(\mathbb{R}, \mathbb{C})$ antara lain didefinisikan sedemikian
\[ (F(f))(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty f(t)e^{i\omega t}dt, \]
sedangkan konvolusi antara dua buah fungsi $f, g \in C(\mathbb{R}, \mathbb{C})$ antara lain adalah $f*g$ sedemikian
\[ (f*g)(t) = \int_{-\infty}^\infty f(t - \tau)g(\tau)d\tau. \]
Andaikan ada dua buah grup, yaitu $(C(\mathbb{R}, \mathbb{C}),~)$ dan $(C(\mathbb{R}, \mathbb{C}), *)$.
\[ (F(f)F(g))(\omega) = (F(f))(\omega)(F(g))(\omega) \]
\[ = \frac{1}{2\pi}\int_{-\infty}^\infty f(t)e^{i\omega t}dt \int_{-\infty}^\infty g(t')e^{i\omega t'}dt' \]
\[ = \frac{1}{2\pi}\int_{-\infty}^\infty \int_{-\infty}^\infty f(t)g(t')e^{i\omega(t + t')}dt'dt \]
\[ = \frac{1}{2\pi}\int_{-\infty}^\infty \int_{-\infty}^\infty f(t)g(\tau - t)e^{i\omega\tau}d\tau\,dt \]
\[ = \frac{1}{2\pi}\int_{-\infty}^\infty \int_{-\infty}^\infty f(t)g(\tau - t)dt\,e^{i\omega\tau}d\tau \]
\[ = \frac{1}{2\pi}\int_{-\infty}^\infty (f*g)(\tau)e^{i\omega\tau}d\tau \]
\[ = \frac{1}{\sqrt{2\pi}}(F(f*g))(\omega). \]
Di samping itu,
\[ (F(f)*F(g))(\omega) = \int_{-\infty}^\infty (F(f))(\omega - u)(F(g))(u)du \]
\[ = \frac{1}{2\pi}\int_{-\infty}^\infty \int_{-\infty}^\infty f(t)e^{i(\omega - u)t}dt \int_{-\infty}^\infty g(t')e^{iut'}dt'du \]
\[ = \frac{1}{2\pi}\int_{-\infty}^\infty \int_{-\infty}^\infty f(t)g(t')e^{i\omega t}\int_{-\infty}^\infty e^{iu(t' - t)}du\,dt\,dt' \]
\[ = \int_{-\infty}^\infty \int_{-\infty}^\infty f(t)g(t')e^{i\omega t}\delta(t' - t)dt'dt \]
\[ = \int_{-\infty}^\infty f(t)g(t)e^{i\omega t}dt \]
\[ = \int_{-\infty}^\infty (fg)(t)e^{i\omega t}dt \]
\[ = \sqrt{2\pi}(F(fg))(\omega). \]
Dari dua perhitungan tersebut, diperoleh kesimpulan bahwa
\[ F(f*g) = \sqrt{2\pi}F(f)F(g) \]
dan
\[ F(fg) = \frac{1}{\sqrt{2\pi}}F(f)*F(g) \]
sehingga homomorfisme dari grup $(C(\mathbb{R}, \mathbb{C}), *)$ ke grup $(C(\mathbb{R}, \mathbb{C}), ~)$ adalah $\sqrt{2\pi}F$, sedangkan homomorfisme dari grup $(C(\mathbb{R}, \mathbb{C}), ~)$ ke grup $(C(\mathbb{R}, \mathbb{C}), *)$ adalah $(1/\sqrt{2\pi})F$.
Terpujilah Kristus.
