I'd like to know how one is supposed to show that the set $ \{ e^{jwt} \}$, where $\omega \in \mathbb{R}$ , is an ortho-normal basis?
So actually, how do I show that for every $w_1 \neq w_2 $: $\int_{-\infty}^{\infty}e^{jw_1t}e^{-jw_2t}dt=0$ ?
Kindly
Sammy
Note first that if $R\in\Bbb R^+$ and $\omega\ne0$,$$\int_{-R}^R\exp(j\omega t)dt=\left[\frac{1}{j\omega}\exp(j\omega t)\right]_{-R}^R=\frac{2\sin(\omega R)}{\omega}=2R\operatorname{sinc}(\omega R).$$The final expression is also valid if $\omega=0$. Since $\frac{1}{\pi}\operatorname{sinc}(\omega)$ is a nascent delta function,$$\int_{-\infty}^\infty\exp(j\omega t)dt=2\pi\delta(\omega).$$In particular,$$\int_{-\infty}^\infty\overline{\exp(j\omega_1t)}\exp(j\omega_2t)dt=2\pi\delta(\omega_1-\omega_2)$$(in your question, you forgot one factor in the integrand should be complex-conjugated). This is our orthonormality condition (well, if we divide each basis element by $\sqrt{2\pi}$).
