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Let's consider the [tex]L^2(\mathbb{R})[/tex] space with an inner product, and the complex sinusoids in the interval [tex](-\infty,+\infty)[/tex].

Is it correct to say that the complex sinusoids form an orthogonal basis for this space?

One would need to have:

[tex]\int_{-\infty}^{+\infty}e^{ipx}e^{-iqx}dx=0[/tex]

for any [tex]p\neq q[/tex]

but if [itex]k=p-q[/itex], that integral is:

[tex]\int_{-\infty}^{+\infty}e^{ikx}dx[/tex]

and that integral is zero only considering itsCauchy Principal Value.

Is this allowed or not?

What rigorous restriction should I include in order to say that those functions are orthogonal?

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# Orthogonality of sinusoids

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