- #1
mnb96
- 715
- 5
Hello,
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 its Cauchy Principal Value.
Is this allowed or not?
What rigorous restriction should I include in order to say that those functions are orthogonal?
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 its Cauchy 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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