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I have a function on [itex][0,\infty)[/itex] which is represented as:

[tex]\sum_{ \stackrel{ \Re( \alpha )\in\mathbb{Q}^+ }{ \Im( \alpha )\in\mathbb{Q} } }{\beta_\alpha e^{-\alpha t}}[/tex]

It seems like this must be a basis for the square integrable functions on [itex][0,\infty)[/itex] with exponential tails. Am I right though in thinking that if the [itex]\alpha[/itex]s are constrained to be real, then it is no longer a basis? What class of functions are spanned then? It also seems like there are many redundant terms in the sum. On the complex side I expect you could restrict [itex]\Im(\alpha)=q[/itex] where [itex]q\in\mathbb{N} \cup 1/\mathbb{N}[/itex]. Is a similar restriction possible on the real side?

Any comments would be much appreciated, as would directions to any papers on the properties of bases like this.

Thanks in advance,

Tom

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# Basis of exponentials on [0,∞)?

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