What Basis Spans Square Integrable Functions with Exponential Tails on [0,∞)?

[cfp]

Hi,
I have a function on $[0,\infty)$ which is represented as:

$$\sum_{ \stackrel{ \Re( \alpha )\in\mathbb{Q}^+ }{ \Im( \alpha )\in\mathbb{Q} } }{\beta_\alpha e^{-\alpha t}}$$

It seems like this must be a basis for the square integrable functions on $[0,\infty)$ with exponential tails. Am I right though in thinking that if the $\alpha$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 $\Im(\alpha)=q$ where $q\in\mathbb{N} \cup 1/\mathbb{N}$. 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

 

[HallsofIvy]

No, I see no reason why requiring that the exponents be real would mean you did not have a basis for "square integrable functions on [0,∞) with exponential tails".

If you allow complex or even pure imaginary exponents, you will have, effectively, a Fourier series which would span much more than "square integrable functions on [0,∞) with exponential tails".

 

[cfp]

Yes, you are correct about pure imaginary exponents spanning a greater space. I'm only really interested in what's needed to span the "square integrable functions on [0,∞) with exponential tails" though, for which real parts must be negative.
Your first claim might be correct, but the proof doesn't seem trivial to me. For example, how would you construct a series of real alphas and betas such that the sum converged to 1 on the interval [a,b] and 0 everywhere else?