How to show that limit is a delta function

Let $\{\phi_{n}(t)\}_{n=1}^{\infty}$ be a complete orthonormal system at $[a,b]$. Then $$\sum\limits_{n=1}^{\infty} \phi_{n}(t)\phi_{n}(s) = \lim\limits_{N \to \infty} \sum\limits_{n=1}^{N} \phi_{n}(t)\phi_{n}(s) = \delta(t-s)$$ How to show that $$\lim\limits_{x \to +0} \sum\limits_{n=1}^{\infty} \phi_{n}(t)\phi_{n}(s)e^{-a_{n}x} = \delta(t-s),$$ in the sense that $$\lim\limits_{x \to +0} \int \sum\limits_{n=1}^{\infty} \phi_{n}(t)\phi_{n}(s)e^{-a_{n}x} f(t) dt = f(s)$$ if series $\sum_{n=1}^{\infty} \phi_n(t)\phi_n(s) e^{-a_n x}$ converge pointwise for $x > 0$ and where $a_{n} \to +\infty$.

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If you ask mathematicians this question, you will have to say limit in what sense? Some sort of distributions of two variables, I guess... – GEdgar Nov 24 '12 at 21:32
@GEdgar thank you for comment, I've improved my post – Nimza Nov 24 '12 at 21:43

Well (assuming real values) $$\int \phi_n(t) f(t) dt = u_n,$$ say, are the coefficients for the orthogonal expansion of $f$ as $\sum_n u_n \phi_n(s) = f(s)$, where this holds in the sense of $L^2$ convergence. Pointwise convergence fails in general. Why should putting some funny exponential factors in there make it converge pointwise?
Exponential factor arises in study of heat equation. For general type of boundary conditions on segment Green function has the form $\sum\limits_{n=1}^{\infty} f_n(x)f_n(y) e^{-a_n t}$. For Dirichlet boundary conditions $f_n$-s are sinuses and exponent saves the day. – Nimza Nov 24 '12 at 22:08