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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?

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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

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