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 $a_{n} \to +\infty$.
How to show that limit is a delta function
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limits
distribution-theory
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1If you ask mathematicians this question, you will have to say limit in what sense? Some sort of distributions of two variables, I guess... – 2012-11-24
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0@GEdgar thank you for comment, I've improved my post – 2012-11-24