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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0@GEdgar thank you for comment, I've improved my post – 2012-11-24
1 Answers
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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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0Exponential 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. – 2012-11-24