Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $a_{n} \to +\infty$.

share|improve this question
1  
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

1 Answer 1

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?

share|improve this answer
    
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.