1
$\begingroup$

I want to write down an arbitrary function $f$ as an (infinite) sum of orthogonal polynomials, e.g. for $f(x) = e^{\sin(x)}$, $f(x) = \sum{a_n T_n}$, where $a_n$ are the coefficients and $T_n$ are the orthogonal polynomials.

So the task is to find the $a_n$, which is easy. But I require the $a_n$ to be left as a nice formula rather than working out the $a_n$ to be horrible numbers like 1.2391000102 etc. etc. or leaving them as integrals. So I guess something like $a_n = \frac13\sin(n)$ or whatever.

When I used Chebyshev polynomials with the discrete orthogonality condition, I ended up with a double sum, so I could not take the limit to infinity by simply replacing the number at the top of sum signs with infinity. It seems I need to choose a function $f(x)$ carefully so that the double sum simplifies, but it is really hard to do so.

Any suggestions, or any ideas for references where such problems are left as questions to the reader (so they probably have nice solutions)? In summary, I'd like a single sum with a nice formula for the $a_n$, and I don't mind what $f$ is or what polynomial system I use.

  • 0
    If you have a generating function for T_n, I might be able to provide a generating function for your series; it might involve functional composition. I haven't tested all of the variants but I have been successful so far; but there might be expressions that stymie my process. I could certainly envision cases that render it useless.2015-10-24

0 Answers 0