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Consider a Sturm-Liouville system over an interval $[a,b]$:

$$(p(x)y')' + (q(x) + w(x) \lambda) y = 0$$

Induced by this Sturm-Liouville system is a set of special functions that form a complete basis of the weighted Hilbert space $(L^2[a,b], w(\mu) d \mu)$.

I have been told of a particular result for conventional Fourier series (sines, cosines; $w(x) = 1$) that I would like to generalise to bases induced by any Sturm-Liouville system:

Let $f \in C([a,b],\mathbb R)$.

If the "periodic extension" of the Fourier expansion of $f$ is $C^m$, then for asymptotically large $n$, the coefficients of the Fourier expansion are of order $O \left(\frac 1 {n^{2+m}}\right).$

I'm not entirely sure if I have written this result correctly, but in any case, what I'm looking for is some method of finding the rate of convergence of a generalised Fourier series.

Is there some technique for finding what conditions on a function $f\in (L^2[a,b], w(\mu) d \mu)$ determine the rate of convergence of its generalised Fourier series?

Myridium
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  • You might find this question interesting http://math.stackexchange.com/questions/1105213/general-fourier-coefficients-and-smoothness?lq=1. For (less abstract) Jacobi polynomials, books on spectral methods might have some results, I am not too sure. For the Chebyshev, one of Jacobi, see Theorem 7.1 https://books.google.com/books?id=h80N5JHm-u4C&lpg=PA53&vq=Therem%207.1&hl=ja&pg=PA52#v=onepage&q&f=false – shall.i.am Sep 01 '15 at 13:43
  • I meant Theorem 7.1 in the book by Trefethen if the decay rate of coefficients are of your interest as the body of the question seems to be saying, and Theorem 7.2 if the convergence rate is what you want as the title suggests. http://www.springer.com/us/book/9783540307259 have some results on convergence rates with other polynomials as well. – shall.i.am Sep 01 '15 at 13:57

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