In Short: I look for a reference to the proof that the spectral coefficients in the Legendre (or Jacobi) expansion are of exponential decay rate.

Longer: If $p_n$ is the $n$-th Legendre polynomial, and the Legendre expansion of a real function $f$ is $f(x) = \sum\limits_{n=0}^{\infty} \hat{f}(n) p_n (x)$, where $\hat{f} (n) = \int\limits_{-1}^1 p_n (x) f(x) dx$, it is a standard result that you find in a lot of textbooks (Szego, Davis, Funaro), in various forms and degrees of formality, that

  1. If $f\in C^{2n+1}$, then $E_n (f) :=\|f- \sum\limits_{n=0}^{N} \hat{f}(n) p_n (x)\|_2 \sim \frac{1}{C^{2n+1}}$ for $C>1$.
  2. If $f\in C^{2n+1}$, then $\lim_{n\to \infty} \frac{\hat{f} (n)}{C^{2n+1}} = O(1)$.

However, if $f\in C^{\infty}$ is analytic in $[-1,1]$, we'd expect $\hat{f}(n)\sim e^{-n}$, or, conversely, $E_n (f) \sim e^{-n}~.$ I found this theorem both in Davis' Interpolation and Approximation (Thm 13.2.2), and Szego's Orthogonal Polynomials (Thm 9.1.1), both without proofs, and references either missing or in German.


This is theorem 2.1 in On the convergence rates of Legendre approximation (2012) [yes, with a proof in English]

  • $\begingroup$ That's brilliant, thank you. I wonder why did they publish the proof of this result. I'd imagine that this is a well known result, and the paper is really about the barycentric Legendre formula. $\endgroup$ – Amir Sagiv Apr 5 '16 at 10:45

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