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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.

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

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  • $\begingroup$ 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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