For the double exponential weight, the squared orthogonal-polynomial coefficients weighted by log-squared degree are controlled by a weighted Sobolev energy, and this rate is sharp.
Levin and Doron S
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Polynomial Approximation in $ L^2 $ of the Double Exponential via Complex Analysis
For the double exponential weight, the squared orthogonal-polynomial coefficients weighted by log-squared degree are controlled by a weighted Sobolev energy, and this rate is sharp.