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The Clenshaw-Curtis quadrature rule approximates an integral $I=\int\limits_{-1}^{1} f(x) \, dx$ by $$I\approx I_n = \sum\limits_{j=1}^N f(x_j)w_j \, ,$$ where the $x_j$'s are the roots of the $N$-th order Chebyshev polynomial, and and $w_j$'s their respective weight. To prove the accuracy of this integration formula, one usually goes by either Fourier representation of $f(x)=f(\cos (\theta))$, or by the "Fourier" expansion of $f$ in the Chebyshev polynomials. See e.g., in the Wiki page.

My Question: Is there a way to prove the accuracy of this formula, which does not rely on spectral/Fourier theory? Specifically, to show that it is exact ($I=I_n$) for polynomials of degree $\leq n$, and to bound its error for $f\in C^n$.

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In "Error bounds for the Clenshaw-Curtis quadrature formulas" the error bounds are obtained without relying on analytic function theory (and no Fourier transforms).

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