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$ .