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For positive quadrature weights summing to the total mass $M$, a quadrature rule exact on polynomials through polynomial degree $d$ satisfies $|I(f)-Q(f)|\le 2M\Vert f-p_d{\Vert }_{\infty }$ for every such polynomial $p_d$. This follows by subtracting the exactly integrated $p_d$ and applying the triangle inequality to the integral and weighted sum. For Gauss-Legendre quadrature with $n+1$ nodes, $M=2$ and $d=2n+1$. Exponentially decaying Chebyshev series coefficients therefore yield a bound proportional to $e^{-2Kn}$ when $|c_j|\le e^{-Kj}$.

Put $x=\cos \theta$ in the weighted inner product. Since $T_r(\cos \theta )=\cos (r\theta )$,The last equality follows by the product-to-sum trigonometric identity and integration. Thus these are nonzero mutually orthogonal polynomials. The first $n+1$ are linearly independent, and the vector space $P_n$ has dimension $n+1$; therefore $\boxed{P_n=\mathrm{span}(T_0,\dots ,T_n)}$.

For any $q\in P_n$, the function of the real parameter $t$has a minimum at $t=0$. Its derivative there vanishes, proving $\boxed{(u-p_n,q)=0}$. This is precisely the defining orthogonality of an orthogonal projection, or a least-squares polynomial in an orthogonal-polynomial basis. Conversely, this condition and the Pythagorean theorem for inner product spaces give $\Vert u-q{\Vert }_2^2=\Vert u-p_n{\Vert }_2^2+\Vert p_n-q{\Vert }_2^2$, proving the minimum and its uniqueness.

Expand $p_n={\sum }_{r=0}^n{a}_r{T}_r$ in the basis already established and take the inner product with $T_s$. Orthogonality leaves $a_s(T_s,T_s)=(u,T_s)$, soIn particular $a_0={\pi }^{-1}{\int }_0^{\pi }u(\cos \theta )d\theta$ and $a_r=2{\pi }^{-1}{\int }_0^{\pi }u(\cos \theta )\cos (r\theta )d\theta$ for $r\ge 1$. These are the corresponding Chebyshev series coefficients with the constant coefficient convention used in this question.

Truncate the Chebyshev series at the full degree of precision of a quadrature rule:Since $|T_j(x)|\le 1$ on $[-1,1]$, the geometric series estimate givesThe same bound proves uniform convergence of the original Chebyshev series, so its sum and the tail estimate are justified. Gauss-Legendre quadrature is exact on $s_{2n+1}$. Its positive quadrature weights sum to $2$, henceTherefore one explicit choice proving exponential convergence of Gaussian quadrature isUsing exactness through polynomial degree $2n+1$ doubles the exponent compared with truncating at polynomial degree $n$.