Quadrature formulas for Fourier coefficients (Q2389570)

\textit{C. A. Micchelli} and \textit{T. J. Rivlin} [IBM J. Res. Develop. 16, 372--379 (1972; Zbl 0288.65013)] discovered the remarkable fact that the quadrature \[ \int^1_{-1}T_n(t)f(t)\frac{dt}{\sqrt{1-t^2}}\approx \frac{\pi}{n2^n}f'[\xi_1, \dots,\xi_n] \] is exact for all algebraic polynomials of degree \(\leq 3n-1\), where \[ T_n(t):=\cos(n\text{arc}\cos t)=\frac{1}{2^{n-1}}(t-\xi_1)\dots(t-\xi_n) \] is the \(n\)th Chebyshev polynomial and \(f'[\xi_1,\dots,\xi_n]\) is the divided difference of the derivative \(f'\) at the points \(\xi_1,\dots,\xi_n\). The present authors consider formulas of the form \[ \int^b_aP_k(t)f(t)\mu(t)dt \approx \sum^n_{j=1}\sum^{v_j-1}_{i=0}c_{ji}f^{(i)}(x_j),\quad a<x_1< \cdots<x_n< b, \] where \(P_k(t):=t^k+\dots\) is a polynomial of degree \(k\), \(\mu(t)\) is a weight function, and the \(\nu_j\) are given natural numbers (multiplicities). They construct new Gaussian formulas for the Fourier coefficients of a function \(f\) based on the values of \(f\) and its derivatives. Furthermore, they also show the uniqueness of such formulas.