A discretized Fourier orthogonal expansion in orthogonal polynomials on a cylinder (Q606665)

In \textit{Y. Xu} [Adv. Appl. Math. 36, No. 4, 388--420 (2006; Zbl 1133.42010)] a discretized expansion in terms of the Radon projections of \(f\) on the closed unit disk \(B^{2}\) in \(\mathbb R^2\) was defined and its convergence was studied. Here the author combines that approach with the Gaussian quadrature associated with the Chebyshev polynomials of the second kind and defines a discretized Fourier orthogonal expansion in orthogonal polynomials on \(B^{2}\times [-1,1].\) Its norm is shown to be of asymptotic order \(m(\log (m+1))^2\).