Szegő-type quadrature formulas

Angular Correlation Using Rogers-Szegő-Chaos

“…Rather than exploit the computational efficiency of methods such as quadrature integral approximations on the unit circle [50][51][52], the integrals are computed as Riemann sums. Therefore, it is necessary to determine an appropriate number of elements that provides an adequate numerical approximation, while remaining computationally feasible.…”

Zeros of quasi-paraorthogonal polynomials and positive quadrature

“…I n (f ) of the form ( 26) to be exact in subspaces L n−1+k , 0 ≤ k ≤ n of the form (27) and not in L n+k . It was proved in [38] for k = n − 1 and in the context of Laurent polynomials in [19,Theorem 2.6] for any k. For completeness we include a proof using the notation and the definitions introduced in Section 2 inspired by the results obtained in [38] for k = n − 1. The result can be stated as follows:…”

“…Hence, from the unicity of the solution, p = S q, and so, P = Q * . For the second part, observe that (19) follows directly from (20) by block Gaussian elimination.…”

“…[19, Theorem 2.3] and there always exist two positive q.f. that are exact in a maximal domain of dimension 2n [19, Corollary 2.9] when supp(µ) = T. Also an analogue of a Jacobi-type theorem (see Theorem 4.1 below) was obtained [19,Theorem 2.6].…”

Orthogonal Rational Functions on the Unit Circle with Prescribed Poles not on the Unit Circle

“…and that no n-point interpolatory quadrature formula with nodes on T and positive weights can be exact in a larger (2n − 1)-dimensional space of the form L α n · L α (n−1) * or L α n * · L α n−1 . They are however exact in a subspace of dimension 2n − 1 of a different form as shown in [39] and [17] for Laurent polynomials and in [11] for the rational case.…”