The concept of quasi-orthogonality was introduced in 1923 by M. Riesz in connection with moment problems, and then by L. Fejér and J.A. Shohat in 1933 and 1937 respectively, in the context or quadrature formulas.
It is very well known that the polynomials that play a similar role on the unit circle to orthogonal polynomials on the real line are paraorthogonal polynomials. The purpose of this talk is to adequately extend the concept of quasi-orthogonality to the unit circle by introducing in the literature the concept of quasi-paraorthogonal polynomials. Applications to the construction of positive quadrature formulas on the unit circle with preassigned nodes will be analyzed. The results of this talk were published in [1] as a part of a joint work with Carlos Díaz Mendoza (La Laguna University) and Adhemar Bultheel (KU Leuven).

[1] A. Bultheel, R. Cruz-Barroso and C. Díaz-Mendoza, Zeros of quasi-paraorthogonal polynomials and positive quadrature. Journal of Computational and Applied Mathematics 407 (2022) 114039.