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

Abstract

Abstract. Orthogonal rational functions (ORF) on the unit circle generalize orthogonal polynomials (poles at infinity) and Laurent polynomials (poles at zero and infinity). In this paper we investigate the properties of and the relation between these ORF when the poles are all outside or all inside the unit disk, or when they can be anywhere in the extended complex plane outside the unit circle. Some properties of matrices that are the product of elementary unitary transformations will be proved and some connections with related algorithms for direct and inverse eigenvalue problems will be explained.

Forecasting Laurent Polynomial in the Chern–Simons Current of V3 Loop Time Series

“…We visualize all transition states in developmental biology by using modified path integral along ribbon graph produced from Laurent polynomial [35,36] in complex projective plane. In the new quantum field theory for biology with Khovanov cohomology and Grothendieck topology, we define Laurent polynomial [37,38] in the knotted time series data with the characteristic class over 35 amino acids in V3 loop HIV viral glycoprotein by taking the functor from the ribbon graph of the tensor network to categories of knots and links in the secondary protein structure.…”