Space Of Laurent Polynomials Research Articles

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Given a sequence of moments $\{c_{n}\}_{n\in\ze}$ associated with an Hermitian linear functional $\mathcal{L}$ defined in the space of Laurent polynomials, we study a new functional $\mathcal{L}_{\Omega}$ which is a perturbation of $\mathcal{L}$ in such a way that a finite number of moments are perturbed. Necessary and sufficient conditions are given for the regularity of $\mathcal{L}_{\Omega}$, and a connection formula between the corresponding families of orthogonal polynomials is obtained. On the other hand, assuming $\mathcal{L}_{\Omega}$ is positive definite, the perturbation is analyzed through the inverse Szego transformation. Keywords: Orthogonal polynomials on the unit circle, perturbation of moments, inverse Szego transformation. To cite this article: E. Fuentes, L.E. Garza, On a finite moment perturbation of linear functionals and the inverse Szego transformation, Rev. Integr. Temas Mat. 34 (2016), No. 1, 39–58.

Hermite Interpolation on the Unit Circle Considering up to the Second Derivative

The paper is devoted to study the Hermite interpolation problem on the unit circle. The interpolation conditions prefix the values of the polynomial and its first two derivatives at the nodal points and the nodal system is constituted by complex numbers equally spaced on the unit circle. We solve the problem in the space of Laurent polynomials by giving two different expressions for the interpolation polynomial. The first one is given in terms of the natural basis of Laurent polynomials and the remarkable fact is that the coefficients can be computed in an easy and efficient way by means of the Fast Fourier Transform (FFT). The second expression is a barycentric formula, which is very suitable for computational purposes.

Quantum continuous gl∞: Semiinfinite construction of representations

We begin a study of the representation theory of quantum continuous gl∞, which we denote by E. This algebra depends on two parameters and is a deformed version of the enveloping algebra of the Lie algebra of difference operators acting on the space of Laurent polynomials in one variable. Fundamental representations of E are labeled by a continuous parameter u∈C. The representation theory of E has many properties familiar from the representation theory of gl∞: vector representations, Fock modules, and semiinfinite constructions of modules. Using tensor products of vector representations, we construct surjective homomorphisms from E to spherical double affine Hecke algebras SḦN for all N. A key step in this construction is an identification of a natural basis of the tensor products of vector representations with Macdonald polynomials. We also show that one of the Fock representations is isomorphic to the module constructed earlier by means of the K-theory of Hilbert schemes.

Perturbations on the subdiagonals of Toeplitz matrices

Let L be an Hermitian linear functional defined on the linear space of Laurent polynomials. It is very well known that the Gram matrix of the associated bilinear functional in the linear space of polynomials is a Toeplitz matrix. In this contribution we analyze some linear spectral transforms of L such that the corresponding Toeplitz matrix is the result of the addition of a constant in a subdiagonal of the initial Toeplitz matrix. We focus our attention in the analysis of the quasi-definite character of the perturbed linear functional as well as in the explicit expressions of the new monic orthogonal polynomial sequence in terms of the first one.

Gromov-Witten theory of orbicurves, the space of tri-polynomials and symplectic field theory of Seifert fibrations

We compute, with symplectic field theory (SFT) techniques, the Gromov-Witten theory of \({\mathbb{P}^1_{\alpha_1,\ldots,\alpha_a}}\), i.e., the complex projective line with a orbifold points. A natural subclass of these orbifolds, the ones with polynomial quantum cohomology, gives rise to a family of (polynomial) Frobenius manifolds and integrable systems of Hamiltonian PDEs, which extend the (dispersionless) bigraded Toda hierarchy (Carlet, The extended bigraded toda hierarchy. arXiv preprint arXiv:math-ph/0604024). We then define a Frobenius structure on the spaces of polynomials in three complex variables of the form F(x, y, z) = −xyz + P 1(x) + P 2(y) + P 3(z) which contains as special cases the ones constructed on the space of Laurent polynomials (Dubrovin, Geometry of 2D topologica field theories. Integrable systems and quantum groups, Springer Lecture Notes in Mathematics 1620:120–348, 1996; Milanov and Tseng, The space of Laurent polynomials, \({\mathbb{P}^1}\)-orbifolds, and integrable hierarchies. preprint arXiv:math/0607012v3 [math.AG]). We prove a mirror theorem stating that these Frobenius structures are isomorphic to the ones found before for polynomial \({\mathbb{P}^1}\)-orbifolds. Finally we link rational SFT of Seifert fibrations over \({\mathbb{P}^1_{a,b,c}}\) with orbifold Gromov-Witten invariants of the base, extending a known result (Bourgeois, A Morse-Bott approach to contact homology. Ph.D. dissertation, Stanford University, 2002) valid in the smooth case.

Quadrature formulas on the unit circle with prescribed nodes and maximal domain of validity

In this paper we investigate the Szegő–Radau and Szegő–Lobatto quadrature formulas on the unit circle. These are ( n + m ) -point formulas for which m nodes are fixed in advance, with m = 1 and m = 2 respectively, and which have a maximal domain of validity in the space of Laurent polynomials. This means that the free parameters (free nodes and positive weights) are chosen such that the quadrature formula is exact for all powers z j , − p ≤ j ≤ p , with p = p ( n , m ) as large as possible.

The spaces of Laurent polynomials, Gromov-Witten theory of ℙ1-orbifolds and integrable hierarchies

Let Mk,m be the space of Laurent polynomials in one variable x+ t1x + . . . tk+mx , where k,m ≥ 1 are fixed integers and tk+m 6= 0. According to B. Dubrovin [11], Mk,m can be equipped with a semi-simple Frobenius structure. In this paper we prove that the corresponding descendent and ancestor potentials of Mk,m (defined as in [16]) satisfy Hirota quadratic equations (HQE for short). Let Ck,m be the orbifold obtained from P by cutting small discs D1 ∼= {|z| ≤ ǫ} and D2 ∼= {|z−1| ≤ ǫ} around z = 0 and z = ∞ and gluing back the orbifolds D1/Zk and D2/Zm in the obvious way. We show that the orbifold quantum cohomology of Ck,m coincides with Mk,m as Frobenius manifolds. Modulo some yet-to-be-clarified details, this implies that the descendent (respectively the ancestor) potential of Mk,m is a generating function for the descendent (respectively ancestor) orbifold Gromov–Witten invariants of Ck,m. There is a certain similarity between our HQE and the Lax operators of the Extended bi-graded Toda hierarchy, introduced by G. Carlet in [7]. Therefore, it is plausible that our HQE characterize the tau-functions of this hierarchy and we expect that the Extended bi-graded Toda hierarchy governs the Gromov–Witten theory of Ck,m.

Sequences of orthogonal Laurent polynomials, bi-orthogonality and quadrature formulas on the unit circle

In this paper, the construction of orthogonal bases in the space of Laurent polynomials on the unit circle is considered. As an application, a connection with the so-called bi-orthogonal systems of trigonometric polynomials is established and quadrature formulas on the unit circle based on Laurent polynomials are studied.

Sequences of orthogonal Laurent polynomials, bi-orthogonality and quadrature formulas on the unit circle

In this paper, the construction of orthogonal bases in the space of Laurent polynomials on the unit circle is considered. As an application, a connection with the so-called bi-orthogonal systems of trigonometric polynomials is established and quadrature formulas on the unit circle based on Laurent polynomials are studied.

Orthogonal Laurent polynomials and two-point Padé approximants associated with Dawson's integral

Starting from the coefficients of the power expansions for the Dawson's integral both around the origin and infinity, a linear functional acting on the space of Laurent polynomials is defined. In this paper, properties of this functional are studied in connection with certain sequences of orthogonal Laurent polynomials and two-point Padé approximants.

Kernels on the unit circle. Orthogonality

In the study of orthogonal polynomials on the unit circle T , one can only state the orthogonality of the kernel polynomials in very few situations (Tasis, 1989). Nevertheless, in the study of orthogonal polynomials on the unit circle one can pose the problem in the same terms as in the real line: Given a regular and hermitian functional u in the space of Laurent polynomials one can consider its modified form by a polynomial of degree one, ( z − α) u. Since this new functional is not hermitian for any complex α, we cannot consider orthogonality in the usual sense. Instead we speak about biorthogonality (Suárez, 1993) with respect to ( z − α) u. In the present paper we focus our attention on this last problem. In Section 1 we study the regularity of ( z − α) u under the assumption on the regularity of u. One of the main results is Theorem 1.3 in which we prove that the right monic orthogonal polynomial sequence related to ( z − α) u is the sequence of monic kernel polynomials K ̃ n(z,α) . In case of orthogonality in the usual sense we also obtain, in Theorems 1.8 and 1.9, the functionals and the measures of orthogonality in the positive definite case. In Section 2 we give the differential equation satisfied by the sequence K n ( z, α) nϵ N in the semiclassical case. Finally, in the last section, we study some properties concerning the roots of kernel polynomials and, in order to compute the roots, we obtain that the zeros of K n ( z, α) are the eigenvalues of a certain matrix.

A generalization of the associated functional to the Lebesgue measure

In this paper we analyse a linear functional defined in the space of Laurent polynomials which can be considered as a generalization of the Lebesgue functional. We study the regularity and the semiclassical character of the functional and we construct the corresponding sequence of orthogonal polynomials. Also, we obtain the differential equation that this family and the associated polynomials of first order satisfy.