The Riemann–Hilbert problem for certain poly domains and its connection to the Riemann problem

Mohammed, Alip

doi:10.1016/j.jmaa.2008.01.053

Cited by 9 publications

(3 citation statements)

References 19 publications

(49 reference statements)

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“…Here we discuss integral expressions for the second kind functions and Plemej type formulae. We follow the papers by Mohammed [72] and [73].…”

Section: Proposition 24mentioning

confidence: 99%

“…Proof. For the proof we need some observations regarding [72] and [73]. Obviously, the MVOLPUT belong to the Wiener algebra of the D-dimensional unit torus,…”

Section: Proposition 24mentioning

confidence: 99%

“…We assume that d µ = μ(θ) d θ with μ ∈ W D , and consequently, given that we are dealing with Laurent polynomials, Φ α μ, Φα μ ∈ W D . Following [72,73] for any Hölder continuous complex function on the unit torus ϕ ∈ C α (T D , C), 0 < α < 1, one defines its integral transformations φ (σ) and its boundary integral conjugates φ (σ) * as follows…”

Section: Proposition 24mentioning

confidence: 99%

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Multivariate Orthogonal Laurent Polynomials and Integrable Systems

Ariznabarreta¹,

Mañas²

2022

Publ. Res. Inst. Math. Sci.

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An ordering for Laurent polynomials in the algebraic torus (C * ) D , inspired by the Cantero-Moral-Velázquez approach to orthogonal Laurent polynomials in the unit circle, leads to the construction of a moment matrix for a given Borel measure in the unit torus T D . The Gauss-Borel factorization of this moment matrix allows for the construction of multivariate biorthogonal Laurent polynomials in the unit torus which can be expressed as last quasi-determinants of bordered truncations of the moment matrix. The associated second kind functions are expressed in terms of the Fourier series of the given measure. Persymmetries and partial persymmetries of the moment matrix are studied and Cauchy integral representations of the second kind functions are found as well as Plemej type formulae. Spectral matrices give string equations for the moment matrix which model the three-term relations as well as the Christoffel-Darboux formulae.Christoffel type perturbations of the measure given by the multiplication by Laurent polynomials are studied. Sample matrices on poised sets of nodes, which belong to the algebraic hypersurface of the perturbing Laurent polynomial, are used for the finding of a Christoffel formula that expresses the perturbed orthogonal Laurent polynomials in terms of a last quasi-determinant of a bordered sample matrix constructed in terms of the original orthogonal Laurent polynomials. Poised sets exist only for nice Laurent polynomials which are analyzed from the perspective of Newton polytopes and tropical geometry. Then, an algebraic geometrical characterization of nice Laurent polynomial perturbation and poised sets is given; full column rankness of the corresponding multivariate Laurent-Vandermonde matrices and a product of different prime nice Laurent polynomials leads to such sets. Some examples are constructed in terms of perturbations of the Lebesgue-Haar measure.Discrete and continuous deformations of the measure lead to a Toda type integrable hierarchy, being the corresponding flows described through Lax and Zakharov-Shabat equations; bilinear equations and vertex operators are found. Varying size matrix nonlinear partial difference and differential equations of the 2D Toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials. The discrete flows are connected with a Gauss-Borel factorization of the Jacobi type matrices and its quasi-determinants alow for expressions for the multivariate orthogonal polynomials in terms of shifted quasi-tau matrices, which generalize those that relate the Baker functions with ratios of Miwa shifted τ-functions in the 1D scenario. It is shown that the discrete and continuous flows are deeply connected and determine nonlinear partial differencedifferential equations that involve only one site in the integrable lattice behaving as a Kadomstev-Petviashvili type system.

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“…Here we discuss integral expressions for the second kind functions and Plemej type formulae. We follow the papers by Mohammed [72] and [73].…”

Section: Proposition 24mentioning

confidence: 99%

“…Proof. For the proof we need some observations regarding [72] and [73]. Obviously, the MVOLPUT belong to the Wiener algebra of the D-dimensional unit torus,…”

Section: Proposition 24mentioning

confidence: 99%

Section: Proposition 24mentioning

confidence: 99%

See 1 more Smart Citation