The Christoffel–Darboux Kernel
نویسنده
چکیده
A review of the uses of the CD kernel in the spectral theory of orthogonal polynomials, concentrating on recent results.
منابع مشابه
On the Christoffel-Darboux kernel for random Hermitian matrices with external source
Bleher and Kuijlaars, and Daems and Kuijlaars showed that the correlation functions of the eigenvalues of a random matrix from unitary ensemble with external source can be expressed in terms of the ChristoffelDarboux kernel for multiple orthogonal polynomials. We obtain a representation of this Christoffel-Darboux kernel in terms of the usual orthogonal polynomials.
متن کاملPfaffians, Determinants, and Multivariable Christoffel–darboux Kernels
Abstract. We obtain expressions for the Christoffel–Darboux kernel of antisymmetric multivariable orthogonal polynomials as determinants and pfaffians. These kernels include correlation functions of orthogonal polynomial ensembles (with β = 2). In subsequent work, our results are applied in combinatorics (enumeration of marked shifted tableaux) and number theory (representation of integers as s...
متن کاملA Christoffel-Darboux formula for multiple orthogonal polynomials
Bleher and Kuijlaars recently showed that the eigenvalue correlations from matrix ensembles with external source can be expressed by means of a kernel built out of special multiple orthogonal polynomials. We derive a Christoffel-Darboux formula for this kernel for general multiple orthogonal polynomials. In addition, we show that the formula can be written in terms of the solution of the Rieman...
متن کاملMultiple orthogonal polynomials of mixed type and non-intersecting Brownian motions
We present a generalization of multiple orthogonal polynomials of type I and type II, which we call multiple orthogonal polynomials of mixed type. Some basic properties are formulated, and a Riemann-Hilbert problem for the multiple orthogonal polynomials of mixed type is given. We derive a Christoffel-Darboux formula for these polynomials using the solution of the Riemann-Hilbert problem. The m...
متن کاملMultivariable Christoffel–Darboux Kernels and Characteristic Polynomials of Random Hermitian Matrices
We study multivariable Christoffel–Darboux kernels, which may be viewed as reproducing kernels for antisymmetric orthogonal polynomials, and also as correlation functions for products of characteristic polynomials of random Hermitian matrices. Using their interpretation as reproducing kernels, we obtain simple proofs of Pfaffian and determinant formulas, as well as Schur polynomial expansions, ...
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تاریخ انتشار 2008