Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userIdentification of the theory of multidimensional orthogonal polynomials with the theory of symmetric interacting Fock spaces with finite dimensional one particle space
307
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The identification mentioned in the title allows a formulation of the multidi mensional Favard Lemma different from the ones currently used in the literature and which exactly parallels the original one dimensional formulation in the sense that the positive Jacobi sequence is replaced by a sequence of positive Hermitean (square) matrices and the real Jacobi sequence by a sequence of Hermitean matri ces of the same dimension. Moreover, in this identification, the multidimensional extension of the compatibility condition for the positive Jacobi sequence becomes the condition which guarantees the existence of the creator in an interacting Fock space. The above result opens the way to the program of a purely algebraic clas sification of probability measures on $mathbb{R}^d$ with finite moments of any order. In this classification the usual Boson Fock space over $mathbb{C}^d$ is characterized by the fact that the positive Jacobi sequence is made up of identity matrices and the real Jacobi sequences are identically zero. The quantum decomposition of classical real valued random variables with all moments is one of the main ingredients in the proof.
rate research
Read More
We have two constructions of the level-$(0,1)$ irreducible representation of the quantum toroidal algebra of type $A$. One is due to Nakajima and Varagnolo-Vasserot. They constructed the representation on the direct sum of the equivariant K-groups of the quiver varieties of type $hat{A}$. The other is due to Saito-Takemura-Uglov and Varagnolo-Vasserot. They constructed the representation on the q-deformed Fock space introduced by Kashiwara-Miwa-Stern. In this paper we give an explicit isomorphism between these two constructions. For this purpose we construct simultaneous eigenvectors on the q-Fock space using nonsymmetric Macdonald polynomials. Then the isomorphism is given by corresponding these vectors to the torus fixed points on the quiver varieties.
We obtain asymptotics in n for the n-dimensional Hankel determinant whose symbol is the Gaussian multiplied by a step-like function. We use Riemann-Hilbert analysis of the related system of orthogonal polynomials to obtain our results.
$SL^infty$ denotes the space of functions whose square function is in $L^infty$, and the subspaces $SL^infty_n$, $ninmathbb{N}$, are the finite dimensional building blocks of $SL^infty$. We show that the identity operator $I_{SL^infty_n}$ on $SL^infty_n$ well factors through operators $T : SL^infty_Nto SL^infty_N$ having large diagonal with respect to the standard Haar system. Moreover, we prove that $I_{SL^infty_n}$ well factors either through any given operator $T : SL^infty_Nto SL^infty_N$, or through $I_{SL^infty_N}-T$. Let $X^{(r)}$ denote the direct sum $bigl(sum_{ninmathbb{N}_0} SL^infty_nbigr)_r$, where $1leq r leq infty$. Using Bourgains localization method, we obtain from the finite dimensional factorization result that for each $1leq rleq infty$, the identity operator $I_{X^{(r)}}$ on $X^{(r)}$ factors either through any given operator $T : X^{(r)}to X^{(r)}$, or through $I_{X^{(r)}} - T$. Consequently, the spaces $bigl(sum_{ninmathbb{N}_0} SL^infty_nbigr)_r$, $1leq rleq infty$, are all primary.
In 1975, P.R. Chernoff used iterates of the Laplacian on $mathbb{R}^n$ to prove an $L^2$ version of the Denjoy-Carleman theorem which provides a sufficient condition for a smooth function on $mathbb{R}^n$ to be quasi-analytic. In this paper, we prove an exact analogue of Chernoffs theorem for all rank one Riemannian symmetric spaces (of noncompact and compact types) using iterates of the associated Laplace-Beltrami operators.
The paper contains a generalization of known properties of Chebyshev polynomials of the second kind in one variable to polynomials of $n$ variables based on the root lattices of compact simple Lie groups $G$ of any type and of rank $n$. The results, inspired by work of H. Li and Y. Xu where they derived cubature formulae from $A$-type lattices, yield Gaussian cubature formulae for each simple Lie group $G$ based on interpolation points that arise from regular elements of finite order in $G$. The polynomials arise from the irreducible characters of $G$ and the interpolation points as common zeros of certain finite subsets of these characters. The consistent use of Lie theoretical methods reveals the central ideas clearly and allows for a simple uniform development of the subject. Furthermore it points to genuine and perhaps far reaching Lie theoretical connections.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
