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- Title
Meixner class of orthogonal polynomials of a non-commutative monotone Lévy noise.
- Authors
Lytvynov, Eugene; Rodionova, Irina
- Abstract
Let (Xt)t≥0 denote a non-commutative monotone Lévy process. Let ω=(ω(t))t≥0 denote the corresponding monotone Lévy noise, i.e. formally ω(t)=ddtXt. A continuous polynomial of ω is an element of the corresponding non-commutative L2-space L2(τ) that has the form ∑i=0n〈ω⊗i,f(i)〉, where f(i)∈C0(ℝ+i). We denote by CP the space of all continuous polynomials of ω. For f(n)∈C0(ℝ+n), the orthogonal polynomial 〈P(n)(ω),f(n)〉 is defined as the orthogonal projection of the monomial 〈ω⊗n,f(n)〉 onto the subspace of L2(τ) that is orthogonal to all continuous polynomials of ω of order ≤n−1. We denote by OCP the linear span of the orthogonal polynomials. Each orthogonal polynomial 〈P(n)(ω),f(n)〉 depends only on the restriction of the function f(n) to the set {(t1,…,tn)∈ℝ+n|t1≥t2≥⋯≥tn}. The orthogonal polynomials allow us to construct a unitary operator J:L2(τ)→𝔽, where 𝔽 is an extended monotone Fock space. Thus, we may think of the monotone noise ω as a distribution of linear operators acting in 𝔽. We say that the orthogonal polynomials belong to the Meixner class if CP=OCP. We prove that each system of orthogonal polynomials from the Meixner class is characterized by two parameters: λ∈ℝ and η≥0. In this case, the monotone Lévy noise has the representation ω(t)=∂t†+λ∂t†∂t+∂t+η∂t†∂t∂t. Here, ∂t† and ∂t are the (formal) creation and annihilation operators at t∈ℝ+ acting in 𝔽.
- Subjects
SET theory; NONCOMMUTATIVE function spaces; MONOTONE operators; POLYNOMIALS; CONTINUOUS functions; MATHEMATICAL proofs
- Publication
Infinite Dimensional Analysis, Quantum Probability & Related Topics, 2018, Vol 21, Issue 2, pN.PAG
- ISSN
0219-0257
- Publication type
Article
- DOI
10.1142/S021902571850011X