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Nobuhiro Asai

Publications and source records attributed to Nobuhiro Asai.

13 recordsLinked to original sources
arXiv2025

Combinatorial Aspects of Weighted Free Poisson Random Variables

This paper will be devoted to study weighted (deformed) free Poisson random variables from the viewpoint of orthogonal polynomials and statistics of non-crossing partitions. A family of weighted (deformed) free Poisson random variables will be defined in a sense by the sum of weighted (deformed) free creation, annihilation, scalar, and intermediate operators with certain parameters on a weighted (deformed) free Fock space together with the vacuum expectation. We shall provide a combinatorial moment formula of non-commutative Poisson random variables. This formula gives us a very nice combinatorial interpretation to two parameters of weights. One can see that the deformation treated in this paper interpolates free and boolean Poisson random variables, their distributions and moments, and yields some conditionally free Poisson distribution by taking limit of the parameter.

math.PR↗
arXiv2025

A Poisson Type Operator Deformed by Generalized Fibonacci Numbers and Its Combinatorial Moment Formula

We introduce a two-parameter deformation of the classical Poisson distribution from the viewpoint of noncommutative probability theory, by defining a $(q,t)$-Poisson type operator (random variable) on the $(q,t)$-Fock space \cite{Bl12} (See also \cite{BY06, AY20}). From the analogous viewpoint of the classical Poisson limit theorem in probability theory, we are naturally led to a family of orthogonal polynomials, which we call the $(q,t)$-Charlier polynomials. These generalize the $q$-Charlier polynomials of Saitoh-Yoshida \cite{SY00a, SY00b} and reflect deeper combinatorial symmetries through the additional deformation parameter $t$. A central feature of this paper is the derivation of a combinatorial moment formula of the $(q,t)$-Poisson type operator and the $(q,t)$-Poisson distribution. This is accomplished by means of a card arrangement technique, which encodes set partitions together with crossing and nesting statistics. The resulting expression naturally exhibits a duality between these statistics, arising from a structure rooted in generalized Fibonacci numbers. Our approach provides a concrete framework where methods in combinatorics and theory of orthogonal polynomials are used to investigate the probabilistic properties arising from the $(q,t)$-deformation.

math.CO↗
arXiv2016

Radial Bargmann representation for the Fock space of type B

Let $ν_{α,q}$ be the probability and orthogonality measure for the $q$-Meixner-Pollaczek orthogonal polynomials, which has appeared in \cite{BEH15} as the distribution of the $(α,q)$-Gaussian process (the Gaussian process of type B) over the $(α,q)$-Fock space (the Fock space of type B). The main purpose of this paper is to find the radial Bargmann representation of $ν_{α,q}$. Our main results cover not only the representation of $q$-Gaussian distribution by \cite{LM95}, but also of $q^2$-Gaussian and symmetric free Meixner distributions on $\mathbb R$. In addition, non-trivial commutation relations satisfied by $(α,q)$-operators are presented.

math.FA↗
arXiv2001

Integral Transform and Segal-Bargmann Representation Associated to q-Charlier Polynomials

Let $μ_p^{(q)}$ be the q-deformed Poisson measure in the sense of Saitoh Yoshida and $ν_p$ be the measure given by Equation \eqref{eq:nu-q}. In this short paper, we introduce the q-deformed analogue of the Segal-Bargmann transform associated with $μ_p^{(q)}$. We prove that our Segal-Bargmann transform is a unitary map of $L^2(μ_p^{(q)})$ onto the q-deformed Hardy space ${\cal H}^2(ν_q)$. Moreover, we give the Segal-Bargmann representation of the multiplication operator by $x$ in $L^2(μ_p^{(q)})$, which is a linear combination of the q-creation, q-annihilation, q-number, and scalar operators.

math.CA↗
arXiv2001

A note on general setting of white noise triple and positive generalized functions

Let $\ce^{*}$ be the space of tempered distributions and $\m$ be the standard Gaussian measure on $\ce^{*}$. Being motivated by the distribution theory on infinite dimensional space by Cochran, Kuo and Sengupta (CKS) \cite{cks}, Asai, Kubo and Kuo (AKK) have recently determined the best possible class $C_{+,{1\over 2},1}^{(2)}$ of functions $u$ to constract white noise triple, [\ce]_u\subset L^2(\ce^{*},\m) \subset [\ce]^{*}_u, and to characterize white noise test function space $[\ce]_u$ and generalized function space $[\ce]_u^{*}$ in the series of papers \cite{akk1}, \citeakk2}, \cite{akk3}, \cite{akk4}, \cite{akk5}. The notion of Legendre transformation plays important roles to examine relationships between the growth order of holomorphic functions (S-transform) and the CKS-space of white noise test and generalized functions. It is well-known that a positive generalized function is induced by a Hida measure $ν$ (generalized measure). A Hida measure can be characterized by integrability conditions on a function inducing the above triple (\cite{akk5}). See also \cite{kuo99-1}, \cite{kuo99-2}, \cite{ob99} for an overview of other recent developments in white noise analysis.

math.FA↗
arXiv2001

Segal-Bargmann Transforms of One-mode Interacting Fock Spaces Associated with Gaussian and Poisson Measures

Let $μ_{g}$ and $μ_{p}$ denote the Gaussian and Poisson measures on ${\Bbb R}$, respectively. We show that there exists a unique measure $\widetildeμ_{g}$ on ${\Bbb C}$ such that under the Segal-Bargmann transform $S_{μ_g}$ the space $L^2({\Bbb R},μ_g)$ is isomorphic to the space ${\cal H}L^2({\Bbb C}, \widetildeμ_{g})$ of analytic $L^2$-functions on ${\Bbb C}$ with respect to $\widetildeμ_{g}$. We also introduce the Segal-Bargmann transform $S_{μ_p}$ for the Poisson measure $μ_{p}$ and prove the corresponding result. As a consequence, when $μ_{g}$ and $μ_{p}$ have the same variance, $L^2({\Bbb R},μ_g)$ and $L^2({\Bbb R},μ_p)$ are isomorphic to the same space ${\cal H}L^2({\Bbb C}, \widetildeμ_{g})$ under the $S_{μ_g}$ and $S_{μ_p}$-transforms, respectively. However, we show that the multiplication operators by $x$ on $L^2({\Bbb R}, μ_g)$ and on $L^2({\Bbb R}, μ_p)$ act quite differently on ${\cal H}L^2({\Bbb C}, \widetildeμ_{g})$.

math.PR↗
arXiv2001

CKS-space in terms of growth functions

A class of growth functions $u$ is introduced to construct Hida distributions and test functions. The Legendre transform $\ell_{u}$ of $u$ is used to define a sequence $\a(n)=(\ell_{u}(n) n!)^{-1}, n\geq 0$, of positive numbers. From this sequence we get a CKS-space. Under various conditions on $u$ we show that the associated sequence $\{\a(n)\}$ satisfies those conditions for carrying out the white noise distribution theory on the CKS-space. We show that $u$ and its dual Legendre transform $u^{*}$ are growth functions for test and generalized functions, respectively, in the characterization theorems.

math.FA↗
arXiv2001

Roles of Log-concavity, log-convexity, and growth order in white noise analysis

In this paper we will develop a systematic method to answer the questions $(Q1)(Q2)(Q3)(Q4)$ (stated in Section 1) with complete generality. As a result, we can solve the difficulties $(D1)(D2)$ (discussed in Section 1) without uncertainty. For these purposes we will introduce certain classes of growth functions $u$ and apply the Legendre transform to obtain a sequence which leads to the weight sequence $\{\a(n)\}$ first studied by Cochran et al. \cite{cks}. The notion of (nearly) equivalent functions, (nearly) equivalent sequences and dual Legendre functions will be defined in a very natural way. An application to the growth order of holomorphic functions on $\ce_c$ will also be discussed.

math.FA↗
arXiv2001

General characterization theorems and intrinsic topologies in white noise analysis

Let $u$ be a positive continuous function on $[0, \infty)$ satisfying the conditions: (i) $\lim_{r\to\infty} r^{-1/2}\log u(r)=\infty$, (ii) $\inf_{r\geq 0} u(r)=1$, (iii) $\lim_{r\to \infty}\break r^{-1}\log u(r)<\infty$, (iv) the function $\log u(x^{2}), x\geq 0$, is convex. A Gel'fand triple $[\ce]_{u} \subset (L^{2}) \subset [\ce]_{u}^{*}$ is constructed by making use of the Legendre transform of $u$ discussed in \cite {akk3}. We prove a characterization theorem for generalized functions in $[\ce]_{u}^{*}$ and also for test functions in $[\ce]_{u}$ in terms of their $S$-transforms under the same assumptions on $u$. Moreover, we give an intrinsic topology for the space$[\ce]_{u}$ of test functions and prove a characterization theorem for measures. We briefly mention the relationship between our method and a recent work by Gannoun et al.\cite{ghor}. Finally, conditions for carrying out white noise operator theory and Wick products are given.

math.FA↗
arXiv2001

Bell numbers, log-concavity, and log-convexity

Let $\{b_{k}(n)\}_{n=0}^{\infty}$ be the Bell numbers of order $k$. It is proved that the sequence $\{b_{k}(n)/n!\}_{n=0}^{\infty}$ is log-concave and the sequence $\{b_{k}(n)\}_{n=0}^{\infty}$ is log-convex, or equivalently, the following inequalities hold for all $n\geq 0$, $$1\leq {b_{k}(n+2) b_{k}(n) \over b_{k}(n+1)^{2}} \leq {n+2 \over n+1}.$$ Let $\{\a(n)\}_{n=0}^{\infty}$ be a sequence of positive numbers with $\a(0)=1$. We show that if $\{\a(n)\}_{n=0}^{\infty}$ is log-convex, then $$\a (n) \a (m) \leq \a(n+m), \quad \forall n, m\geq 0.$$ On the other hand, if $\{\a(n)/n!\}_{n=0}^{\infty}$ is log-concave, then $$\a (n+m) \leq {n+m \choose n} \a (n) \a (m), \quad \forall n, m\geq 0.$$ In particular, we have the following inequalities for the Bell numbers $$b_{k}(n) b_{k}(m) \leq b_{k}(n+m) \leq {n+m \choose n} b_{k}(n) b_{k}(m), \quad \forall n, m\geq 0.$$ Then we apply these results to white noise distribution theory.

math.CO↗
arXiv2001

Characterization of Hida Measures in white noise analysis

The main purpose of this work is to prove Theorem 4.4, so-called, the characterization theorem of Hida measures (generalized measures). As examples of such measures, we shall present the Poisson noise measure and the Grey noise measure in Example 4.5 and 4.6, respectively.

math.FA↗
arXiv2001

Characterization of Product Measures by Integrability Condition

It is natural to ask whether "positivity" of white noise operators can be discussed in some sense and characterized. To answer this question, we consider the Gel'fand triple over the Complex Gaussian space $(\ce'_c,\m_c)$, i.e. $\ce'_c=\ce'+i\ce'$ equipped with the product measure $\m_c=\m'\times\m'$ where $\m'$ is the Gaussian measure on $\ce'$ with variance 1/2 (Section \ref{sec:2-2}). Following AKK's Legendre transform technique, we have $\cw_{u_1,u_2}\subset L^2(\ce'_c,\m_c)\subset [\cw]^{*}_{u_1,u_2}$ for functions $u_1,u_2\in C_{+,1/2}$ satisfying (U0)(U2)(U3). Several examples for $u_1, u_2$ are given in Section \ref{sec:2-3}. We remark that Ouerdiane \cite{oue} studied a special case $u_1(r^2)=u_2(r^2)=\exp(k^{-1}r^k)$, where $1\leq k\leq 2$. In Section \ref{sec:3}, the characterization theorem for measures can be extended to the case of positive product Radon measures on $\ce'\times \ce'$. In addition, the notion of pseudo-positive operators is naturally introduced via kernel theorem and characterized by an integrability condition. Lemma \ref{lem:3-2} plays crucial roles in Section \ref{sec:3}.

math.FA↗