Roles of Log-Concavity, Log-Convexity, and Growth Order in White Noise Analysis

Asai, Nobuhiro; Kubo, Izumi; Kuo, Hui-Hsiung

doi:10.1142/s0219025701000498

Cited by 17 publications

(19 citation statements)

References 13 publications

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“…Relationships with the Work by Gannoun et al [9] In the rest of this paper, we shall discuss some of similarities and differences between our papers [6] [7] and Gannoun et al [9] (GHOR for simplicity). We refer the readers to consult the papers [4][6] [7] for more technical and delicate differences, which will not be mentioned in this paper.…”

Section: Characterization Theorems and Quantum White Noisesmentioning

confidence: 90%

“…The main purpose of this work is to realize quantum Gaussian and Poisson white noises in terms of multiple Wiener-Itô integrals, and show that such realizations cannot be achieved by J-transform and its holomorphy, but can be done by S X -transform depending on the exponential function φ X ξ , which determines a unitary isomorphism between Boson Fock space and L 2 (E * , µ X ), X = G, P . In Appendix A, some connections between [6][7] and [9] will be discussed. …”

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confidence: 99%

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Gaussian and Poisson white noises with related characterization theorems

Asai¹,

Kubo²,

Kuo³

2003

Finite and Infinite Dimensional Analysis in Honor of Leonard Gross

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Dedicated to Professor Leonard Gross on the occasion of his 70th birthday.Abstract. Let µ G and µ P be a Gaussian measure and a Poisson measure on E * , respectively. Let at and a * t be respectively annihilation and creation operators at a point t ∈ R. In the theory of quantum white noise, it is known that at is a continuous linear operator from Γu(E C ) into itself and a * t is a continuous linear operator from Γu(E C ) * into itself. In paticular, at + a * t and at+a * t +a * t at+I are called the quantum Gaussian white noise and the quantum Poisson white noise, respectively. The main purpose of this work is to realize quantum Gaussian and Poisson white noises in terms of multiple Wiener-Itô integrals, and show that such realizations cannot be achieved by J-transform and its holomorphy, but can be done by S X -transform depending on the exponential function φ X ξ , which determines a unitary isomorphism between Boson Fock space and L 2 (E * , µ X ), X = G, P . In Appendix A, some connections between [6][7] and [9] will be discussed.

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Section: Characterization Theorems and Quantum White Noisesmentioning

confidence: 90%

mentioning

confidence: 99%

Gaussian and Poisson white noises with related characterization theorems

Asai¹,

Kubo²,

Kuo³

2003

Finite and Infinite Dimensional Analysis in Honor of Leonard Gross

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“…Let

{false{α_{n} false}}_{n \geq 0}

be a sequence of positive numbers. In previous studies and closely related references therein, for studying the spaces of test and generalized functions and their characterization theorems in white noise distribution theory, the following conditions for the sequence

{false{α_{n} false}}_{n \geq 0}

are required:

α_{0} = 1, \inf_{n \geq 0} false(α_{n} σ^{n} false) > 0, \lim_{n \to \infty} {false(\frac{α_{n}}{n!} false)}^{1 false/ n} = 0, \lim_{n \to \infty} {false(\frac{1}{n! α_{n}} false)}^{1 false/ n} = 0,

\underset{n \to \infty}{lim sup} {false[\frac{n!}{α_{n}} \inf_{x > 0} \frac{G_{α} false(x false)}{x^{n}} false]}^{1 false/ n} < \infty, \underset{n \to \infty}{lim sup} {false[n! α_{n} \inf_{x > 0} \frac{G_{1 false/ α} false(x false)}{x^{n}} false]}^{1 false/ n} < \infty,

the sequence {({}, \frac{α_{n}}{n!})}_{n \geq 0} is logarithmically concave,

the sequence {({}, \frac{1}{n! α_{n}})}_{n \geq 0}

…”

Section: An Application To White Noise Distribution Theorymentioning

confidence: 99%

“…In Kubo et al, it was pointed out that the condition implies . In Asai et al,, p83 it was concluded that the essential conditions for distribution theory on a CKS‐space are the first three in and the conditions , and .…”

Section: An Application To White Noise Distribution Theorymentioning

confidence: 99%

Some properties and an application of multivariate exponential polynomials

Niu

Lim

et al. 2020

Math Methods in App Sciences

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In the paper, the authors introduce a notion “multivariate exponential polynomials” which generalize exponential numbers and polynomials, establish explicit formulas, inversion formulas, and recurrence relations for multivariate exponential polynomials in terms of the Stirling numbers of the first and second kinds with the help of the Faà di Bruno formula, two identities for the Bell polynomials of the second kind, and the inversion theorem for the Stirling numbers of the first and second kinds, construct some determinantal inequalities and product inequalities for multivariate exponential polynomials with the aid of some properties of completely monotonic functions and other known results, derive the logarithmic convexity and logarithmic concavity for multivariate exponential polynomials, and finally find an application of multivariate exponential polynomials to white noise distribution theory by confirming that multivariate exponential polynomials satisfy conditions for sequences required in white noise distribution theory.

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“…A nice example is the recent use of log-convex sequences in quantum physics for constructing generalized coherent states in models with discrete nonlinear spectra [25,36]. Then, the log-convexity of certain sequences (of generalized Bell numbers) enabled their use as important examples in white noise theory [5,6,27]. The log-convex sequences also play an important role in probability-the log-convexity of a sequence P {X = n} is a sufficient condition for a discrete random variable X to be infinitelydivisible [30,53].…”

Section: Introductionmentioning

confidence: 99%