WHITE NOISE LÉVY–MEIXNER PROCESSES THROUGH A TRANSFER PRINCIPAL FROM ONE-MODE TO ONE-MODE TYPE INTERACTING FOCK SPACES

2010 ◽  
Vol 13 (03) ◽  
pp. 435-460 ◽  
Author(s):  
LUIGI ACCARDI ◽  
ABDESSATAR BARHOUMI ◽  
ANIS RIAHI
Keyword(s):  
White Noise ◽  
Fock Space ◽  
Delta Function ◽  
Fock Spaces ◽  
Jacobi Fields ◽  
Derivative Formula ◽  
Scalar Products ◽  
The Fourier Transform ◽  
The One ◽  
Interacting Fock Space

Consider the Lévy–Meixner one-mode interacting Fock space {ΓLM, 〈 ⋅, ⋅ 〉LM}. Inspired by a derivative formula appearing in 〈 ⋅, ⋅ 〉LM, we define scalar products 〈 ⋅, ⋅ 〉LM , nin symmetric n-particle spaces. Then, we introduce a class of one-mode type interacting Fock spaces [Formula: see text] naturally associated to the one-dimensional infinitely divisible distributions with Lévy–Meixner type {μr; r > 0}. The Fourier transform in generalized joint eigenvectors of a family [Formula: see text] of Lévy–Meixner Jacobi fields provides a way to explicit a unitary isomorphism 𝔘LMbetween [Formula: see text] and the so-called Lévy–Meixner white noise space [Formula: see text]. We derive a chaotic decomposition property of the quadratic integrable functionals of the Lévy–Meixner white noise processes in terms of an appropriate Wick tensor product. For their stochastic regularity, we give explicit form and sharp estimate of the associated Donsker's delta function.

ON THE QUADRATIC HEISENBERG GROUP

2010 ◽  
Vol 13 (04) ◽  
pp. 551-587 ◽  
Author(s):  
LUIGI ACCARDI ◽  
HABIB OUERDIANE ◽  
HABIB REBEÏ
Keyword(s):  
White Noise ◽  
Heisenberg Group ◽  
Lie Group ◽  
Unitary Operator ◽  
Fock Space ◽  
Local Chart ◽  
Rational Expression ◽  
The One ◽  
Interacting Fock Space ◽  
Group Law

In this paper we introduce the quadratic Weyl operators canonically associated to the one mode renormalized square of white noise (RSWN) algebra as unitary operator acting on the one mode interacting Fock space {Γ, {ωn, n ∈ ℕ}, Φ} where {ωn, n ∈ ℕ} is the principal Jacobi sequence of the nonstandard (i.e. neither Gaussian nor Poisson) Meixner classes. We deduce the quadratic Weyl relations and construct the quadratic analogue of the Heisenberg Lie group with one degree of freedom. In particular, we determine the manifold structure of the group and introduce a local chart containing the identity on which the group law has a simple rational expression in the chart coordinates (see Theorem 6.3).

scholarly journals Interacting Fock spaces and subproduct systems

2020 ◽  
Vol 23 (03) ◽  
pp. 2050017
Author(s):  
Malte Gerhold ◽  
Michael Skeide
Keyword(s):  
Selfadjoint Operator ◽  
Operator Algebras ◽  
Fock Space ◽  
Fock Spaces ◽  
Full Generality ◽  
Necessary And Sufficient Criteria ◽  
Definition Of ◽  
Interacting Fock Space ◽  
Necessary And Sufficient

We present a new more flexible definition of interacting Fock space that allows to resolve in full generality the problem of embeddability. We show that the same is not possible for regularity. We apply embeddability to classify interacting Fock spaces by squeezings. We give necessary and sufficient criteria for when an interacting Fock space has only bounded creators, giving thus rise to new classes of non-selfadjoint and selfadjoint operator algebras.

scholarly journals Fourier transforms in generalized Fock spaces

1990 ◽  
Vol 13 (3) ◽  
pp. 431-441
Author(s):  
John Schmeelk
Keyword(s):  
Fourier Transform ◽  
Taylor Series ◽  
Fourier Transforms ◽  
Fock Space ◽  
Natural Generalization ◽  
Fock Spaces ◽  
Test Functions ◽  
Tempered Distributions ◽  
Tempered Distribution ◽  
The Fourier Transform

A classical Fock space consists of functions of the form,Φ↔(ϕ0,ϕ1,…,ϕq,…),whereϕ0∈Candϕq∈L2(R3q),q≥1. We will replace theϕq,q≥1withq-symmetric rapid descent test functions within tempered distribution theory. This space is a natural generalization of a classical Fock space as seen by expanding functionals having generalized Taylor series. The particular coefficients of such series are multilinear functionals having tempered distributions as their domain. The Fourier transform will be introduced into this setting. A theorem will be proven relating the convergence of the transform to the parameter,s, which sweeps out a scale of generalized Fock spaces.

SCALING LIMIT FOR GIBBS STATES OF JOHNSON GRAPHS AND RESULTING MEIXNER CLASSES

2003 ◽  
Vol 06 (01) ◽  
pp. 139-143 ◽  
Author(s):  
AKIHITO HORA
Keyword(s):  
Fock Space ◽  
Gibbs State ◽  
Scaling Limit ◽  
Temperature Limit ◽  
Zero Temperature ◽  
Johnson Graph ◽  
Meixner Polynomials ◽  
Johnson Graphs ◽  
The One ◽  
Interacting Fock Space

Asymptotic behavior of spectral distribution of the adjacency operator on the Johnson graph with respect to the Gibbs state is discussed in infinite volume and zero temperature limit. The limit picture is drawn on the one-mode interacting Fock space associated with Meixner polynomials.

ON THE FILTERING PROPERTIES OF THE EMPIRICAL MODE DECOMPOSITION

2010 ◽  
Vol 02 (04) ◽  
pp. 397-414 ◽  
Author(s):  
ZHAOHUA WU ◽  
NORDEN E. HUANG
Keyword(s):  
White Noise ◽  
Empirical Mode Decomposition ◽  
Wavelet Decomposition ◽  
Mathematical Expression ◽  
Delta Function ◽  
Time Frequency ◽  
Mode Decomposition ◽  
Energy Domain ◽  
The Fourier Transform ◽  
Special Case

The empirical mode decomposition (EMD) based time-frequency analysis has been used in many scientific and engineering fields. The mathematical expression of EMD in the time-frequency-energy domain appears to be a generalization of the Fourier transform (FT), which leads to the speculation that the latter may be a special case of the former. On the other hand, the EMD is also known to behave like a dyadic filter bank when used to decompose white noise. These two observations seem to contradict each other. In this paper, we study the filtering properties of EMD, as its sifting number changes. Based on numerical results of the decompositions using EMD of a delta function and white noise, we conjecture that, as the (pre-assigned and fixed) sifting number is changed from a small number to infinity, the EMD corresponds to filter banks with a filtering ratio that changes accordingly from 2 (dyadic) to 1; the filter window does not narrow accordingly, as the sifting number increases. It is also demonstrated that the components of a delta function resulted from EMD with any prescribed sifting number can be rescaled to a single shape, a result similar to that from wavelet decomposition, although the shape changes, as the sifting number changes. These results will lead to further understandings of the relations of EMD to wavelet decomposition and FT.

scholarly journals CONSTRUCTIVE UNIVERSAL CENTRAL LIMIT THEOREMS BASED ON INTERACTING FOCK SPACES

2005 ◽  
Vol 08 (04) ◽  
pp. 631-650 ◽  
Author(s):  
L. ACCARDI ◽  
V. CRISMALE ◽  
Y. G. LU
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Probability Measure ◽  
Central Limit ◽  
Limit Theorems ◽  
Fock Space ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Fock Spaces ◽  
Interacting Fock Space

Cabana-Duvillard and lonescu11 have proved that any symmetric probability measure with moments of any order can be obtained as central limit theorem of self-adjoint, weakly independent and symmetrically distributed (in a quantum souse) random variables. Results of this type will be called "universal central limit theorem". Using Interacting Fock Space (IFS) techniques we extend this result in two directions: (i) we prove that the random variables can be taken to be generalized Gaussian in the sense of Accardi and Bożejko3 and we give a realization of such random variables as sums of creation, annihilation and preservation operators acting on an appropriate IFS; (ii) we extend the above-mentioned result to the nonsymmetric case. The nontrivial difference between the symmetric and the nonsymmetric case is explained at the end of the introduction below.

scholarly journals The q-Gamma White Noise

2016 ◽  
Vol 66 (1) ◽  
pp. 81-90
Author(s):  
Hakeem A. Othman
Keyword(s):  
White Noise ◽  
Fock Space ◽  
Infinite Dimensional ◽  
Chaos Decomposition ◽  
Noise Measure ◽  
Interacting Fock Space

Abstract For 0 < q < 1 and 0 < α < 1, we construct the infinite dimensional q-Gamma white noise measure γα,q by using the Bochner-Minlos theorem. Then we give the chaos decomposition of an L2 space with respect to the measure γα,q via an isomorphism with the 1-mode type interacting Fock space associated to the q-Gamma measure.

scholarly journals OPERATORS OF GAMMA WHITE NOISE CALCULUS

2000 ◽  
Vol 03 (03) ◽  
pp. 303-335 ◽  
Author(s):  
YURI G. KONDRATIEV ◽  
EUGENE W. LYTVYNOV
Keyword(s):  
White Noise ◽  
Fock Space ◽  
Noise Analysis ◽  
Gamma Field ◽  
Chaos Decomposition ◽  
The Creation ◽  
The Fourier Transform ◽  
White Noise Calculus ◽  
Adjoint Operators ◽  
Natural Way

The paper is devoted to the study of Gamma white noise analysis. We define an extended Fock space ℱ ext (ℋ) over ℋ= L2(ℝd, dσ) and show how to include the usual Fock space ℱ(ℋ) in it as a subspace. We introduce in ℱ ext (ℋ) operators a(ξ)=∫ℝddxξ(x)a(x), ξ∈ S, with [Formula: see text], where [Formula: see text] and ∂x are the creation and annihilation operators at x. We show that (a(ξ))ξ∈S is a family of commuting self-adjoint operators in ℱ ext (ℋ) and construct the Fourier transform in generalized joint eigenvectors of this family. This transform is a unitary I between ℱ ext (ℋ) and the L2-space L2(S', dμ G ), where μ G is the measure of Gamma white noise with intensity σ. The image of a(ξ) under I is the operator of multiplication by <·,ξ>, so that a(ξ)'s are Gamma field operators. The Fock structure of the Gamma space determined by I coincides with that discovered in Ref. 22. We note that I extends in a natural way the multiple stochastic integral (chaos) decomposition of the "chaotic" subspace of the Gamma space. Next, we introduce and study spaces of test and generalized functions of Gamma white noise and derive explicit formulas for the action of the creation, neutral, and Gamma annihilation operators on these spaces.

Quantum Decomposition Associated with the q-Deformed Lévy–Meixner White Noise

2020 ◽  
Vol 27 (02) ◽  
pp. 2050011
Author(s):  
Anis Riahi ◽  
Amine Ettaieb
Keyword(s):  
White Noise ◽  
Fock Space ◽  
Vacuum Expectation ◽  
White Noise Process ◽  
Noise Process ◽  
The One

In this paper we start with a new detailed construction of the one-mode type q-Lévy-Meixner Fock space [Formula: see text] which serves to obtain the quantum decomposition associated with the q-deformed Lévy-Meixner white noise processes. More precisely, based on the notion of quantum decomposition and the orthogonalization of polynomials of noncommutative q-Lévy-Meixner white noise [Formula: see text], we study the chaos property of the noncommutative L2-space with respect to the vacuum expectation τ. Next, we determine the distribution of the q-Lévy-Meixner operator J(χD) = ⟨ω, χD⟩ and as a consequence we give some useful properties of the q-Lévy-Meixner white noise process.

GAUSSIAN TYPE INTERACTING FOCK SPACES

2008 ◽  
Vol 11 (04) ◽  
pp. 475-494 ◽  
Author(s):  
YUN GANG LU
Keyword(s):  
Linear Combination ◽  
Fock Space ◽  
Field Operator ◽  
Vacuum Expectation ◽  
Fock Spaces ◽  
Gaussian Type ◽  
Interacting Fock Space

We prove in this paper that the vacuum expectation of any product of field operator of an interacting Fock space with the interactions λn's is driven by pair partitions if and only if each interaction λn is a linear combination of permutation operators.

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