Orthogonal polynomials, Laguerre Fock space, and quasi-classical asymptotics

Engliš, Miroslav; Ali, S. Twareque

doi:10.1063/1.4927343

Cited by 4 publications

(3 citation statements)

References 24 publications

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“…Evidently, there are many interesting open challenges left which will directly follow our results. By utilizing the Hankel determinant method, or any other existing mechanism in the literature [26,[28][29][30] one may study the orthogonal polynomials, which are associated with our coherent states. The investigation may end up with some known orthogonal polynomials, or some fascinating q-orthogonal polynomials.…”

Section: Discussionmentioning

confidence: 99%

On completeness of coherent states in noncommutative spaces with the generalised uncertainty principle

Dey

2017

Physical and Mathematical Aspects of Symmetries

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Coherent states are required to form a complete set of vectors in the Hilbert space by providing the resolution of identity. We study the completeness of coherent states for two different models in a noncommutative space associated with the generalised uncertainty relation by finding the resolution of unity with a positive definite weight function. The weight function, which is sometimes known as the Borel measure, is obtained through explicit analytic solutions of the Stieltjes and Hausdorff moment problem with the help of the standard techniques of inverse Mellin transform.

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Section: Discussionmentioning

confidence: 99%

On completeness of coherent states in noncommutative spaces with the generalised uncertainty principle

Dey

2017

Physical and Mathematical Aspects of Symmetries

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“…Reproducing kernels using Laguerre polynomials. In this section we briefly show how the same sort of analysis may be done using the real Laguerre polynomials, again successively obtaining three reproducing kernel Hilbert spaces -the first consisting of functions of a real variable, the second of a complex variable and the third a quaternionic Hilbert space of a quaternionic variable (see, also [15,20,21]). The generalized real Laguerre polynomials are defined for any α > −1 by Once again, while ∞ n=0 |L α n (x)| 2 = ∞, one has, for any ε ∈ (0, 1) (see, for example, [20,21]), (5.16)…”

Section: 2mentioning

confidence: 99%

“…In this section we build reproducing kernels and reproducing kernel Hilbert spaces starting from real orthogonal polynomials, then with their complexified versions and quaternionic extensions. Even though some of these kernels have been considered in the literature, in different contexts [5,15], we work these out here as examples of our general construction of QCS and reproducing kernel spaces. 5.1.…”

Section: Some Examples Of Reproducing Kernel Hilbert Spaces Arising F...mentioning

confidence: 99%

General construction of reproducing kernels on a quaternionic Hilbert space

Thirulogasanthar

Ali

2017

Rev. Math. Phys.

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Abstract. A general theory of reproducing kernels and reproducing kernel Hilbert spaces on a right quaternionic Hilbert space is presented. Positive operator valued measures and their connection to a class of generalized quaternionic coherent states are examined. A Naimark type extension theorem associated with the positive operator valued measures is proved in a right quaternionic Hilbert space. As illustrative examples, real, complex and quaternionic reproducing kernels and reproducing kernel Hilbert spaces arising from Hermite and Laguerre polynomials are presented. In particular, in the Laguerre case, the Naimark type extension theorem on the associated quaternionic Hilbert space is indicated.

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Non-central limit theorems for random fields subordinated to gamma-correlated random fields

Leonenko¹,

Ruiz-Medina²,

Taqqu³

2017

Bernoulli

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A reduction theorem is proved for functionals of Gamma-correlated random fields with long-range dependence in d-dimensional space. In the particular case of a nonlinear function of a chi-squared random field with Laguerre rank equal to one, we apply the Karhunen-Loéve expansion and the Fredholm determinant formula to obtain the characteristic function of its Rosenblatt-type limit distribution. When the Laguerre rank equals one and two, we obtain the multiple Wiener-Itô stochastic integral representation of the limit distribution. In both cases, an infinite series representation in terms of independent random variables is constructed for the limit random variables. The Lancaster-Sarmanov random fieldsWe now introduce here the class of Lancaster-Sarmanov random fields with given onedimensional marginal distributions and general covariance structure. Denote by L 2 (Ω, F, P ) the Hilbert space of zero-mean second-order random variables defined on the complete probability space (Ω, F, P ). For a probability density function p on the interval (l, r), with −∞ ≤ l < r ≤ ∞, we consider the Hilbert space L 2 ((l, r), p(u)du) of equivalence classes of Lebesgue measurable functions h : (l, r) → R satisfying r l h 2 (u) p(u) du < ∞, p(u) ≥ 0.

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Orthogonal polynomials, Laguerre Fock space, and quasi-classical asymptotics

Cited by 4 publications

References 24 publications

On completeness of coherent states in noncommutative spaces with the generalised uncertainty principle

On completeness of coherent states in noncommutative spaces with the generalised uncertainty principle

General construction of reproducing kernels on a quaternionic Hilbert space

Non-central limit theorems for random fields subordinated to gamma-correlated random fields

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