Function Spaces of Polyanalytic Functions

Abreu, Luís Daniel; Feichtinger, Hans G.

doi:10.1007/978-3-319-01806-5_1

Cited by 70 publications

(54 citation statements)

References 59 publications

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“…To illustrate such situation, take F (z) := zz − 1, a polyanalytic function of order 2. Now, if we look at the so-called true polyanalytic Fock spaces [7] , [8] which will be denoted here by A 2 l (C) , l = 0, 1, ..., m. These spaces are related to the polyanalytic Fock space F m+1 (C) by the orthogonal decomposition ( [4], [7], [8]):…”

Section: Polyanalytic Functions On Cmentioning

confidence: 99%

“…More information on these spaces and applications to signal analysis and physics can be found in [4], [6], [10] and references therein.…”

Section: Polyanalytic Functions On Cmentioning

confidence: 99%

“…denotes the Laguerre polynomial [14] and m ∧ n = min (m, n) . To be more precise, the {Φ m n (z)} constitute an orthonormal basis of a true polyanalytic space attached to a fixed mth Landau level [4]. Here, we proceed by modifying the coefficients (1.1) by a factor e −nε .…”

Section: Introductionmentioning

confidence: 99%

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Epsilon coherent states with polyanalytic coefficients for the harmonic oscillator

Mouayn

2017

Anal.Math.Phys.

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We construct a new class of coherent states indexed by points z of the complex plane and depending on two positive parameters m and ε > 0 by replacing the coefficients z n / √ n! of the canonical coherent states by polyanalytic functions. These states solve the identity of the states Hilbert space of the harmonic oscillator at the limit ε → 0 + and obey a thermal stability property. Their wavefunctions are obtained in a closed form and their associated Bargmann-type transform is also discussed.

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Section: Polyanalytic Functions On Cmentioning

confidence: 99%

“…More information on these spaces and applications to signal analysis and physics can be found in [4], [6], [10] and references therein.…”

Section: Polyanalytic Functions On Cmentioning

confidence: 99%

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Epsilon coherent states with polyanalytic coefficients for the harmonic oscillator

Mouayn

2017

Anal.Math.Phys.

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“…. , ∞ (see (4.2)) are polyanalytic functions of order n (see, for example [2]). More precisely, the subspace H n of H(C), consisting of such polyanalytic functions spanned by the complex Hermite polynomials with a fixed degree is the nth polyanalytic sector (corresponding to the nth Landau level, also called a true [39] or pure [28] polyanalytic space) and the direct sum of the first k such spaces is the kth polyanalytic space.…”

Section: Deformed Complex Hermite Polynomialsmentioning

confidence: 99%

D-Pseudo-Bosons, Complex Hermite Polynomials, and Integral Quantization

Ali¹,

Багарелло²,

Gazeau

2015

SIGMA

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Abstract. The D-pseudo-boson formalism is illustrated with two examples. The first one involves deformed complex Hermite polynomials built using finite-dimensional irreducible representations of the group GL(2, C) of invertible 2 × 2 matrices with complex entries. It reveals interesting aspects of these representations. The second example is based on a pseudo-bosonic generalization of operator-valued functions of a complex variable which resolves the identity. We show that such a generalization allows one to obtain a quantum pseudo-bosonic version of the complex plane viewed as the canonical phase space and to understand functions of the pseudo-bosonic operators as the quantized versions of functions of a complex variable.

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“…Appl. Clifford Algebras boundary value problems, their study is closely connected with the theory of singular integral equations [17,30] and has a wide range of applications in other fields, such as in the theory of cracks and elasticity [22,30], in quantum mechanics and of statistical physics [9] as well as in the theory of linear and nonlinear partial differential equations [14], in the theory of orthogonal polynomials and asymptotic analysis [12], and in the theory of time-frequency analysis [1]. In addition, their methods and related problems have been extended to null-solutions to complex partial differential equations (PDEs) on the complex plane like poly-analytic functions, meta-analytic functions, and poly-harmonic functions, see, e.g.…”

mentioning

confidence: 99%

Riemann–Hilbert Problems for Monogenic Functions on Upper Half Ball of $${\mathbb {R}}^4$$ R 4

Ku¹,

Wang²,

He³

et al. 2017

Adv. Appl. Clifford Algebras

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In this paper we are interested in finding solutions to Riemann-Hilbert boundary value problems, for short Riemann-Hilbert problems, with variable coefficients in the case of axially monogenic functions defined over the upper half unit ball centred at the origin in four-dimensional Euclidean space. Our main idea is to transfer Riemann-Hilbert problems for axially monogenic functions defined over the upper half unit ball centred at the origin of four-dimensional Euclidean spaces into Riemann-Hilbert problems for analytic functions defined over the upper half unit disk of the complex plane. Furthermore, we extend our results to axially symmetric null-solutions of perturbed generalized Cauchy-Riemann equations. Mathematics SubjectClassification. Primary: 30E25, 35Q15, 31A25, 31B20; Secondary: 31B10, 35J56, 35J58.2494 M. Ku et al.Adv. Appl. Clifford Algebras boundary value problems, their study is closely connected with the theory of singular integral equations [17,30] and has a wide range of applications in other fields, such as in the theory of cracks and elasticity [22,30], in quantum mechanics and of statistical physics [9] as well as in the theory of linear and nonlinear partial differential equations [14], in the theory of orthogonal polynomials and asymptotic analysis [12], and in the theory of time-frequency analysis [1]. In addition, their methods and related problems have been extended to null-solutions to complex partial differential equations (PDEs) on the complex plane like poly-analytic functions, meta-analytic functions, and poly-harmonic functions, see, e.g. [5]. Moreover, also recently, the boundary value theory of null-solutions to complex PDEs has been explored on special domains of the complex plane, like the unit disk, upper half plane, half disks, circular rings, triangles, and sectors, and so on, see, e.g. [5,6,36].In parallel, there have been many attempts of generalizing the classical boundary value theory of Riemann-Hilbert problems into higher dimensions, mainly by considering two principal ways, i.e., the theory of several complex variables and Clifford analysis, in particular quaternionic analysis. The latter is an elegant generalization of the classical theory of complex analysis. It concentrates on the study of the theory of so-called monogenic functions, and refines real harmonic analysis in the sense that its principal operator, the Dirac or generalized Cauchy-Riemann operator, factorizes the higher-dimensional Laplace operator, see, e.g. [7,13,18]. In the setting of quaternionic and Clifford analysis, Riemann-Hilbert problems were discussed by many authors, see, e.g. ], the authors studied Riemann-Hilbert problems with constant coefficients. Their solutions were given explicitly in terms of Cauchy-type integral operators together with power series expansions. Moreover, these problems are closely connected with applications, like the theory of fluid mechanics [18] and signal processing in higher dimensions [16,32]. To the authors' knowledge in Refs. [20,21] the authors made the first ...

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Function Spaces of Polyanalytic Functions

Cited by 70 publications

References 59 publications

Epsilon coherent states with polyanalytic coefficients for the harmonic oscillator

Epsilon coherent states with polyanalytic coefficients for the harmonic oscillator

D-Pseudo-Bosons, Complex Hermite Polynomials, and Integral Quantization

Riemann–Hilbert Problems for Monogenic Functions on Upper Half Ball of $${\mathbb {R}}^4$$ R 4

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