Regular subspaces of a quaternionic Hilbert space from quaternionic Hermite polynomials and associated coherent states

Thirulogasanthar, K.; Ali, S. Twareque

doi:10.1063/1.4774963

Cited by 18 publications

(50 citation statements)

References 28 publications

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“…The few known properties of

H_{m, n}^{Q} false(q, \overset{q}{true} false)

are derived from their analogues of

H_{m, n} false(z, \overset{z}{true} false)

by means of

H_{m, n}^{Q} false(q, \overset{q}{true} false) = U_{q} ((), \begin{matrix} center \\ array H_{m, n} (z, \bar{z}) & array 0 \\ array 0 & array \overset{true‾}{H_{m, n} (z, \bar{z})} \end{matrix}) U_{q}^{*},

where

q \in double-struckH

is identified here to its matrix representation

q = U_{q} ((), \begin{matrix} center \\ array z & array 0 \\ array 0 & array \bar{z} \end{matrix}) U_{q}^{*}

for some U q ∈ S U (2) (see Thirulogasanthar and Twareque). In the sequel, we discuss some basic properties of these polynomials using their different representations.…”

Section: Some Auxiliary Results For Hmnqfalse(qtrueq¯false)mentioning

confidence: 99%

“…Such polynomials have found several interesting applications in various branches of mathematics, physics, and technology. They are used as basic tools in studying the complex Markov process, the nonlinear analysis of travelling wave tube amplifiers, the singular values of the Cauchy transform, the coherent states,() combinatory,() the poly‐analytic functions, and signal processing. ()…”

Section: Introductionmentioning

confidence: 99%

“…A natural extension of

H_{m, n} false(z, \overset{z}{true} false)

to the quaternionic setting is defined by Thirulogasanthar and Twareque:

H_{m, n}^{Q} false(q, \overset{q}{true} false) = true \sum_{k = 0}^{m \land n} {false(- 1 false)}^{k} k! ((), \binom{m}{k}) ((), \binom{n}{k}) q^{m - k} {\overset{q}{true}}^{n - k}, q \in double-struckH .

…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On some analytic properties of slice poly‐regular Hermite polynomials

Hamyani

Ghanmi

2018

Math Methods in App Sciences

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We consider a quaternionic analogue of the univariate complex Hermite polynomials and study some of their analytic properties in some detail. We obtain their integral representation as well as the operational formulas of exponential and Burchnall types they obey. We show that they form an orthogonal basis of the slice Hilbert space L2false(LI;e−false|q|2dλIfalse) of all quaternionic‐valued functions defined the whole quaternions space and subject to norm boundedness with respect to the Gaussian measure on a given slice as well as of the full left quaternionic Hilbert space L2false(double-struckH;e−false|q|2dλfalse) of square integrable functions on quaternions with respect to the Gaussian measure on the whole double-struckH≡R4. We also provide different types of generating functions. Remarkable identities, including quadratic recurrence formulas of Nielsen type, are also derived.

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“…The few known properties of

H_{m, n}^{Q} false(q, \overset{q}{true} false)

are derived from their analogues of

H_{m, n} false(z, \overset{z}{true} false)

by means of

H_{m, n}^{Q} false(q, \overset{q}{true} false) = U_{q} ((), \begin{matrix} center \\ array H_{m, n} (z, \bar{z}) & array 0 \\ array 0 & array \overset{true‾}{H_{m, n} (z, \bar{z})} \end{matrix}) U_{q}^{*},

where

q \in double-struckH

is identified here to its matrix representation

q = U_{q} ((), \begin{matrix} center \\ array z & array 0 \\ array 0 & array \bar{z} \end{matrix}) U_{q}^{*}

for some U q ∈ S U (2) (see Thirulogasanthar and Twareque). In the sequel, we discuss some basic properties of these polynomials using their different representations.…”

Section: Some Auxiliary Results For Hmnqfalse(qtrueq¯false)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On some analytic properties of slice poly‐regular Hermite polynomials

Hamyani

Ghanmi

2018

Math Methods in App Sciences

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“…Precursors of this work, involving quaternionic coherent states and wavelets over quaternionic fields, may be be found in [7][8][9]11].…”

Section: Coherent States and Waveletsmentioning

confidence: 99%

On Some Quaternionic Coherent States and Wavelets

Ali

2015

Trends in Mathematics

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As a natural extension of the affine groups of the reals and the complexes, leading to one and two-dimensional wavelets, we look at the quaternionic affine group and its representations on both complex and quaternionic Hilbert spaces. We then study the problem of coherent states and wavelets built from these representations.Mathematics Subject Classification (2010). Primary 81R30; Secondary 57S20.

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“…For instant dµ can be taken as a Radon measure or dµ = dλdω, where dλ is a Lebesgue measure on C and dω is a Harr measure on SU (2). For details we refer the reader to, for example, [18] (page 12).…”

Section: Mathematical Preliminariesmentioning

confidence: 99%

S-spectrum and associated continuous frames on quaternionic Hilbert spaces

Khokulan

Thirulogasanthar

Muraleetharan

2015

Journal of Geometry and Physics

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Abstract. As needed for the construction of rank n continuous frames on a right quaternionic Hilbert space the so-called S-spectrum of a right quaternionic operator is studied. Using the S-spectrum, as for the case of complex Hilbert spaces, along the lines of the arguments of Ann.Phys., 222 (1993), 1-37., various classes of rank n continuous frames and their equivalencies on a right quaternionic Hilbert space are presented.

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Regular subspaces of a quaternionic Hilbert space from quaternionic Hermite polynomials and associated coherent states

Cited by 18 publications

References 28 publications

On some analytic properties of slice poly‐regular Hermite polynomials

On some analytic properties of slice poly‐regular Hermite polynomials

On Some Quaternionic Coherent States and Wavelets

S-spectrum and associated continuous frames on quaternionic Hilbert spaces

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