Poly-Fock Spaces

Vasilevski, Nikolai

doi:10.1007/978-3-0348-8403-7_28

Cited by 64 publications

(65 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Also, the generalization of the theory of Segal-Bargmann spaces during the last decade by Askour et al (cf [3]) and Vasilevski (cf [35]) led recently to ubiquitous developments in the theory of Gabor-Window Fourier analysis [1,2,5]. From the border view of the Heisenberg group, it was recently shown in [13] that the Schrödinger representation and alike play an important role in the study of pseudo-differential operators establishing structural properties between the Weyl calculus and the Landau-Weyl calculus.…”

Section: Motivation and Main Resultsmentioning

confidence: 99%

Fock spaces, Landau operators and the time-harmonic Maxwell equations

Constales¹,

Faustino²,

Kraußhar³

2011

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

We investigate the representations of the solutions to Maxwell's equations based on the combination of hypercomplex function-theoretical methods with quantum mechanical methods. Our approach provides us with a characterization for the solutions to the time-harmonic Maxwell system in terms of series expansions involving spherical harmonics resp. spherical monogenics. Also, a thorough investigation for the series representation of the solutions in terms of eigenfunctions of Landau operators that encode n-dimensional spinless electrons is given. This new insight should lead to important investigations in the study of regularity and hypo-ellipticity of the solutions to Schrödinger equations with natural applications in relativistic quantum mechanics concerning massive spinor fields.

show abstract

Section: Motivation and Main Resultsmentioning

confidence: 99%

Fock spaces, Landau operators and the time-harmonic Maxwell equations

Constales¹,

Faustino²,

Kraußhar³

2011

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

show abstract

“…From this explicit formula we see that the function F (z) = e −πixξ e π|z| 2 /2 V g f (z) satisfies the diffential equation (27) ∂ N +1 ∂z N +1 F = 0 . In the established terminology, F is poly-analytic of order N + 1 and F is a version of the poly-analytic Bargmann transform of f [1,4,32]. Precisely given g = N n=0 √ π n n!c n h n with associated polynomial P (z) = c n z n , we define the poly-analytic Bargmann transform of f with respect to g to be…”

Section: Polyanalytic Functionsmentioning

confidence: 99%

“…with positive terms, (32) is a convex linear combination of points in the unit disc of C. As 1 is an extreme point of the unit disc, (32) holds if and only if it is a convex combination of 1's. Therefore, I(ξ) = 0 if and only if e 2πia k ξ = 1, k = 2, .…”

Section: Polyanalytic Functionsmentioning

confidence: 99%

Zeros of the Wigner distribution and the short-time Fourier transform

2019

View full text Add to dashboard Cite

We study the question under which conditions the zero set of a (cross-) Wigner distribution W (f, g) or a short-time Fourier transform is empty. This is the case when both f and g are generalized Gaussians, but we will construct less obvious examples consisting of exponential functions and their convolutions. The results require elements from the theory of totally positive functions, Bessel functions, and Hurwitz polynomials. The question of zero-free Wigner distributions is also related to Hudson's theorem for the positivity of the Wigner distribution and to Hardy's uncertainty principle. We then construct a class of step functions S so that the Wigner distribution W (f, 1 (0,1) ) always possesses a zero f ∈ S ∩ L p for p < ∞, but may be zero-free for f ∈ S ∩ L ∞ . The examples show that the question of zeros of the Wigner distribution may be quite subtle and relate to several branches of analysis.

show abstract

“…We should note that the expression of the transforms B m coincides with the expression of a family of isometric operators linking the space L 2 (R) with the true-poly-Fock spaces introduced by Vasilevski [11], which coincide with spaces A m (C). Thereby, the present work constitutes another way to arrive at the result of Theorem 2.5 in [11], by using a coherent states method exploiting tools of the L 2 -spectral theory of the Schrödinger operator given in (1.3).…”

Section: Introductionmentioning

confidence: 98%