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A Reproducing Kernel and Toeplitz Operators in the Quantum Plane

Stephen Bruce Sontz (2013)

Communications in Mathematics

We define and analyze Toeplitz operators whose symbols are the elements of the complex quantum plane, a non-commutative, infinite dimensional algebra. In particular, the symbols do not come from an algebra of functions. The process of forming operators from non-commuting symbols can be considered as a second quantization. To do this we construct a reproducing kernel associated with the quantum plane. We also discuss the commutation relations of creation and annihilation operators which are defined...

Berezin and Berezin-Toeplitz quantizations for general function spaces.

Miroslav Englis (2006)

Revista Matemática Complutense

The standard Berezin and Berezin-Toeplitz quantizations on a Kähler manifold are based on operator symbols and on Toeplitz operators, respectively, on weighted L2-spaces of holomorphic functions (weighted Bergman spaces). In both cases, the construction basically uses only the fact that these spaces have a reproducing kernel. We explore the possibilities of using other function spaces with reproducing kernels instead, such as L2-spaces of harmonic functions, Sobolev spaces, Sobolev spaces of holomorphic...

Boundary behaviour of holomorphic functions in Hardy-Sobolev spaces on convex domains in ℂⁿ

Marco M. Peloso, Hercule Valencourt (2010)

Colloquium Mathematicae

We study the boundary behaviour of holomorphic functions in the Hardy-Sobolev spaces p , k ( ) , where is a smooth, bounded convex domain of finite type in ℂⁿ, by describing the approach regions for such functions. In particular, we extend a phenomenon first discovered by Nagel-Rudin and Shapiro in the case of the unit disk, and later extended by Sueiro to the case of strongly pseudoconvex domains.

Complétude des noyaux reproduisants dans les espaces modèles

Emmanuel Fricain (2002)

Annales de l’institut Fourier

Soit ( λ n ) n 1 une suite de Blaschke du disque unité 𝔻 et Θ une fonction intérieure. On suppose que la suite de noyaux reproduisants k Θ ( z , λ n ) : = 1 - Θ ( λ n ) ¯ Θ ( z ) 1 - λ n ¯ z n 1 est complète dans l’espace modèle K Θ p : = H p Θ H 0 p ¯ , 1 < p < + . On étudie, dans un premier temps, la stabilité de cette propriété de complétude, à la fois sous l’effet de perturbations des fréquences ( λ n ) n 1 mais également sous l’effet de perturbations de la fonction Θ . On retrouve ainsi un certain nombre de résultats classiques sur les systèmes d’exponentielles. Puis, si on suppose de plus que la suite ...

Décomposition atomique des espaces de Bergman.

Frédéric Symesak (1995)

Publicacions Matemàtiques

The aim of this paper is to establish the theorem of atomic decomposition of weighted Bergman spaces Ap(Ω), where Ω is a domain of finite type in C2. We construct a kernel function H(z,w) which is a reproducing kernel for Ap(Ω) and we prove that the associated integral operator H is bounded in Lp(Ω).

Distance formulae and invariant subspaces, with an application to localization of zeros of the Riemann ζ -function

Nikolai Nikolski (1995)

Annales de l'institut Fourier

It is proved that a subspace of a holomorphic Hilbert space is completely determined by their distances to the reproducing kernels. A simple rule is established to localize common zeros of a subspace of the Hardy space of the unit disc. As an illustration we show a series of discs of the complex plan free of zeros of the Riemann ζ -function.

Duality on vector-valued weighted harmonic Bergman spaces

Salvador Pérez-Esteva (1996)

Studia Mathematica

We study the duals of the spaces A p α ( X ) of harmonic functions in the unit ball of n with values in a Banach space X, belonging to the Bochner L p space with weight ( 1 - | x | ) α , denoted by L p α ( X ) . For 0 < α < p-1 we construct continuous projections onto A p α ( X ) providing a decomposition L p α ( X ) = A p α ( X ) + M p α ( X ) . We discuss the conditions on p, α and X for which A p α ( X ) * = A q α ( X * ) and M p α ( X ) * = M q α ( X * ) , 1/p+1/q = 1. The last equality is equivalent to the Radon-Nikodým property of X*.

Dunkl-Gabor transform and time-frequency concentration

Saifallah Ghobber (2015)

Czechoslovak Mathematical Journal

The aim of this paper is to prove two new uncertainty principles for the Dunkl-Gabor transform. The first of these results is a new version of Heisenberg’s uncertainty inequality which states that the Dunkl-Gabor transform of a nonzero function with respect to a nonzero radial window function cannot be time and frequency concentrated around zero. The second result is an analogue of Benedicks’ uncertainty principle which states that the Dunkl-Gabor transform of a nonzero function with respect to...

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