For an open set V C Cn, denote by Mα （V） the family of α-analytic functions that obey a boundary maximum modulus principle. We prove that, on a bounded ＂harmonically fat＂ domain Ω C Cn, a function f ∈M a（Ω／...
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For an open set V C Cn, denote by Mα （V） the family of α-analytic functions that obey a boundary maximum modulus principle. We prove that, on a bounded ＂harmonically fat＂ domain Ω C Cn, a function f ∈M a（Ω／f-1（0）） automatically sat- isfies f ∈M a（Ω）, if it is Caj-1smooth in the z/variable, α ∈ Zn＋ up to the boundary. For a submanifold U C Cn, denote by ma（U）, the set of functions locally approximable by α-analytic functions where each approximating member and its reciprocal （off the singularities） obey the boundary maximum modulus principle. We prove, that for a C3-smooth hypersurface, Ω, a member of ma （Ω）, cannot have constant modulus near a point where the Levi form has a positive eigenvalue, unless it is there the trace of a polyanalytic function of a simple form. The result can be partially generalized to C4-smooth submanifolds of higher codimension, at least near points with a Levi cone condition.
In this paper, we investigate a singular integral operator with polyanalytic Cauchy kernel. In particular, we will prove that the higher order Lipschitz classes (of order 1+alpha) behave invariant under the action of ...
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In this paper, we investigate a singular integral operator with polyanalytic Cauchy kernel. In particular, we will prove that the higher order Lipschitz classes (of order 1+alpha) behave invariant under the action of that operator.
We construct a polyanalytic extension of analytic (Poisson) wavelet frames, leading to a new system of wavelet superframes. Superframes have been introduced in signal analysis as a tool for the multiplexing of signals...
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We construct a polyanalytic extension of analytic (Poisson) wavelet frames, leading to a new system of wavelet superframes. Superframes have been introduced in signal analysis as a tool for the multiplexing of signals-encoding several signals as a single one with the purpose of sharing a communication channel. In this paper, a new system of vector-valued wavelets is constructed by selecting as elements of the analyzing vector the first n elements from an explicit orthogonal basis of the space of admissible functions depending on a parameter . We show that if the resulting discrete affine system indexed by the set is a wavelet superframe, then the estimate holds. This is proved by defining a polyanalytic Bergman transform, a unitary map which rephrases the problem in terms of sampling in polyanalytic Bergman spaces of the upper half-plane. Besides the applications in multiplexing, polyanalytic superframes lead to discrete counterparts of multitapered wavelet representations used in spectrum estimation and in high resolution time-frequency representation algorithms.
Nevanlinna contours (and domains) were introduced by K. Yu. Fedorovskii in connection with the problem of uniform approximation of continuous functions by polyanalytic polynomials;also, these contours are related to p...
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Nevanlinna contours (and domains) were introduced by K. Yu. Fedorovskii in connection with the problem of uniform approximation of continuous functions by polyanalytic polynomials;also, these contours are related to pseudocontinuation of analytic functions, to the theory of model spaces, etc. An example of a nonrectifiable Nevanlinna contour is constructed in this paper for the first time.
Using Gabor analysis, we give a complete characterization of all lattice sampling and interpolating sequences in the Fock space of polyanalytic functions, displaying a "Nyquist rate" which increases with n, ...
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Using Gabor analysis, we give a complete characterization of all lattice sampling and interpolating sequences in the Fock space of polyanalytic functions, displaying a "Nyquist rate" which increases with n, the degree of polyanaliticity of the space. Such conditions are equivalent to sharp lattice density conditions for certain vector-valued Gabor systems, namely superframes and Gabor super-Riesz sequences with Hermite windows, and in the case of superframes they were studied recently by Grochenig and Lyubarskii. The proofs of our main results use variations of the Janssen-Ron-Shen duality principle and reveal a duality between sampling and interpolation in polyanalytic spaces, and multiple interpolation and sampling in analytic spaces. To connect these topics we introduce the polyanalytic Bargmann transform, a unitary mapping between vector-valued Hilbert spaces and polyanalytic Fock spaces, which extends the Bargmann transform to polyanalytic spaces. Motivated by this connection, we discuss a vector-valued version of the Gabor transform. These ideas have natural applications in the context of multiplexing of signals. We also point out that a recent result of Man. Casazza and Landau, concerning density of Gabor frames, has important consequences for the Grochenig-Lyubarskii conjecture on the density of Gabor frames with Hermite windows. (C) 2009 Elsevier Inc. All rights reserved.
For a bounded anti-analytic symbol, we show that the Hankel operator on spaces of polyanalytic functions is in the Hilbert-Schmidt class if and only if its symbol is in the Dirichlet space. Its Hilbert-Schmidt norm is...
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For a bounded anti-analytic symbol, we show that the Hankel operator on spaces of polyanalytic functions is in the Hilbert-Schmidt class if and only if its symbol is in the Dirichlet space. Its Hilbert-Schmidt norm is computed in terms of the Dirichlet integral.
We consider the q-analytic functions on a given planar domain Omega, square integrable with respect to a weight. This gives us a q-analytic Bergman kernel, which we use to extend the Bergman metric to this context. We...
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We consider the q-analytic functions on a given planar domain Omega, square integrable with respect to a weight. This gives us a q-analytic Bergman kernel, which we use to extend the Bergman metric to this context. We recall that f is q-analytic if (partial derivative) over bar (q) f = 0 for the given positive integer q. polyanalytic Bergman spaces and kernels appear naturally in time-frequency analysis of Gabor systems of Hermite functions as well as in the mathematical physics of the analysis of Landau levels. We obtain asymptotic formulae in the bulk for the q-analytic Bergman kernel in the setting of the power weights e(-2mQ), as the positive real parameter m tends to infinity. This is only known previously for q = 1, by the work of Tian, Yau, Zelditch, and Catlin. Our analysis, however, is inspired by the more recent approach of Berman, Berndtsson, and Sjostrand, which is based on ideas from microlocal analysis. We remark here that since a q-analytic function may be identified with a vector-valued holomorphic function, the Bergman space of q-analytic functions may be understood as a vector-valued holomorphic Bergman space supplied with a certain singular local metric on the vectors. Finally, we apply the obtained asymptotics for q = 2 to the bianalytic Bergman metrics, and after suitable blow-up, the result is independent of Q for a wide class of potentials Q. We interpret this as an instance of geometric universality. (C) 2014 Elsevier Inc. All rights reserved.
Gabor frames with Hermite functions are equivalent to sampling sequences in true Fock spaces of polyanalytic functions. In the L-2-case, such an equivalence follows from the unitarity of the polyanalytic Bargmann tran...
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Gabor frames with Hermite functions are equivalent to sampling sequences in true Fock spaces of polyanalytic functions. In the L-2-case, such an equivalence follows from the unitarity of the polyanalytic Bargmann transform. We will introduce Banach spaces of polyanalytic functions and investigate the mapping properties of the polyanalytic Bargmann transform on modulation spaces. By applying the theory of coorbit spaces and localized frames to the Fock representation of the Heisenberg group, we derive explicit polyanalytic sampling theorems which can be seen as a polyanalytic version of the lattice sampling theorem discussed by J.M. Whittaker in Chapter 5 of his book Interpolatory Function Theory.
Connections between the boundary behaviour of polyanalytic functions and the structure of the boundary are investigated. In particular, a Jordan domain with Lipschitz boundary is constructed which is regular for the D...
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Connections between the boundary behaviour of polyanalytic functions and the structure of the boundary are investigated. In particular, a Jordan domain with Lipschitz boundary is constructed which is regular for the Dirichlet problem in the class of bianalytic functions..
We introduce poly-Bergman type spaces on the Siegel domain Dn ⊂ C n, and prove that they are isomorphic to tensor products of one-dimensional spaces generated by orthogonal polynomials of two kinds: Laguerre and Hermi...
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We introduce poly-Bergman type spaces on the Siegel domain Dn ⊂ C n, and prove that they are isomorphic to tensor products of one-dimensional spaces generated by orthogonal polynomials of two kinds: Laguerre and Hermite polynomials. The linear span of all poly-Bergman type spaces is dense in the Hilbert space L2(Dn,dμλ), where dμλ = (Imzn -|z1|2 |z n-1|2)λdx1dy1 dx ndyn and λ > -1.
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