Polyanalytic Functions Research Articles

Riemann boundary-value problem for doubly-periodic bianalytic functions

In this paper, we study Riemann boundary-value problem for doubly-periodic bianalytic functions. By the decomposition of doubly-periodic polyanalytic functions, the problem is transformed into two equivalent and independent Riemann boundary-value problems of doubly-periodic analytic functions, which has been discussed according to growth order of functions at the origin by Jianke Lu. Finally, we obtain the explicit expression of solutions and the conditions of solvability for the doubly-periodic bianalytic functions.

The Plemelj–Privalov theorem in polyanalytic function theory

In this paper we prove that the higher order Lipschitz classes behave invariant under the action of a singular integral operator naturally arising in polyanalytic function theory. This result provides a generalization of the well-known theorem by Joseph Plemelj [16] and Ivan Privalov [17].

A Cauchy transform for polyanalytic functions on fractal domains

A polyanalytic Cauchy transform over fractal curves is introduced and discussed, which allows us to obtain, in particular, integral representation formulas for smooth and polyanalytic functions in the presence of fractal boundaries.

Epsilon coherent states with polyanalytic coefficients for the harmonic oscillator

We construct a new class of coherent states indexed by points z of the complex plane and depending on two positive parameters m and $$ \varepsilon >0$$ by replacing the coefficients $$z^{n}/\sqrt{n!}$$ 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 $$\varepsilon \rightarrow 0^{+}$$ and obey a thermal stability property. Their wavefunctions are obtained in a closed form and their associated Bargmann-type transform is also discussed.

C 1 approximation of functions by solutions of second-order elliptic systems on compact sets in ℝ2

This paper is a brief survey of the recent results in problems of approximating functions by solutions of homogeneous elliptic systems of PDEs on compact sets in the plane in the norms of Cm spaces, m ≥ 0. We focus on general second-order systems. For such systems the paper complements the recent survey by M. Mazalov, P. Paramonov, and K. Fedorovskiy (2012), where the problems of Cm approximation of functions by holomorphic, harmonic, and polyanalytic functions as well as by solutions of homogeneous elliptic PDEs with constant complex coefficients were considered.

Uniqueness theorem for linear elliptic equation of the second order with constant coefficients

The interior uniqueness theorem for analytic functions was generalized by M. B. Balk to the case of polyanalytic functions of order n. He proved that if the zeros of a polyanalytic function have an accumulation point of order n, then this function is identically zero. In this paper the interior uniqueness theorem is generalized to the solution to a linear homogeneous second order differential equation of elliptic type with constant coefficients.

The Weyl–Heisenberg ensemble: hyperuniformity and higher Landau levels

Weyl–Heisenberg ensembles are a class of determinantal point processes associated with the Schrödinger representation of the Heisenberg group. Hyperuniformity characterizes a state of matter for which (scaled) density fluctuations diminish towards zero at the largest length scales. We will prove that Weyl–Heisenberg ensembles are hyperuniform. Weyl–Heisenberg ensembles include as a special case a multi-layer extension of the Ginibre ensemble modeling the distribution of electrons in higher Landau levels, which has recently been object of study in the realm of the Ginibre-type ensembles associated with polyanalytic functions. In addition, the family of Weyl–Heisenberg ensembles includes new structurally anisotropic processes, where point-statistics depend on the different spatial directions, and thus provide a first means to study directional hyperuniformity.

Criteria of univalence and fully –accessibility for p–harmonic and p–analytic functions

In this article, we present univalence criteria for polyharmonic and polyanalytic functions. Our approach yields a new criterion for a polyharmonic functions to be fully -accessible. Several examples are presented to illustrate the use of these criteria.

Level Sets of Certain Subclasses of α-analytic Functions

For an open set V subset of C-n, denote by M-alpha(V) the family of a-analytic functions that obey a boundary maximum modulus principle. We prove that, on a bounded harmonically fat domain Omega subset of C-n, a function f is an element of M-alpha (Omega\f(-1)(0)) automatically satisfies f is an element of M-alpha(Omega), if it is C alpha j-1-smooth in the z(j) variable, alpha is an element of Z(+)(n) up to the boundary. For a submanifold U subset of C-n, denote by M-alpha(U), the set of functions locally approximable by a-analytic functions where each approximating member and its reciprocal (off the singularities) obey the boundary maximum modulus principle. We prove, that for a C-3-smooth hypersurface, Omega, a member of M-alpha(Omega), 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 C-4-smooth submanifolds of higher codimension, at least near points with a Levi cone condition. (Less)

Polyharmonic Bergman spaces and Bargmann type transforms

In the main result of the paper we prove the decomposition of polyharmonic Bergman spaces over the upper-half plane into spaces of polyanalytic functions. Then, we introduce the decomposition of polyharmonic Bergman spaces into the orthogonal sum of its true polyharmonic Bergman subspaces and we state isometric isomorphisms between the different true polyharmonic Bergman spaces. This allows us to define the k-th harmonic Hilbert component of a polyharmonic Bergman function and to prove closed formulas for the reproducing kernel functions of the true polyharmonic and the polyharmonic Bergman spaces. The harmonic complex Fourier transform is introduced in order to give an explicit description of the cartesian and the Laguerre harmonic components of the images of a Bargmann type transform for the true polyharmonic Bergman spaces. Finally, it is proved that the polyharmonic Bergman space of order j is isometric isomorphic to 2j copies of the corresponding Hardy space.

Cauchy Integral Operators Involving Higher Order Lipschitz Classes in the Poly-Analytic Function Theory

In this note, we consider poly-analytic Cauchy integral operators introduced in a rather natural way on the higher order Lipschitz classes. We prove some boundary properties of the function represented by such an integral operator in the general context of rectifiable Jordan curves.

A note on Riemann problems for single‐periodic polyanalytic functions

In this article, a new canonical function has been established to deal with Riemann boundary‐value problem of periodic analytic functions discussed in . In comparison with the corresponding result in , the expression of solution obtained here is much simpler. Then, we demonstrate the equivalence of solutions for the homogeneous Riemann problem. What's more, we obtain the precise rank of matrix of coefficients for the system of linear algebraic equations (4.35) in . Those results can simplify the discussion of Riemann problem of single‐periodic polyanalytic functions in .

On the existence of polyanalytic functions

We give a sharp estimate for the codimension of the poly‐Bergman space in the poly‐Bergman space over the punctured domain. It is established the behaviour at the infinity point of polyanalytic Bergman functions on the complement of closed disks. In the main result of the paper, we prove that for and the j‐polyanalytic Bergman space over the domain U is trivial precisely when the complement of U has at most one point and at most two points or three points lying in a circle, respectively. We point out the differences between the domains over which the Bergman space and the non‐analytic poly‐Bergman space are trivial.

A uniqueness theorem for linear elliptic equations with dominating derivative with respect to $$\bar z$$

The interior uniqueness theorem for analytic functions was generalized by M.B. Balk to the case of polyanalytic functions of order n. He proved that, if the zeros of a polyanalytic function have an accumulation point of order n, then this function is identically zero. M.F. Zuev generalized this result to the case of metaanalytic functions. In this paper, we generalize the interior uniqueness theorem to solutions of linear homogeneous elliptic differential equations of order n with analytic coefficients whose senior derivative is the n-th power of the Cauchy–Riemann operator.

Cauchy and Poisson formulas for polyanalytic functions and applications

We obtain new Cauchy and Poisson integral formulas for polyanalytic functions. As an application, we establish mean value theorems for polyanalytic and real polyharmonic functions in a disk. We also give applications to sharp estimates of generalized maximum modulus principle type for associated functions, and, in particular, to estimates for rational functions (components) in the problem of singularity separation for polyrational functions.

Local Maximum Modulus Property for Polyanalytic Functions

Let \(\Omega \subset {\mathbb {C}}\) be an open subset and let \({\mathcal {F}}\) be a space of functions defined on \(\Omega \). \({\mathcal {F}}\) is said to have the local maximum modulus property if: for every \(f\in {\mathcal {F}},p_0\in \Omega ,\) and for every sufficiently small domain \(D\subset \Omega ,\) with \(p_0\in D,\) it holds true that \(\max _{z\in \overline{D}}\left| f(z)\right| = \max _{z\in \Sigma \cup \partial D}\left| f(z)\right| ,\) where \(\Sigma \subset \Omega \) denotes the set of points at which \(\left| f\right| \) attains strict local maximum. This property fails for \({\mathcal {F}}=C^{\infty }.\) We verify it however for the set of complex-valued functions whose real and imaginary parts are real analytic. We show by example that the property cannot be improved upon whenever \({\mathcal {F}}\) is the set of n-analytic functions on \(\Omega \), \(n\ge 2,\) in the sense that locality cannot be removed as a condition and independently \(\Sigma \) cannot be removed from the conclusion.

On Toeplitz operators and localization operators

This note is a contribution to a problem of Lewis Coburn concerning the relation between Toeplitz operators and Gabor-Daubechies localization operators. We will show that, for any localization operator with a general window w 2 F2(C) (the Fock space of analytic functions square-integrable on the complex plane), there exists a di¤erential operator of in nite order D, with constant coe¢ cients explicitly determined by w; such that the localization operator with symbol f coincides with the Toeplitz operator with symbol Df . This extends results of Coburn, Lo and Engli s, who obtained similar results in the case where w is a polynomial window. Our technique of proof combines their methods with a direct sum decomposition in true polyanalytic Fock spaces. Thus, polyanalytic functions are used as a tool to prove a theorem about analytic functions. 1. Introduction Let Lp(C) (1 p <1) be the weighted L space of functions with norm kukLp(C) = Z C ju(z)je 2 pjzj2dz 1 p ; dz being the area measure on C and L1(C) the space of measurable functions on C such that ju(z)je 2 jzj2 is bounded, endowed with the norm kukL1(C) = sup z2C ju(z)je 2 jzj2 : Let H be a Hilbert space contained in L2(C), with reproducing kernel K( ; z). For f 2 L1(C), the Toeplitz operator Toepf with symbol f(z) is the operator acting on H de ned by (Toepfg)( ) = Z C f(z)g(z)K( ; z)e jzj 2 dz; 8 g 2 H: Date: March 27, 2013. 1991 Mathematics Subject Classi cation. Primary 47B32,30H20; Secondary 81R30,81S30.

On the structure of polyharmonic Bergman-type spaces over the unit disk

Let be a pair of nonnegative integers. We deduce explicit representations for the projections onto spaces of square integrable -polyanalytic functions in terms of the two-sided compression of the Beurling–Ahlfors transform to the unit disk. We show that the space of square integrable -polyanalytic functions is a reproducing kernel Hilbert space and we deduce representations for one-to-one bounded operators from the Bergman space onto the true poly-Bergman spaces and from the harmonic Bergman space onto the true polyharmonic Bergman spaces. Moreover, we prove a decomposition theorem for polyharmonic functions in terms of their harmonic components. Finally, for positive integers we establish an isometry between a subspace of the true polyharmonic Bergman space of order with codimension and the true polyharmonic Bergman space of order .

The Hilbert–Schmidt Norm of Hankel Operators on Poly-Bergman Spaces

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.

On Riemann problem of automorphic polyanalytic functions connected with a rotation group

In this article, we will discuss a Riemann-type boundary-value problem for automorphic polyanalytic functions, which is connected with a rotation group on the complex plane. By the decomposition of automorphic polyanalytic functions, the problem is transformed into equivalent and independent Riemann problems of automorphic analytic functions. And then, we obtain the explicit expression of solution and the conditions of solvability for the Riemann problem of automorphic polyanalytic functions.