Polyanalytic Functions Research Articles

Short-time Fourier transform and superoscillations

In this paper we investigate new results on the theory of superoscillations using time-frequency analysis tools and techniques such as the short-time Fourier transform (STFT) and the Zak transform. We start by studying how the short-time Fourier transform acts on superoscillation sequences. We then apply the supershift property to prove that the short-time Fourier transform preserves the superoscillatory behavior by taking the limit. It turns out that these computations lead to interesting connections with various features of time-frequency analysis such as Gabor spaces, Gabor kernels, Gabor frames, 2D-complex Hermite polynomials, and polyanalytic functions. We treat different cases depending on the choice of the window function moving from the general case to more specific cases involving the Gaussian and the Hermite windows. We consider also an evolution problem with an initial datum given by superoscillation multiplied by the time-frequency shifts of a generic window function. Finally, we compute the action of STFT on the approximating sequences with a given Hermite window.

Density of Complex and Quaternionic Polyanalytic Polynomials in Polyanalytic Fock Spaces

In this paper we consider the polyanalytic Fock spaces both in the complex and in the quaternionic case. In this latter case, the polyanalytic functions are considered in the slice regular case, and we shall treat Fock spaces of the first and of the second kind. In all these spaces we prove quantitative results in the approximation by polyanalytic polynomials. The quantitative approximation results are given in terms of higher order Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{p}$$\\end{document}-moduli of smoothness.

Weighted Bergman kernels for nearly holomorphic functions on bounded symmetric domains

We identify the standard weighted Bergman kernels of spaces of nearly holomorphic functions, in the sense of Shimura, on bounded symmetric domains. This also yields a description of the analogous kernels for spaces of “invariantly-polyanalytic” functions — a generalization of the ordinary polyanalytic functions on the ball which seems to be the most appropriate one from the point of view of holomorphic invariance. In both cases, the kernels turn out to be given by certain spherical functions, or equivalently Heckman-Opdam hypergeometric functions, and a conjecture relating some of these to a Faraut-Koranyi hypergeometric function is formulated based on the study of low rank situations. Finally, analogous results are established also for compact Hermitian symmetric spaces, where explicit formulas in terms of multivariable Jacobi polynomials are given.

Harmonic and polyanalytic functional calculi on the S-spectrum for unbounded operators

Harmonic and polyanalytic functional calculi have been recently defined for bounded commuting operators. Their definitions are based on the Cauchy formula of slice hyperholomorphic functions and on the factorization of the Laplace operator in terms of the Cauchy–Fueter operator {mathcal{D}} and of its conjugate overline{{mathcal{D}}}. Thanks to the Fueter extension theorem, when we apply the operator {mathcal{D}} to slice hyperholomorphic functions, we obtain harmonic functions and via the Cauchy formula of slice hyperholomorphic functions, we establish an integral representation for harmonic functions. This integral formula is used to define the harmonic functional calculus on the S-spectrum. Another possibility is to apply the conjugate of the Cauchy–Fueter operator to slice hyperholomorphic functions. In this case, with a similar procedure we obtain the class of polyanalytic functions, their integral representation, and the associated polyanalytic functional calculus. The aim of this paper is to extend the harmonic and the polyanalytic functional calculi to the case of unbounded operators and to prove some of the most important properties. These two functional calculi belong to so called fine structures on the S-spectrum in the quaternionic setting. Fine structures on the S-spectrum associated with Clifford algebras constitute a new research area that deeply connects different research fields such as operator theory, harmonic analysis, and hypercomplex analysis.

Theory of Certain Non-Univalent Analytic Functions

ABSTRACT We investigate the non-univalent function’s properties reminiscent of the theory of univalent starlike functions. Let the analytic function ψ ( z ) = ∑ i = 1 ∞ A i z i , A 1 ≠ 0 be univalent in the unitdisk. Non-univalent functions may be found in the class ℱ ( ψ ) of analytic functions f of the form f ( z ) = z + ∑ k = 2 ∞ a k z k satisfying (zf′ (z)/f (z) – 1) ≺ ψ(z). Such functions, like the Ma and Minda classes k=2 of starlike functions, also have nice geometric properties. For these functions, growth and distortion theorems have been established. Further, we obtain bounds for some sharp coefficient functionals and establish the Bohr and Rogosinki phenomenon for the class ℱ ( ψ ) . Non-analytic functions that share properties of analytic functions are known as poly-analytic functions. Moreover, we compute Bohr and Rogosinski’s radius for poly-analytic functions with analytic counterparts in the class ℱ ( ψ ) or classes of Ma-Minda starlike and convex functions.

Lattice sums for double periodic polyanalytic functions

In 1892, Lord Rayleigh estimated the effective conductivity of rectangular arrays of disks and proved, employing the Eisenstein summation, that the lattice sum S_2 is equal to pi for the square array. Further, it became clear that such equality can be treated as a necessary condition of the macroscopic isotropy of composites governed by the Laplace equation. This yielded the description of two-dimensional conducting composites by the classic elliptic functions, including the conditionally convergent Eisenstein series. In 1935, Natanzon used a polyharmonic function to solve the plane elasticity problem. This paper is devoted to the extension of the classic lattice sums to the lattice sums for double periodic (pseudoperiodic) polyanalytic functions. The exact relations and computationally effective formulae between the polyanalytic and classic lattice sums are established. Polynomial representations of the lattice sums are obtained. They are a source of new exact formulae for the lattice sums.

Properties of a polyanalytic functional calculus on the S‐spectrum

AbstractThe Fueter mapping theorem gives a constructive way to extend holomorphic functions of one complex variable to monogenic functions, that is, null solutions of the generalized Cauchy–Riemann operator in , denoted by . This theorem is divided in two steps. In the first step, a holomorphic function is extended to a slice hyperholomorphic function. The Cauchy formula for this type of functions is the starting point of the S‐functional calculus. In the second step, a monogenic function is obtained by applying the Laplace operator in four real variables, namely, Δ, to a slice hyperholomorphic function. The polyanalytic functional calculus, that we study in this paper, is based on the factorization of . Instead of applying directly the Laplace operator to a slice hyperholomorphic function, we apply first the operator and we get a polyanalytic function of order 2, that is, a function that belongs to the kernel of . We can represent this type of functions in an integral form and then we can define the polyanalytic functional calculus on S‐spectrum. The main goal of this paper is to show the principal properties of this functional calculus. In particular, we study a resolvent equation suitable for proving a product rule and generate the Riesz projectors.

On Polyanalytic Functions in Several Complex Variables

We give a characterisation of the polyanalytic type subspaces of the Hilbert spaces mathcal {H}, being the weighted L_2 function spaces on a connected simply connected domains D subset mathbb {C}^n. The typical examples considered in the paper are the unit ball mathbb {B}^n and the whole space mathbb {C}^n. Our approach is based on the use of the two tuples of operators a=(a1,a2,…,an)andb=(b1,b2,…,bn),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\mathbf {\\mathfrak {a}}= (\\mathfrak {a}_1, \\ \\mathfrak {a}_2, \\ \\ldots , \\ \\mathfrak {a}_n) \\quad \ extrm{and} \\quad \\mathbf {\\mathfrak {b}}= (\\mathfrak {b}_1, \\ \\mathfrak {b}_2, \\ \\ldots , \\ \\mathfrak {b}_n), \\end{aligned}$$\\end{document}which act invariantly in some linear space and satisfy therein the commutation relations aj,bℓ=δj,ℓI,aj,aℓ=0,bj,bℓ=0,j,ℓ=1,2,...,n.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left[ \\mathfrak {a}_j, \\mathfrak {b}_{\\ell }\\right] = \\delta _{j,\\ell }I, \\quad \\left[ \\mathfrak {a}_j, \\mathfrak {a}_{\\ell }\\right] = 0, \\quad \\left[ \\mathfrak {b}_j, \\mathfrak {b}_{\\ell }\\right] = 0, \\quad j,\\ell = 1,2,...,n. \\end{aligned}$$\\end{document}We assume further that a common invariant domain of the above operators is a dense linear subspace in a Hilbert space mathcal {H}, and impose several additional conditions. The exact formulation is given in the Extended Fock space construction. Further we specify the obtained description to different concrete realizations of the above operators and Hilbert spaces mathcal {H}, illustrating a variety of possibilities that may occur in the characterisation of the polyanalytic type spaces in several complex variables.

An extension of the complex–real (C–R) calculus to the bicomplex setting, with applications

AbstractIn this paper, we extend notions of complex ‐calculus to the bicomplex setting and compare the bicomplex polyanalytic function theory to the classical complex case. Applications of this theory include two bicomplex least mean square algorithms, which extend classical real and complex least mean square algorithms.

A Hörmander–Fock space

ABSTRACT In a recent paper, we used a basic decomposition property of polyanalytic functions of order 2 in one complex variable to characterize solutions of the classical ∂ ¯ -problem for given analytic and polyanalytic data. Our approach suggested the study of a special reproducing kernel Hilbert space that we call the Hörmander–Fock space that will be further investigated in this paper. The main properties of this space are encoded in a specific moment sequence denoted by η = ( η n ) n ≥ 0 leading to a special entire function E ( z ) that is used to express the kernel function of the Hörmander–Fock space. We present also an example of a special function belonging to the class Mittag-Leffler (ML) introduced recently by Alpay et al. and apply a Bochner–Minlos type theorem to this function, thus motivating further connections with the theory of stochastic processes.

Hörmander’s L^2-Method, {overline{partial }}-Problem and Polyanalytic Function Theory in One Complex Variable

In this paper we consider the classical {overline{partial }}-problem in the case of one complex variable both for analytic and polyanalytic data. We apply the decomposition property of polyanalytic functions in order to construct particular solutions of this problem and obtain new Hörmander type estimates using suitable powers of the Cauchy-Riemann operator. We also compute particular solutions of the {overline{partial }}-problem for specific polyanalytic data such as the Itô complex Hermite polynomials and polyanalytic Fock kernels.

A polyanalytic functional calculus of order 2 on the 𝑆-spectrum

The Fueter theorem provides a two step procedure to build an axially monogenic function, i.e. a null-solution of the Cauchy-Riemann operator inR4\mathbb {R}^4, denoted byD\mathcal {D}. In the first step a holomorphic function is extended to a slice hyperholomorphic function, by means of the so-called slice operator. In the second step a monogenic function is built by applying the Laplace operatorΔ\Deltain four real variables to the slice hyperholomorphic function. In this paper we use the factorization of the Laplace operator, i.e.Δ=D¯D\Delta = \mathcal {\overline {D}} \mathcal {D}to split the previous procedure. From this splitting we get a class of functions that lies between the set of slice hyperholomorphic functions and the set of axially monogenic functions: the set of axially polyanalytic functions of order 2, i.e. null-solutions ofD2\mathcal {D}^2. We show an integral representation formula for this kind of functions. The formula obtained is fundamental to define the associated functional calculus on theSS-spectrum.

Generalized Appell polynomials and Fueter–Bargmann transforms in the polyanalytic setting

This paper deals with some special integral transforms in the setting of quaternionic valued slice polyanalytic functions. In particular, using the polyanalytic Fueter mappings, it is possible to construct a new family of polynomials which are called the generalized Appell polynomials. Furthermore, the range of the polyanalytic Fueter mappings on two different polyanalytic Fock spaces is characterized. Finally, we study the polyanalytic Fueter–Bargmann transforms.

Reproducing Kernel Hilbert Spaces of Polyanalytic Functions of Infinite Order

In this paper we introduce reproducing kernel Hilbert spaces of polyanalytic functions of infinite order. First we study in details the counterpart of the Fock space and related results in this framework. In this case the kernel function is given by \(\displaystyle e^{z\overline{w}+\overline{z}w}\) which can be connected to kernels of polyanalytic Fock spaces of finite order. Segal–Bargmann and Berezin type transforms are also considered in this setting. Then, we study the reproducing kernel Hilbert spaces of complex-valued functions with reproducing kernel \(\displaystyle \frac{1}{(1-z\overline{w})(1-\overline{z}w)}\) and \(\displaystyle \frac{1}{1-2\mathrm{Re}\, z\overline{w}}\). The corresponding backward shift operators are introduced and investigated.

Approximation by polyanalytic functions in Hölder spaces

The problem of approximation of functions on plane compact sets by polyanalytic functions of order higher than two in the Hölder spaces C m C^m , m ∈ ( 0 , 1 ) m\in (0,1) , is significantly more complicated than the well-studied problem of approximation by analytic functions. In particular, the fundamental solutions of the corresponding operators belong to all the indicated Hölder spaces, but this does not lead to the triviality of the approximation conditions. In the model case of polyanalytic functions of order 3, approximation conditions and a constructive approximation method generalizing the Vitushkin localization method are studied. Some unsolved problems are formulated.

Affine density, von Neumann dimension and a problem of Perelomov

We provide a solution to Perelomov's 1972 problem concerning the existence of a phase transition (known in signal analysis as ‘Nyquist rate’) determining the basis properties of certain affine coherent states labelled by Fuchsian groups. As suggested by Perelomov, the transition is given according to the hyperbolic volume of the fundamental region. The solution is a more general form (in phase space) of the PSL(2,R) variant of a 1989 conjecture of Kristian Seip about wavelet frames, where the same value of ‘Nyquist rate’ is obtained as the trace of a certain localization operator. The proof consists of first connecting the problem to the theory of von Neumann algebras, by introducing a new class of projective representations of PSL(2,R) acting on non-analytic Bergman-type spaces. We then adapt to this setting a new method for computing von Neumann dimensions, due to Sir Vaughan Jones. Our solution contains necessary conditions in the form of a ‘Nyquist rate’ dividing frames from Riesz sequences of coherent states and sampling from interpolating sequences. They hold for an infinite sequence of spaces of polyanalytic functions containing the eigenspaces of the Maass operator and their orthogonal sums. Within mild boundaries, we show that our result is best possible, by characterizing our sequence of function spaces as the only invariant spaces under the non-analytic PSL(2,R)-representations.

Slice-by-slice and global smoothness of slice regular and polyanalytic functions

The concept of slice regular function over the real algebra \(\mathbb {H}\) of quaternions is a generalization of the notion of holomorphic function of a complex variable. Let \(\varOmega \subset \mathbb {H}\) be a domain, i.e., a non-empty connected open subset of \(\mathbb {H}=\mathbb {R}^4\). Suppose that \(\varOmega\) intersects \(\mathbb {R}\) and is invariant under rotations of \(\mathbb {H}\) around \(\mathbb {R}\). A function \(f:\varOmega \rightarrow \mathbb {H}\) is slice regular if it is of class \(\mathcal {C}^1\) and, for all complex planes \(\mathbb {C}_I\) spanned by 1 and a quaternionic imaginary unit I (\(\mathbb {C}_I\) is a ‘complex slice’ of \(\mathbb {H}\)), the restriction \(f_I\) of f to \(\varOmega _I=\varOmega \cap \mathbb {C}_I\) satisfies the Cauchy–Riemann equations associated with I, i.e., \(\overline{\partial }_If_I=0\) on \(\varOmega _I\), where \(\overline{\partial }_I=\frac{1}{2}\big (\frac{\partial }{\partial \alpha }+I\frac{\partial }{\partial \beta }\big )\). Given any positive natural number n, a function \(f:\varOmega \rightarrow \mathbb {H}\) is called slice polyanalytic of order n if it is of class \(\mathcal {C}^n\) and \(\overline{\partial }_I^{\,n}f_I=0\) on \(\varOmega _I\) for all I. We define global slice polyanalytic functions of order n as the functions \(f:\varOmega \rightarrow \mathbb {H}\), which admit a decomposition of the form \(f(x)=\sum _{h=0}^{n-1}\overline{x}^hf_h(x)\) for some slice regular functions \(f_0,\ldots ,f_{n-1}\). Global slice polyanalytic functions of any order n are slice polyanalytic of the same order n. The converse is not true: for each \(n\ge 2\), we give examples of slice polyanalytic functions of order n, which are not global. The aim of this paper is to study the continuity and the differential regularity of slice regular and global slice polyanalytic functions viewed as solutions of the slice-by-slice differential equations \(\overline{\partial }_I^{\,n}f_I=0\) on \(\varOmega _I\) and as solutions of their global version \({\overline{\vartheta }\,}^nf=0\) on \(\varOmega \setminus \mathbb {R}\). Our quaternionic results extend to the slice monogenic case.

Mean approximation in Bergman spaces of polyanalytic functions

Mean approximation in Bergman spaces of polyanalytic functions

Approximation by Convolution Polyanalytic Operators in the Complex and Quaternionic Compact Unit Balls

In this paper, by using the convolution method, we obtain quantitative results in terms of various moduli of smoothness for approximation of polyanalytic functions by polyanalytic polynomials in the complex unit disc. Then, by introducing the polyanalytic Gauss–Weierstrass operators of a complex variable, we prove that they form a contraction semigroup on the space of polyanalytic functions defined on the compact unit disk. The quantitative approximation results in terms of moduli of smoothness are then extended to the case of slice p-polyanalytic functions on the quaternionic unit ball. Moreover, we show that also in the quaternionic case the Gauss–Weierstrass operators of a quaternionic variable form a contraction semigroup on the space of polyanalytic functions defined on the compact unit ball.

On the polyanalytic short-time Fourier transform in the quaternionic setting

<p style='text-indent:20px;'>In this paper, we consider a quaternionic short-time Fourier transform (QSTFT) with normalized Hermite functions as windows. It turns out that such a transform is based on the recent theory of slice polyanalytic functions on quaternions. Indeed, we will use the notions of true and full slice polyanalytic Fock spaces and Segal-Bargmann transforms. We prove new properties of this QSTFT including a Moyal formula, a reconstruction formula and a Lieb's uncertainty principle. These results extend a recent paper of the authors which studies a QSTFT having a Gaussian function as a window.</p>