Polyanalytic Functions Research Articles

Hilbert boundary value problems of polyanalytic functions on the unit circumference†

In this article, the Hilbert boundary value problem (BVP) for bianalytic functions with different factors on the unit circumference is first investigated in two different ways. The different expressions of the solutions and the conditions of solvability are obtained and the corresponding equivalence among them is verified. Then, the general Hilbert BVP for polyanalytic functions is studied by transforming it into the equivalent Hilbert BVP for a system of analytic functions, and the expressions of the solution and the solvability condition dependent on the so-called canonical matrix is obtained. †Dedicated to Professor Guochun Wen on the occasion of his 75th birthday.

On some properties and examples of Nevanlinna domains

The properties of Nevanlinna domains are considered. These domains arise in the problems of approximation by polyanalytic functions. Several analytic and geometric properties (both new and earlier known) of Nevanlinna domains are described. In particular, a new method for constructing Nevanlinna domains with boundaries belonging to the class C1 is proposed, and new examples of such domains whose boundaries do not belong to the class C1,α for α ∈ (0, 1) are presented. This method is based on the property of pseudocontinuation of a conformal mapping from the unit disk onto a Nevanlinna domain.

Existence of solution in weighted Sobolev space W n,p (D; dμ(z; ∂D)) to Riemann–Hilbert problem with piece-wise continuous boundary data for non-homogeneous polyanalytic equation of order n†

The given non-homogeneous polyanalytic differential equation is transformed into an equivalent system of integro–differential equations before its general solution in the weighted Sobolev space W n,p (D; d μ(z, ∂D)) is sought using functional analytic tools. The boundary value problem is then solved by first transforming it to two analogous problems for polyanalytic functions. The latter was already solved by the author in a separate study.

Riemann boundary value problems of polyanalytic functions and metaanalytic functions on the closed curves†

In this article, Riemann boundary value problems (BVPs) of polyanalytic functions and metaanalytic functions on the closed curve are investigated. Using the theorem of decomposition of polyanalytic functions, the expression of solution and the condition of solvability for Riemann BVP of polyanalytic functions are obtained by reducing the problem to the equivalent Riemann BVP of analytic functions. Then the expression of solution and the condition of solvability for Riemann BVP of metaanalytic functions are obtained by reducing the problem into the equivalent Riemann BVP of polyanalytic functions.

ON RIEMANN BOUNDARY VALUE PROBLEM FOR POLYANALYTIC FUNCTIONS ON THE REAL AXIS

In this article, Riemann boundary value problem with different factors for polyanalytic functions on the real axis is studied. The expression of solution and sufficient and necessary condition for solvability of the non-homogeneous Riemann boundary value problem are obtained.

Estimate of the Norm of a Polyanalytic Function via the Norm of Its Polyharmonic Component

We obtain an estimate of the norm of a polyanalytic function belonging to the space Lp(D,α), 1 ≤ p -1, via the norm of its polyharmonic component.

On Boundary Value Problems of Polyanalytic Functions on the Real Axis

In this article, we discuss the Riemann boundary value problems and the Hilbert boundary value problems of polyanalytic functions on the real axis, and both the explicit expressions of solutions and the conditions of solvability are obtained.

A Mixed Boundary Value Problem for Polyanalytic Function of Order n in the Sobolev Space Wn,p (D)

We discuss the construction of a polyanalytic function Φ of order n on a simple bounded domain D. The function satisfies n prescribed generalized Riemann-Hilbert boundary conditions on the boundary ∂D and n generalized jump conditions on a simple closed smooth contour γ contained in D. The boundary conditions are transformed into n classical Riemann-Hilbert problems and the n jump conditions into n Riemann problems of conjugation for some 2n holomorphic functions. These transformed problems are solved using the standard methods from the literature.

A Mixed Boundary Value Problem for Generalized Polyanalytic Functions of Order n in the Sobolev Space W n , p ( D )

Given is the following boundary value problem for a generalized polyanalytic differential equation of order n in a domain D : $${{\\partial ^n w} \\over {\\partial \\bar z^n }} = F\\left( {z, w, {{\\partial ^{m + k} w} \\over {\\partial z^m \\partial \\bar z^k }}} \\right)\\quad {\\rm on}\\enskip D \\eqno (1)$$ $$Bw = g\\quad {\\rm on}\\enskip \\gamma \\cup \\partial D \\eqno (2)$$ $$n \\ge m, k \\in {\\shadN}_0 \\quad m + k \\le n, \\quad (0, 0) \ e (m, k) \ e (0, n), \\quad n \\in {\\shadN}$$ where B is an operator acting on the boundary ‘ D of D and possibly on some closed curve n in D . First the existence of a general solution is established under certain assumptions on the right-hand side of (1). Next, a general method for solving boundary value problems or other related problems (2) is laid down. The latter is possible whenever the corresponding problem for a polyanalytic function admits a unique solution which, in addition, satisfies certain a priori estimates. The method will be illustrated for the case of a mixed boundary value problem.

On the Structure of Spaces of Polyanalytic Functions

Suppose that AmLp(D,α) is the space of all m-analytic functions on the disk D={z:|z| -1. In the paper, we introduce subspaces AkLp0(D,α), k=1,2,...,m, of the space AmLp(D,α) and prove that AmLp(D,α) is the direct sum of these subspaces. These results are used to obtain growth estimates of derivatives of polyanalytic functions near the boundary of arbitrary domains.

The Generalized Riemann Problem of Linear Conjugation for Non-Homogeneous Polyanalytic Equations of Order $n$ in $W_{n,p}(D)$

We consider a non-homogeneous polyanalytic partial differential equation of order n in a simply-connected domain D with smooth boundary \partial D in the complex plane \mathbb C . Initially we transform the given equation into an equivalent system of integro-differential equations and then find the general solution of the former in W_{n,p} (D) . Next we pose and prove the solvability of a generalized Riemann problem of linear conjugation to the differential equation. This is effected by first reducing the Riemann problem to a corresponding one for a polyanalytic function. The latter is solved by first transforming it into n classical Riemann problems of linear conjugation for n holomorphic functions expressed in terms of the analytic functions which define the polyanalytic function. The solution of the classical Riemann problem is available in the literature.

The Generalized Riemann Problem of Linear Conjugation for Polyanalytic Functions of Order $n$ in $W_{n,p}(D)$

We consider a homogeneous polyanalytic differential equation of order n in a simply-connected domain D with a smooth boundary \partial D in the complex plane \mathbb C . We pose and then prove solvability of a generalized Riemann problem of linear conjugation to the differential equation. This is done by reducing the problem into n classical Riemann problems of linear conjugation for holomorphic functions, the solution of which is available in the literature.

Dense Subsets of [formula omitted]1-Solutions to Linear Elliptic Partial Differential Equations

Let Ω⊂RN (N⩾2) be an unbounded domain, and Lm be a homogeneous linear elliptic partial differential operator with constant coefficients. In this paper we show, among other things, that rapidly decreasing L1-solutions to Lm (in Ω) approximate all L1-solutions to Lm (in Ω), provided there exist real numbers Rj→∞, ε⩾0, and a sequence {yj} such that B(yj, ε)∩Ω=∅ and|Λ(yj, Rj, RN\\Ω)|RNj>ε ∀ j,where |·| means the volume andΛ(z, R, D)≔∪x∈B(z, R)∩Dz+t(x−z)|x−z|;t⩽1,for z∈RN, R>0 and D⊂RN. For m=2, we can replace the volume density by the capacity-density. It appears that the problem is related to the characterization of largest sets on which a nonzero polynomial solution to Lm may vanish, along with its (m−1)-derivatives. We also study a similar approximation problem for polyanalytic functions in C.

Representation of the space of polyanalytic functions as a direct sum of orthogonal subspaces. Application to rational approximations

Suppose thatD={z:|z|<1}, L 2 (D) is the space of functions square-integrable over area inD,A k (D) is the set of allk-analytic functions inD, (A 1 (D)=A(D) is the set of all analytic functions inD),A k L 2 (D)=L 2 (D)∩A k (D),A 1 L 2 (D)=AL 2 (D), $$A_k L_2^0 \left( D \right) = \left\{ {f {\text{:}} f(z) = \frac{{\partial ^{k - 1} }}{{\partial z^{k - 1} }}\left( {\left( {1 - z\bar z} \right)^{k - 1} F\left( z \right)} \right); F \in A\left( D \right), f \in A_k L_2 \left( D \right)} \right\}$$ . It is proved that the subspacesA k L 2 0 (D),k=1, 2,..., are orthogonal to one another and the spaceA m L 2 (D) is the direct sum of such subspaces fork=1, 2,...,m. The kernel of the orthogonal projection operator from the spaceA m L 2 (D) onto its subspacesA k L 2 0 (D) is obtained. These results are applied to the study of the properties of polyrational functions of best approximation in the metricL 2 (D).

The Generalized Riemann-Hilbert Boundary Value Problem for Non-Homogeneous Polyanalytic Differential Equation of Order $n$ in the Sobolev Space $W_{n,p}(D)$

Given is a nonlinear non-homogeneous polyanalytic differential equation of order n in a simply-connected domain D in the complex plane. Initially we prove (under certain conditions) the existence of its general solution in W_{n,p}(D) by first transforming it into a system of integro-differential equations. Next we prove the solvability of a generalized Riemann-Hilbert problem for the differential equation. This is effected by first reducing the boundary value problem posed to a corresponding one for a polyanalytic function. The latter is then transformed into n classical Riemann-Hilbert problems for holomorphic functions, whose solutions are known in the literature.

The boundary behavior of components of polyharmonic functions

We consider the following representation of polyharmonic functions on the unit ballDm: 450-1 where the Φj are harmonic onDm. We study the relation between uniform boundary properties ofƒ (its smoothness and growth while approaching the boundary) and the same properties of the terms in this representation. The theorems proved in this paper generalize some results obtained by Dolzhenko in the theory of polyanalytic functions.

A generalised pompeiu formula related to differential equations in Cn with higher order

This paper deals with a generalization of the Pompeiu formula for partial differential equation with higher order in the n dimensional complex space. In the centre of the investigation lies a construction of a fundamental solution. The problem of constructing such a fundamental solution can be reduced to solving a special initial value problem. This solution will be turned out to be a generalized Riemann-Vekua-function. With the help of the fundamental solution an integral representation for functions being smooth enough can be found. Functions, which are generalizations to higher dimensions of the polyanalytic or polyharmonic functions in the complex plane, can be represented by the boundary values of the function and some of its derivatives.

New mean value theorems for polyanalytic functions

New mean value theorems for polyanalytic functions

Projections on Lp-spaces of polyanalytic functions

The fundamental result: for an arbitrary bounded, simply connected domain Ω in ℂ, the subspace Ln,m p(Ω) of the space Lp(Ω, σ) (σ is the plane Lebesgue measure, p ≥ 1), consisting of the (m, n)-analytic functions in Ω, is complemented in LP(Ω, σ) (a function f is said to be (m, n)-analytic if (∂m+n/∂¯Zm∂Zn)f=0 in Ω). Consequently, by virtue of a theorem of J. Lindenstrauss and A. Pelczyński, the space Ln,m P(Ω) is linearly homeomorphic to lP. In particular, for m=n=1 we obtain that the space of all harmonic LP-functions in Ω is complemented in LP(Ω, σ). This result has been known earlier only for smooth domains.

Toeplitz Operators on Polyanalytic Functions and Klee's Combinatorial Identity

Mathematische NachrichtenVolume 166, Issue 1 p. 5-15 Article Toeplitz Operators on Polyanalytic Functions and Klee's Combinatorial Identity Hartmut Wolf, Hartmut Wolf Technische Universität Chemnitz Fachbereich Mathematik PSF 964 D-09126 Chemnitz GermanySearch for more papers by this author Hartmut Wolf, Hartmut Wolf Technische Universität Chemnitz Fachbereich Mathematik PSF 964 D-09126 Chemnitz GermanySearch for more papers by this author First published: 1994 https://doi.org/10.1002/mana.19941660102Citations: 1AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume166, Issue11994Pages 5-15 RelatedInformation