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),

Schwarz problem for higher‐order complex partial differential equations in the upper half plane

Linear and nonlinear elliptic complex partial differential equations of higher‐order are considered under Schwarz conditions in the upper‐half plane. Firstly, using the integral representations for the solutions of the inhomogeneous polyanalytic equation with Schwarz conditions, a class of integral operators is introduced together with some of their properties. Then, these operators are used to transform the problem for linear equations into singular integral equations. In the case of nonlinear equations such a transformation yields a system of integro‐differential equations. Existence of the solutions of the relevant boundary value problems for linear and nonlinear equations are discussed via Fredholm theory and fixed point theorems, respectively.     

单位圆周上多解析函数的Riemann边值问题(英文)

本文研究了单位圆周上具不同回子的Riemann边值问题的可解性.利用转化法,得到了可解条件和解的表达式,将现有的相同冈了的情况作为特例.     

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 16 . In comparison with the corresponding result in 16 , 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 16 . Those results can simplify the discussion of Riemann problem of single‐periodic polyanalytic functions in 16 .     

On the Structure of Spaces of Polyanalytic Functions

Suppose that A_mL_p(D,) is the space of all m-analytic functions on the disk D={z:|z| < 1} which are pth power integrable over area with the weight (1-|z|²), > -1. In the paper, we introduce subspaces A_kL_p ⁰(D,), k=1,2,...,m, of the space A _mL_p(D,) and prove that A _mL_p(D,) 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.     

On boundary properties of the components of polyharmonic functions

The following two classes of functions are introduced forp0: the classCU ^p (G) of uniformly continuous functions of orderp in a domainG , and the classM ^p (G) of functions of the boundedness of orderp inG. Criterions are established for ann-analytic function to belong to each of these classes.Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 821–834, June, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-013766     

Dense Subsets of -Solutions to Linear Elliptic Partial Differential Equations

Let Ω ^N (N2) be an unbounded domain, and L_m be a homogeneous linear elliptic partial differential operator with constant coefficients. In this paper we show, among other things, that rapidly decreasing ¹-solutions to L_m (in Ω) approximate all ¹-solutions to L_m (in Ω), provided there exist real numbers R_j→∞, 0, and a sequence {y^j} such that B(y^j, )∩Ω= and where |·| means the volume and

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for z ^N, R>0 and D ^N. 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 L_m may vanish, along with its (m−1)-derivatives. We also study a similar approximation problem for polyanalytic functions in .     

On Riemann problems for single‐periodic polyanalytic functions

In this article, Riemann‐type boundary‐value problem of single‐periodic polyanalytic functions has been investigated. By the decomposition of single‐periodic polyanalytic functions, the problem is transformed into n equivalent and independent Riemann boundary‐value problems of single‐periodic analytic functions, which has been discussed in details according to two growth orders of functions. Finally, we obtain the explicit expression of the solution and the conditions of solvability for Riemann problem of the single‐periodic polyanalytic functions.     

On a Boundary Value Problem for Pairs of Polyanalytic Functions

The necessary and sufficient conditions of Fredholmness and the index formula for a boundary value problems for a pair of polyanalytic functions, containing differential operators as well as operators with Carleman shift and conjugations, are obtained. Submitted: May 16, 2007. Accepted: November 5, 2007.     

Banach Gabor frames with Hermite functions: polyanalytic spaces from the Heisenberg group

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.