Riemann–Hilbert Problem for Automorphic Functions and the Schottky–Klein Prime Function

The construction of analogues of the Cauchy kernel is crucial for the solution of Riemann–Hilbert problems on compact Riemann surfaces. A formula for the Cauchy kernel can be given as an infinite sum over the elements of a Schottky group, and this sum is often used for the explicit evaluation of the kernel. In this paper a new formula for a quasi-automorphic analogue of the Cauchy kernel in terms of the Schottky–Klein prime function of the associated Schottky double is derived. This formula opens the door to finding new ways to evaluate the analogue of the Cauchy kernel in cases where the infinite sum over a Schottky group is not absolutely convergent. Application of this result to the solution of the Riemann–Hilbert problem with a discontinuous coefficient for symmetric automorphic functions is discussed. Received: March 10, 2007. Accepted: April 11, 2007.     

Impedance Passive and Conservative Boundary Control Systems

The external Cayley transform is used for the conversion between the linear dynamical systems in scattering form and in impedance form. We use this transform to define a class of formal impedance conservative boundary control systems (colligations), without assuming a priori that the associated Cauchy problems are solvable. We give sufficient and necessary conditions when impedance conservative colligations are internally well-posed boundary nodes; i.e., when the associated Cauchy problems are solvable and governed by C ₀ semigroups. We define a “strong” variant of such colligations, and we show that “strong” impedance conservative boundary colligation is a slight generalization of the “abstract boundary space” construction for a symmetric operator in the Russian literature. Many aspects of the theory is illustated by examples involving the transmission line and the wave equations. Received: August 21, 2006. Accepted: October 22, 2006.     

The Generalized Clifford-Gegenbauer Polynomials Revisited

By applying a method introduced by De Bie and Sommen in Clifford superanalysis, the orthogonality relations of the generalized Clifford–Gegenbauer polynomials of wavelet analysis are extended. Moreover, this new approach allows for proving new important properties of these polynomials, such as an annihilation equation, a differential equation and an expression in terms of the Jacobi polynomials on the real line. This paper is dedicated to the memory of our friend and colleague Jarolim Bureš     

On Fundamental Solutions in Clifford Analysis

Euclidean Clifford analysis is a higher dimensional function theory offering a refinement of classical harmonic analysis. The theory is centred around the concept of monogenic functions, which constitute the kernel of a first order vector valued, rotation invariant, differential operator ?{\underline{\partial}} called the Dirac operator, which factorizes the Laplacian. More recently, Hermitean Clifford analysis has emerged as a new branch of Clifford analysis, offering yet a refinement of the Euclidean case; it focusses on a subclass of monogenic functions, i.e. the simultaneous null solutions, called Hermitean (or h−) monogenic functions, of two Hermitean Dirac operators ?_z{\partial_{\underline{z}}} and ?_z^f{\partial_{\underline{z}^\dagger}} which are invariant under the action of the unitary group, and constitute a splitting of the original Euclidean Dirac operator. In Euclidean Clifford analysis, the Clifford–Cauchy integral formula has proven to be a corner stone of the function theory, as is the case for the traditional Cauchy formula for holomorphic functions in the complex plane. Also a Hermitean Clifford–Cauchy integral formula has been established by means of a matrix approach. Naturally Cauchy integral formulae rely upon the existence of fundamental solutions of the Dirac operators under consideration. The aim of this paper is twofold. We want to reveal the underlying structure of these fundamental solutions and to show the particular results hidden behind a formula such as, e.g. ?E = d{\underline{\partial}E = \delta}. Moreover we will refine these relations by constructing fundamental solutions for the differential operators issuing from the Euclidean and Hermitean Dirac operators by splitting the Clifford algebra product into its dot and wedge parts.     

On the Notion of the Bochner–Martinelli Integral for Domains with Rectifiable Boundary

In this paper we discuss the notion of the Bochner–Martinelli kernel for domains with rectifiable boundary in , by expressing the kernel in terms of the exterior normal due to Federer (see [17,18]). We shall use the above mentioned kernel in order to prove both Sokhotski–Plemelj and Plemelj–Privalov theorems for the corresponding Bochner–Martinelli integral, as well as a criterion of the holomorphic extendibility in terms of the representation with Bochner–Martinelli kernel of a continuous function of two complex variables. Explicit formula for the square of the Bochner–Martinelli integral is rediscovered for more general surfaces of integration extending the formula established first by Vasilevski and Shapiro in 1989. The proofs of all these facts are based on an intimate relation between holomorphic function theory of two complex variables and some version of quaternionic analysis. Submitted: September 6, 2006. Accepted: November 1, 2006.     

A Hilbert Transform for Matrix Functions on Fractal Domains

We consider H?lder continuous circulant (2 × 2) matrix functions G¹₂{{\bf G}^1_2} defined on the fractal boundary Γ of a Jordan domain Ω in \mathbbR²ⁿ{\mathbb{R}^{2n}}. The main goal is to establish a Hilbert transform for such functions, within the framework of Hermitian Clifford analysis. This is a higher dimensional function theory centered around the simultaneous null solutions of two first order vector valued differential operators, called Hermitian Dirac operators. In Brackx et al. (Bull Braz Math Soc 40(3): 395–416, 2009) a Hermitian Cauchy integral was constructed by means of a matrix approach using circulant (2 × 2) matrix functions, from which a Hilbert transform was derived in Brackx et al. (J Math Anal Appl 344: 1068–1078, 2008) for the case of domains with smooth boundary. However, crucial parts of the method are not extendable to the case where the boundary of the considered domain is fractal. At present we propose an alternative approach which will enable us to define a new Hermitian Hilbert transform in that case. As a consequence, we give necessary and sufficient conditions for the Hermitian monogenicity of a circulant matrix function G¹₂{{\bf G}^1_2} in the interior and exterior of Ω, in terms of its boundary value g¹₂=G¹₂|_G{{\bf g}^1_2={\bf G}^1_2|_\Gamma}, extending in this way also results of Abreu Blaya et al. (Bound. Value Probl. 2008: 2008) (article ID 425256), (article ID 385874), where Γ is required to be Ahlfors–David regular.     

Boundedness, regularity and smoothness of universal Taylor series

Let Ω be an unbounded simply connected domain in satisfying some topological assumptions; for example let Ω be an open half-plane. We show that there exists a bounded holomorphic function on Ω which extends continuously on and is a universal Taylor series in Ω in the sense of Luh and Chui–Parnes with respect to any center. Our proof uses Arakeljan’s Approximation Theorem. Further we strengthen results of G. Costakis [2] concerning universal Taylor series with respect to one center in the sense of Luh and Chui–Parnes in the complement G of a compact connected set. We prove that such functions can be smooth on the boundary of G and be zero at ∞. If the universal approximation is also valid on ∂G, then the function can not be smooth on ∂G, but it may vanish at ∞. Our results are generic in natural Fréchet spaces of holomorphic functions. Received: 29 September 2005; revised: 21 February 2006     

A priori estimates for a solution to the first initial-boundary-value problem for a system of fully nonlinear parabolic equations

A priori estimates for a solution to a system of fully nonlinear parabolic equations are obtained in a bounded domain under the condition that the solution vanishes on the boundary of the domain. The method of obtaining a priori estimates is based on the possibility of reducing the problem under consideration to the Cauchy problem for a scalar equation on a manifold without boundary in some linear space. Bibliography: 9 titles. Translated fromProblemy Matematicheskogo Analiza, No. 17, 1997, pp. 46–71.     

Jump problem and removable singularities for monogenic functions

In this article the jump problem for monogenic functions (Clifford holomorphicity) on the boundary of a Jordan domain in Euclidean spaces is investigated. We shall establish some criteria that imply the uniqueness of the solution in terms of a natural analogue of removable singularities in the plane to ℝⁿ⁺¹ (n ≥ 2). Sufficient conditions to extend monogenically continuous Clifford algebra valued functions across a hypersurface are proved. Communicated by Jenny Harrison     

Hermitean Clifford-Hermite Polynomials

Orthogonal Clifford analysis in flat m–dimensional Euclidean space focusses on monogenic functions, i.e. null solutions of the rotation invariant vector valued Dirac operator , where ( ) forms an orthogonal basis for the quadratic space underlying the construction of the Clifford algebra . When allowing for complex constants and taking the dimension to be even: m = 2n, the same set of generators produces the complex Clifford algebra , which we equip with a Hermitean Clifford conjugation and a Hermitean inner product. Hermitean Clifford analysis then focusses on the simultaneous null solutions of two mutually conjugate Hermitean Dirac operators, naturally arising in the present context and being invariant under the action of a realization of the unitary group U (n). In this so–called Hermitean setting Clifford–Hermite polynomials are constructed, starting from a Rodrigues formula involving both Dirac operators mentioned. Due to the specific features of the Hermitean setting, four different types of polynomials are obtained, two types of even degree and two types of odd degree. We investigate their properties: recurrence relations, structure, explicit form and orthogonality w.r.t. a deliberately chosen weight function. They also give rise to the definition of the Hermitean Clifford–Hermite functions, and may be used to develop a Hermitean continuous wavelet transform, see [4].     

The Mehler Formula for the Generalized Clifford-Hermite Polynomials

The Mehler formula for the Hermite polynomials allows for an integral representation of the one-dimensional Fractional Fourier transform. In this paper, we introduce a multi-dimensional Fractional Fourier transform in the framework of Clifford analysis. By showing that it coincides with the classical tensorial approach we are able to prove Mehler＇s formula for the generalized Clifford-Hermite polynomials of Clifford analysis.     

Boundary Behavior of Functions in the de Branges–Rovnyak Spaces

This paper deals with the boundary behavior of functions in the de Branges–Rovnyak spaces. First, we give a criterion for the existence of radial limits for the derivatives of functions in the de Branges–Rovnyak spaces. This criterion generalizes a result of Ahern–Clark. Then we prove that the continuity of all functions in a de Branges–Rovnyak space on an open arc I of the boundary is enough to ensure the analyticity of these functions on I. We use this property in a question related to Bernstein’s inequality. Received: May 10, 2007. Revised: August 8, 2007. Accepted: August 8, 2007.     

Interior estimates for a low order finite element method for the Reissner–Mindlin plate model

Interior error estimates are obtained for a low order finite element introduced by Arnold and Falk for the Reissner–Mindlin plates. It is proved that the approximation error of the finite element solution in the interior domain is bounded above by two parts: one measures the local approximability of the exact solution by the finite element space and the other the global approximability of the finite element method. As an application, we show that for the soft simply supported plate, the Arnold–Falk element still achieves an almost optimal convergence rate in the energy norm away from the boundary layer, even though optimal order convergence cannot hold globally due to the boundary layer. Numerical results are given which support our conclusion. This revised version was published online in June 2006 with corrections to the Cover Date.     

Boundary value problems for quaternionic monogenic functions on non-smooth sureaces

In this paper, analogous of the Compound Riemann-Hilbert boundary value problems are investigate for quaternionic monogenic functions. The solution (explicitly) of the problem is established over continuous surface, with little smoothness, which bounds a bounded domain of R³. In particular, smoothness property for high-dimensional Cauchy type integral are computed. We also use Zygmund type estimates to adapt existing one-variable complex results to ilustrate the Hölder-boundedness of the singular integral operator on 2-dimensional Ahlfors regular surfaces. At the end, uniqueness of solution for the Riemann boundary value problem have already built taking as a base the general Operator Theory.     

A Fatou–Bieberbach domain in \mathbb {C}^2 which is not Runge

Since a paper by Rosay and Rudin (Trans. Am. Math. Soc. 310, 47–86, 1988) there has been an open question whether all Fatou–Bieberbach domains are Runge. We give an example of a Fatou–Bieberbach domain Ω in which is not Runge. The domain Ω provides (yet) a negative answer to a problem of Bremermann. Supported by Schweizerische Nationalfonds grant 200021-116165/1.     

Holder Norm Estimate for a Hilbert Transform in Hermitean Clifford Analysis

A Hilbert transform for H?lder continuous circulant (2 × 2) matrix functions, on the d-summable (or fractal) boundary Γ of a Jordan domain Ω in ?²ⁿ , has recently been introduced within the framework of Hermitean Clifford analysis. The main goal of the present paper is to estimate the H?lder norm of this Hermitean Hilbert transform. The expression for the upper bound of this norm is given in terms of the H?lder exponents, the diameter of Γ and a specific d-sum (d > d) of the Whitney decomposition of Ω. The result is shown to include the case of a more standard Hilbert transform for domains with left Ahlfors-David regular boundary.     

Finite-Term Relations for Planar Orthogonal Polynomials

We prove by elementary means that, if the Bergman orthogonal polynomials of a bounded simply-connected planar domain, with sufficiently regular boundary, satisfy a finite-term relation, then the domain is algebraic and characterized by the fact that Dirichlet’s problem with boundary polynomial data has a polynomial solution. This, and an additional compactness assumption, is known to imply that the domain is an ellipse. In particular, we show that if the Bergman orthogonal polynomials satisfy a three-term relation then the domain is an ellipse. This completes an inquiry started forty years ago by Peter Duren. To Peter Duren on the occasion of his seventieth birthday The first author was partially supported by the National Science Foundation Grant DMS- 0350911. Received: October 15, 2006. Revised: January 22, 2007.     

Numerical solution of the differential equation describing the behavior of the zeroth-order boundary function

The asymptotic solution of the integro-differential plasma-sheath equation is considered. This equation is singularly perturbed because of the small coefficient multiplying the highest order (second) derivative. The asymptotic solution is obtained by the boundary function method. Equations are derived for the first two coefficients in the form of both a regular series expansion and an expansion in boundary functions. The equation for the first coefficient of the regular series has only a trivial solution. A numerical algorithm is considered for the solution of the second-order differential equation describing the behavior of the zeroth-order boundary function. The proposed algorithm efficiently solves the boundary-value problem and produces a well-behaved solution of the Cauchy problem. __________ Translated from Prikladnaya Matematika i Informatika, No. 23, pp. 24–35, 2006.     

The edge wave in the problem of diffraction on a boundary with jump of curvature

The problem of the diffraction of creeping waves on a point of transition of the convex boundary to the straight boundary of a domain is investigated. It is assumed that at the point of jump of curvature, the tangent to the boundary is continuous and its derivative has a jump. An expression for the edge wave is obtained and investigated. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 274–287. Translated by N. Ya. Kirpichnikova.     

A hybrid nonoverlapping domain decomposition scheme for advection dominated advection–diffusion problems

Two nonoverlapping domain decomposition algorithms are proposed for convection dominated convection–diffusion problems. In each subdomain, artificial boundary conditions are used on the inflow and outflow boundaries. If the flow is simple, each subdomain problem only needs to be solved once. If there are closed streamlines, an iterative algorithm is needed and the convergence is proved. Analysis and numerical tests reveal that the methods are advantageous when the diffusion parameter ɛ is small. In such cases, the error introduced by the domain decomposition methods is negligible in comparison with the error in the singular layers, and it allows easy and efficient grid refinement in the singular layers. This revised version was published online in August 2006 with corrections to the Cover Date.