Strong asymptotics for Bergman polynomials over non-smooth domains

Let G be a bounded simply-connected domain in the complex plane ℂ, whose boundary Γ:=∂G is a Jordan curve, and let \(\{p_{n}\}_{n=0}^{\infty}\) denote the sequence of Bergman polynomials of G. This is defined as the unique sequence $$p_n(z) = \lambda_n z^n+\cdots, \quad \lambda_n>0,\ n=0,1,2,\ldots, $$ of polynomials that are orthonormal with respect to the inner product $$\langle f,g\rangle := \int_G f(z) \overline{g(z)} \,dA(z), $$ where dA stands for the area measure.We establish the strong asymptotics for p n and λ n , n∈ℕ, under the assumption that Γ is piecewise analytic. This complements an investigation started in 1923 by T. Carleman, who derived the strong asymptotics for Γ analytic, and carried over by P.K. Suetin in the 1960s, who established them for smooth Γ. In order to do so, we use a new approach based on tools from quasiconformal mapping theory. The impact of the resulting theory is demonstrated in a number of applications, varying from coefficient estimates in the well-known class Σ of univalent functions and a connection with operator theory, to the computation of capacities and a reconstruction algorithm from moments.

Can there be infinitely many minimal hypersurfaces with a given boundary in a Riemannian manifold? A number of previous results, positive and negative, already indicated that the answer depends on the definition of surface, on orientability, on stability and minimizing properties of the surface, on the smoothness and geometry of the boundary, and on the ambient manifold. 1. Finiteness for area-minimizing hypersurfaces in certain manifolds. Several years ago, R. Hardt and L. Simon [HS, 1979, §12], following the original work of F. Tomi [T2, 1974], proved that a C 4 ' a Jordan curve T in R 3 bounds only finitely many orientable, area-minimizing immersed manifolds-with-boundary of least area. The immersed manifolds are assumed to minimize area over all immersed manifolds of finite topological type. It has been more or less understood that this finiteness theorem applies as well to hypersurfaces in R n (n < 7), bounded by a compact, connected, C submanifold B of R n . Such finiteness for oriented area-minimizing hypersurfaces fails in general ambient manifolds. For example, in the standard 2-dimensional sphere, there is a continuum of length-minimizing paths from the south pole to the north pole. However, we prove the following finiteness theorem for a noncompact, real-analytic Riemannian manifold. THEOREM A. Fix n < 7, 0 < a < 1. Let N be an n-dimensional, complete, connected, noncompact, real-analytic Riemannian manifold, with sectional curvature bounded above and injectivity radius bounded away from 0. Let B be a compact, oriented (n — 2) -dimensional, C 2 , a submanifold of N, with a positive integer multiplicity assigned to each component. Let M be the space of equivalence classes (under reparameterization) ofC immersions ƒ of compact, orientable, (n — 1)-dimensional C' manifolds-with-boundary M into N, such that f dM is a covering of B with the prescribed multiplicities. Let a = inf{area/: [ƒ,M] G Al}. Then MQ = {[ƒ,M] G M: area/ = a} is finite. When the ambient manifold N is compact, it is shown that infinitely many area-minimizing hypersurfaces with a common, smooth boundary necessarily give an open book decomposition of N (cf. [Wi, G]). It follows that if N = CP 2 , for example, or if N is nonorientable, there are only finitely many area-minimizing hypersurfaces with a common, smooth boundary. Received by the editors April 25, 1985. 1980 Mathematics Subject Classificatian. Primary 53C42; 49F22.

Let Ω be a bounded, simply connected domain in the complex plane, with C ∞ smooth boundary and take u ∈ C∞(∂Ω) real-valued, then there exists a unique real-valued harmonic function for which the restriction to the boundary ∂Ω is precisely u. Let be the conjugate harmonic to U in the sense that F = U+iV is holomorphic in Ω, for which moreover V(a)=0, a ∈ Ω, then the restriction v of V to the boundary ∂Ω is called the Hilbert transform of u. In this article, we extend the principles of this construction to m-dimensional space , more specifically in a Clifford analysis setting, in order to define a Hilbert-like integral transform on the unit sphere S m−1, based upon a specific notion of conjugate harmonic functions using spherical co-ordinates. ‡Dedicated to Richard Delanghe on the occasion of his 65th birthday.

The classical Green's function associated to a simply connected domain in the complex plane is easily expressed in terms of a Riemann mapping function. The purpose of this paper is to express the Green's function of a finitely connected domain in the plane in terms of a single Ahlfors mapping of the domain, which is a proper holomorphic mapping of the domain onto the unit disc that is the analogue of the Riemann map in the multiply connected setting.

Strong Asymptotic Behavior and Weak Convergence of Polynomials Orthogonal on an Arc of the Unit Circle

Near-circularity for the rational Zolotarev problem in the complex plane

We show a Lefschetz fixed point formula for holomorphic functions in a bounded domain D \mathcal {D} with smooth boundary in the complex plane. To introduce the Lefschetz number for a holomorphic map of D \mathcal {D} , we make use of the Bergman kernel of this domain. The Lefschetz number is proved to be the sum of the usual contributions of fixed points of the map in D \mathcal {D} and contributions of boundary fixed points, these latter being different for attracting and repulsing fixed points.

We consider the Dirichlet problem for Laplace's equation on planar domains with corners.A Nyström method is used to solve the corresponding double layer boundary integral equation.The Nyström method is based on the trapezoidal rule with a graded mesh method as in [13], but we adopt a generalized mesh grading.An improved convergence is observed.

In the above discussion of approximation of the parabolic problem the discretization in space was based on a family of finite-dimensional spaces S h ⊂ H 0 1 = H 0 1 (Ω), such that, for some r ≥ 2, the approximation property (1.7) holds. The most natural example of such a family in a plane domain Ω is to take for S h the continuous functions which reduce to polynomials of degree at most r - 1 on the triangles τ of a triangulation Th of Ω of the type described in the beginning of Chapter 1, and which vanish on ∂Ω. However, for r > 2 and in the case of a domain with smooth boundary, it is not possible, in general, to satisfy the homogeneous boundary conditions exactly for this choice. This diffiiculty occurs, of course, already for the elliptic problem, and several methods have been suggested to deal with it. Here we shall only, as an example, consider the following method which was proposed by Nitsche. This will serve as background for our subsequent discussion of the discretization of the parabolic problem. Another example will be considered in Chapter 17 below.

Let D be a simply-connected domain in the complex plane ℂ, and let f = u + iv be a univalent, orientation-preserving, harmonic mapping from D into ℂ. Then f can be written in the form f = h + \(\bar g\) where h and g belong to the linear space H(D) of analytic functions on D. In addition, f can be viewed as a solution of the elliptic partial differential equation $$ {\bar f_{\bar z}} = a{f_z} $$ (1.1) where the function a = g′/h′ belongs to H(D) and satisfies |a(z)| < 1 for all z ∈ D. Hence the mapping f is locally quasiconformal. Conversely, any univalent solution of (1.1) with a analytic and |a| < 1 is an orientation-preserving harmonic mapping of D (see [4]). Observe that if φ is a conformal mapping from D 1 onto D, then f 1 = f o φ is a harmonic mapping defined on D 1 with D 1 with a 1 = a o φ.

1. In our research on approximation by polynomial and rational interpolation in the complex plane we were led to some problems on balayage (sweeping out) which we want to describe. Let D be a bounded simply connected domain in the complex plane C. Let C = C ∪∞ be the extended complex plane and let A be defined by A = C\D, where D = D ∪ ∂D and ∂D is the boundary of D. In the problems below we consider unit (probability) measures α and β such that α(D) = 1 or α(D) = 1, and β(A) = 1 or β(A) = 1. In all cases α′ and β ′ denote the balayage onto ∂D of α and β, respectively. We denote by δ∞ the Dirac measure at infinity, and by δ′ ∞ its balayage onto ∂D. Then δ ′ ∞ is the equilibrium measure for logarithmic potentials of ∂D. We refer to [6] for this and other facts on balayage. In the problems below we are interested in the equality α′ = β ′. In Problem 1 we consider the case β = δ∞.

Let Ω be a domain in the complex plane C whose complement E=\(\overline C \backslash \Omega \) is a subset of the real line (i.e., Ω a Denjoy domain). If each point of E is regular for the Dirichlet problem in Ω, we provide a geometric description of the structure of E near infinity such that the Martin boundary of Ω has one or two “infinite” points.

Let Ω be a domain in the complex plane \(\mathbb{C}\) containing the origin and let \({\cal P}_{n}^{0}(\Omega)\) denote the set of all polynomials P of degree ≤ n satisfying the conditions \(P(0)=0\) and \(P(\mathbb{D})\subset \Omega\), where \(\mathbb{D}=\left\{z\,:\,\,|z|<1\right\}\) is the unit disk in the complex plane. The maximal range, denoted by Ω n , is then defined as the union of all sets \(P(\mathbb{D}), P \in {\cal P}_{n}^{0}\) .We shall construct extremal polynomials and maximal polynomial range for a domain of intersection of two circular disks.

Motivation: In designing linear feedback systems, tracking quality specifications are typically mapped into admissible domains in the complex plane and then the root locus technique is used to synthesize a controller that places closed loop poles in the desired locations. This methodology motivates the first goal of this chapter: to generalize the notion of admissible domains and the root locus technique to LPNI systems. Accomplishing this goal leads to a quasilinear method for PID controller design.

Exploiting the homogeneous structure of a wedge in the complex plane, we compute the spectrum of the anti-linear Ahlfors-Beurling transform acting on the associated Bergman space. Consequently, the similarity equivalence between the Ahlfors-Beurling transform and the Neumann-Poincare operator provides the spectrum of the latter integral operator on a wedge. A localization technique and conformal mapping lead to the first complete description of the essential spectrum of the Neumann-Poincare operator on a planar domain with corners, with respect to the energy norm of the associated harmonic field.