Smooth Jordan Research Articles

The numerical solution of Symm's equation on smooth open arcs by spline Galerkin methods

We consider the classical Fredholm linear integral equation of the first kind with logarithmic kernel on a smooth Jordan open arc. Applying the well-known cosine change of variable, the arc is reparametrized and the problem is transformed into a new integral equation. We investigate the existence of an asymptotic expansion for the error of the Galerkin method with splines on a uniform mesh as test-trial functions. We also analyse a full discretization of the method based on the Galerkin collocation method using high order integration formulae to keep the optimal error estimates of the Galerkin method in weak norms. Asymptotic expansions of the error for this method are provided. Finally, we show how these expansions extend to the computation of the potential. The expansions of the error in powers of the discretization parameter are useful to obtain a posteriori estimates of the error and to apply Richardson extrapolation for acceleration of convergence. © 1999 Elsevier Science Ltd. All rights reserved.

Erdős–Turán Type Theorems on Quasiconformal Curves and Arcs

The theorems of Erdős and Turán mentioned in the title are concerned with the distribution of zeros of a monic polynomial with known uniform norm along the unit interval or the unit disk. Recently, Blatt and Grothmann ( Const. Approx. 7(1991), 19–47), Grothmann (“Interpolation Points and Zeros of Polynomials in Approximation Theory,” Habilitationsschrift, Katholische Universität Eichstätt, 1992), and Andrievskii and Blatt ( J. Approx. Theory 88(1977), 109–134) established corresponding results for polynomials, considered on a system of sufficiently smooth Jordan curves and arcs or piecewise smooth curves and arcs. We extend some of these results to polynomials with known uniform norm along an arbitrary quasiconformal curve or arc. As applications, estimates for the distribution of the zeros of best uniform approximants, values of orthogonal polynomials, and zeros of Bieberbach polynomials and their derivatives are obtained. We also give a negative answer to one conjecture of Eiermann and Stahl (“Zeros of orthogonal polynomials on regular N-gons,” in Lecture Notes in Math. 1574(1994), 187–189).

Numerical Conformal Mapping for Exterior Regions via the Kerzman-Stein Kernel

A simple numerical method is described for computing a conformal map from a domain exterior to a smooth Jordan curve in the complex plane onto the exterior of the unit disk. The numerical method is based on a boundary integral equation similar to the Kerzman-Stein integral equation for interior mappings. Typical examples show that numerical results of high accuracy can be obtained provided that the boundaries are smooth

A Discrepancy Theorem on Quasiconformal Curves

In [7], Blatt and Mhaskar estimated the Erdős-Turan type discrepancy of a signed Borel measure σ on a sufficiently smooth Jordan curve or arc L in terms of the logarithmic potential of σ on a curve enclosing L. We extend this result to a measure σ on an arbitrary quasiconformal curve. As applications, estimates for the distribution of simple zeros of monic polynomials, Fekete points, extreme points of polynomials of best uniform approximation are obtained.

On the approximate solution of nonlinear singular integral equations with positive index

This paper is devoted to investigating a class of nonlinear singular integral equations with a positive index on a simple closed smooth Jordan curve by the collocation method. Sufficient conditions are given for the convergence of this method in Holder space.

Trace formulas for a class of subnormal tuples of operators

Trace formulas are established for the product of commutators related to subnormal tuple of operators (S 1,...,S n ) with minimal normal extension (N 1,...,N n ) satisfying conditions that sp(S j )/sp(N j ) is simply-connected with smooth boundary Jordan curve sp(N i ) and [S * ,S j ]1/2 ∈L 1,j=1, 2,...,n. Some complete unitary invariants related to the trace formulas are found.

THE PLEMELJ-PRIVALOV THEOREM FOR GENERALIZED HÖLDER CLASSES

The Plemelj-Privalov theorem on invariance of the Hölder classes , , with respect to a one-dimensional singular integral with Cauchy kernel on closed smooth Jordan curves is well known. V. V. Salaev posed the problem of describing the class of all closed rectifiable Jordan curves on which the Plemelj-Privalov theorem is valid; it was solved in a paper by Salaev, Guseĭnov, and Seĭfullaev, where a condition completely characterizing the class is given in terms of the planar measure of boundary strips of sets constructed from the curve. The present article is devoted to the solution of Salaev's problem for the generalized Hölder class .

The distribution of zeros of asymptotically extremal polynomials

In this paper, we study the asymptotic behavior of the zeros of a sequence of polynomials whose weighted norms have the same nth root behavior as the weighted norms for certain extremal polynomials. Our results include as special cases several of the previous results of Erdős, Freud, Jentzsch, Szegő and Blatt, Saff, and Simkani. Applications are given concerning the zeros of orthogonal polynomials over a smooth Jordan curve (in particular, on the unit circle) and the zeros of polynomials of best approximation on R to nonentire functions.

Determinant of the Neumann Operator on Smooth Jordan Curves

Determinant of the Neumann Operator on Smooth Jordan Curves

Determinant of the Neumann operator on smooth Jordan curves

Using the method of heat kernel expansion, the determinant of the Neumann operator on an arbitrary smooth Jordan curve is shown to be equal to the circumference.

Testing Analyticity on Rotation Invariant Families of Curves

Let $\Gamma \subset C$ be a piecewise smooth Jordan curve, symmetric with respect to the real axis, which contains the origin in its interior and which is not a circle centered at the origin. Let $\Omega$ be the annulus obtained by rotating $\Gamma$ around the origin. We characterize the curves $\Gamma$ with the property that if $f \in C(\Omega )$ is analytic on $s\Gamma$ for every $s$, $|s| = 1$, then $f$ is analytic in Int $\Omega$.

Testing analyticity on rotation invariant families of curves

Let Γ ⊂ C \Gamma \subset C be a piecewise smooth Jordan curve, symmetric with respect to the real axis, which contains the origin in its interior and which is not a circle centered at the origin. Let Ω \Omega be the annulus obtained by rotating Γ \Gamma around the origin. We characterize the curves Γ \Gamma with the property that if f ∈ C ( Ω ) f \in C(\Omega ) is analytic on s Γ s\Gamma for every s s , | s | = 1 |s| = 1 , then f f is analytic in Int Ω \Omega .

Order of approximation by electrostatic fields due to electrons

LetD be a Jordan domain in the complex plane andA q (D) the Bers space with norm ∥ ∥ q . IfD is the unit disk, it is known that ∥S n 0∥2≥π/18, whereS n =∑ k=1 n l/(z−z nk ) withz nk ∈∂D, so that approximation in ∥ ∥ q ,q<-2, is not possible. In this paper, we give an order of estimate of ∥f−S n ∥ q for 2<q<∞ when ∂D is a sufficiently smooth Jordan curve, and prove that this order of approximation is in general best possible.

The Dirichlet Problem With Denjoy-Perron Integrable Boundary Condition

AbstractThe Poisson integral of a Denjoy-Perron integrable function defined on the boundary of an open disc is harmonic in this disc. Moreover, almost everywhere on the boundary, the nontangential limits of the integral coincide with the boundary condition. This extends the classical result for Lebesgue integrable boundary conditions. By means of conformai maps, a generalization to domains bounded by a sufficiently smooth Jordan curve is also obtained.

Area-minimizing currents bounded by higher multiples of curves

For any positive integern, we give an example of a smooth Jordan curveC in IR4 such that the smallest integral current bounded bynC has less thann/k times the area of smallest integral current bounded bykC (1≤k<n).

Zur Gitterpunktlehre der euklidischen Ebene

On lattice points in the Euclidean plane For a large parameter T, let D(T) denote a domain in the ( x,y)-plane bounded by a smooth (closed) Jordan curve C(T) which is defined by φ( x/ T, y/ T) = 0 where φ( u, v) is an analytic function with grad φc ≠ (0, 0) on C(1). Denote further by A(T) the number of lattice points (of the unit lattice Z 2) in D(T) and by V(T) its area and put P( T) = A( T)- V( T). Then it had been proved by J.G. Van der Corput [5] that P( T)= O( T θ ) with θ< 2 3 , provided that the curvature of C(T) vanishes nowhere, and recently by Y. Colin de Verdière [3] that P( T) = O( T 1-1/ n ) if the curvature of C(T) has only zeroes of order ≦n-2 (n≧3). In this paper, under the hypothesis that C(T) has rational slope in each point with curvature 0, the contribution of these points to P(T) is evaluated explicitely up to an error term O( T θ ) with θ< 2 3 .

Analyticity on rotation invariant families of curves

Let G \mathfrak {G} be a rotation invariant family of smooth Jordan curves contained in Δ \Delta , the open unit disc in C {\mathbf {C}} . For each Γ ∈ G \Gamma \in \mathfrak {G} let D Γ {D_\Gamma } be the simply connected domain bounded by Γ \Gamma . We present various conditions which imply that if f f is a continuous function on Δ \Delta such that for every Γ ∈ G \Gamma \in \mathfrak {G} the function f | Γ f|\Gamma has a continuous extension to D Γ ¯ \overline {{D_\Gamma }} which is analytic in D Γ {D_\Gamma } , then f f is analytic in Δ \Delta .

Zero sets of multiparameter functions and stability of multidimensional systems

New theorems, which can be used to find the zero set of a multiparameter rational function of a complex variable where the distinguished boundary of the parameters' closed domain of definition is composed of piecewise smooth Jordan arcs or curves, are stated and proved. The results are used to derive a new simplified procedure for a multi-dimensional stability test, presented for the continuous-analog case, as well as for the discrete-digital case. Now proofs to Huang's and Strintzis' theorems on multidimensional stability are also provided, as an application of the theorems on zero sets.

A minimum principle for superharmonic functions subject to interface conditions

Let D be a bounded domain in R 2 with smooth boundary. Let B 1, …, B m be non-intersecting smooth Jordan curves contained in D, and let D′ denote the complement of ∪ i − 1 m B i respect to D. Suppose that u ϵ C 2(D′) ∩ C( D ̄ ) and Δu ⩽ 0 in D′ (where Δ is the Laplacian), while across each “interface” B i , i = 1,…, m, there is “continuity of flux” (as suggested by the theory of heat conduction). It is proved here that the presence of the interfaces does not alter the conclusions of the classical minimum principle (for Δu ⩽ 0 in D). The result is extended in several regards. Also it is applied to an elliptic free boundary problem and to the proof of uniqueness for steady-state heat conduction in a composite medium. Finally this minimum principle (which assumes “continuity of flux”) is compared with one due to Collatz and Werner which employs an alternative interface condition.

Indecomposable compact perturbations of the bilateral shift

Recent results of M. Radjabalipour and H. Radjavi assert that the sum of a normal operator N with spectrum on a smooth Jordan curve and a compact operator K in the Macaev ideal S ω {\mathfrak {S}_\omega } is decomposable provided the spectrum of N + K N + K does not fill the interior of the curve. Examples are given to show that this result cannot be essentially improved by taking K in a larger ideal.