Finite Blaschke product interpolation on the closed unit disc

We show how to construct all finite Blaschke product solutions and the minimal scaled Blaschke product solution to the Nevanlinna-Pick interpolation problem in the open unit disc by solving eigenvalue problems of the interpolation data. Based on a result of Jones and Ruscheweyh we note that there always exists a finite Blaschke product of degree at most n−1 that maps n distinct points in the closed unit disc, of which at least one is on the unit circle, into n arbitrary points in the closed unit disc, provided that the points inside the unit circle form a positive semi-definite Pick matrix of full rank. Finally, we discuss a numerical limiting procedure.     

The boundary Carathéodory-Fejér interpolation problem

We give a new solvability criterion for the boundary Carathéodory-Fejér problem: given a point x∈R and, a finite set of target values, to construct a function f in the Pick class such that the first few derivatives of f take on the prescribed target values at x. We also derive a linear fractional parametrization of the set of solutions of the interpolation problem with real target values. The proofs are based on a reduction method due to Julia and Nevanlinna.     

Classes of tuples of commuting contractions satisfying the multivariable von Neumann inequality

We obtain a decomposition for multivariable Schur-class functions on the unit polydisk which, to a certain extent, is analogous to Agler's decomposition for functions from the Schur-Agler class. As a consequence, we show that d-tuples of commuting strict contractions obeying an additional positivity constraint satisfy the d-variable von Neumann inequality for an arbitrary operator-valued bounded analytic function on the polydisk. Also, this decomposition yields a necessary condition for solvability of the finite data Nevanlinna-Pick interpolation problem in the Schur class on the unit polydisk.     

Nevanlinna-Pick meromorphic interpolation: The degenerate case and minimal norm solutions

The Nevanlinna-Pick interpolation problem is studied in the class Sκ of meromorphic functions f with κ poles inside the unit disk D and with ‖f_‖L^∞(T)?1. In the indeterminate case, the parametrization of all solutions is given in terms of a family of linear fractional transformations with disjoint ranges. A necessary and sufficient condition for the problem being determinate is given in terms of the Pick matrix of the problem. The result is then applied to obtain necessary and sufficient conditions for the existence of a meromorphic function with a given pole multiplicity which satisfies Nevanlinna-Pick interpolation conditions and has the minimal possible L^∞-norm on the unit circle T.     

A Naimark Dilation Perspective of Nevanlinna-Pick Interpolation

In this paper a positive real tangential Nevanlinna-Pick interpolation problem with interpolation at operator points is solved. The Naimark dilation theorem together with the state space method from systems theory are used to obtain a parameterization for the set of all solutions. Explicit state space formulas are given for both the singular and non-ingular case. In the proofs the solution of an intermediate isometric extension problem plays an important role.     

Boundary Nevanlinna�CPick interpolation via reduction and augmentation

We give an elementary proof of Sarason??s solvability criterion for the Nevanlinna?CPick problem with boundary interpolation nodes and boundary target values. We also give a concrete parametrization of all solutions of such a problem. The proofs are based on a reduction method due to Julia and Nevanlinna. Reduction of functions corresponds to Schur complementation of the corresponding Pick matrices.     

A two-dimensional inverse problem for the viscoelasticity equation

For the integrodifferential equation that corresponds to the two-dimensional viscoelasticity problem, we study the problem of determining the density, the elasticity coefficient, and the spaceintegral term in the equation. We assume that the sought functions differ from the given constants only inside the unit disk D = {x ∈ ?² | |x| < 1}. As information for solving this inverse problem, we consider the one-parameter family of solutions to the integrodifferential equation corresponding to impulse sources localized on straight lines and, on the boundary of D, there are defined the traces of the solutions for some finite time interval. It is shown that the use of a comparatively small part of the given information about the kinematics and the elements of dynamics of the propagating waves makes it possible to reduce the problem under consideration to three consecutively and uniquely solvable inverse problems that together give a solution to the initial inverse problem.     

On rank invariance of generalized Schwarz-Pick-Potapov block matrices of matrix-valued Carathéodory functions

In view of a multiple Nevanlinna-Pick interpolation problem, we study the rank of generalized Schwarz-Pick-Potapov block matrices of matrix-valued Carathéodory functions. Those matrices are determined by the values of a Carathéodory function and the values of its derivatives up to a certain order. We derive statements on rank invariance of such generalized Schwarz-Pick-Potapov block matrices. These results are applied to describe the case of exactly one solution for the finite multiple Nevanlinna-Pick interpolation problem and to discuss matrix-valued Carathéodory functions with the highest degree of degeneracy.     

双解析函数的一般复合边值问题关于边界曲线的稳定性

开口弧段Γ上的双解析函数的Riemann边值问题与单位圆周L上双解析函数的Hilbert边值问题复合而成的一般复合边值问题,当L与Γ发生微小的光滑摄动后,借助于推广的拉甫伦捷夫近似于圆的共形映射,将星形域映为单位圆域,从而得出摄动后的问题的解的表达式,同时讨论了解的稳定性情况,并给出误差估计.     

Minimum w-entropy interpolants for matricial Carathéodory functions and maximum determinant completions of associated block Pick matrix

The so-called modified block Toeplitz vector approach is used to connect a class of particular solutions G_w for w∈D of a nondegenerate interpolation problem of the Nevanlinna-Pick type with a class of particular solutions F_w of a certain matricial Carathéodory coefficient problem in a transparent way. This will suggest a simple approach to the minimum w-entropy interpolants and the maximum determinant completions of the associated block Pick matrix within the framework of that Nevanlinna-Pick type interpolation problem by using the known assertions corresponding to F_w. It turns out that G_w(w∈D) is exactly or provides us with the unique solution to these two extremal problems in a manner.     

Nevanlinna-Pick Interpolation for {\mathbb{C}} + BH^\infty

We study the Nevanlinna-Pick problem for a class of subalgebras of H ^∞. This class includes algebras of analytic functions on embedded disks, the algebras of finite codimension in H ^∞ and the algebra of bounded analytic functions on a multiply connected domain. Our approach uses a distance formula that generalizes Sarason’s [23] work. We also investigate the difference between scalar-valued and matrix-valued interpolation through the use of C ^*-envelopes. This research was partially supported by the NSF grant DMS 0300128. This research was completed as part of my Ph.D. dissertation at the University of Houston.     

Sobolev Trace Theorem and the Dirichlet Problem in the Unit Disk

We give a direct and elementary proof for the trace theorem in L _p -based Sobolev spaces, when the domain is the unit disk. We also consider the Dirichlet boundary problem for the Laplace equation, where the boundary value is a function in the Besov space. The Poisson kernel enables us to solve this problem in the unit disk more easily than in a general domain.     

The impact of finite precision arithmetic and sensitivity on the numerical solution of partial differential equations

In this paper, we address some fundamental issues concerning “time marching” numerical schemes for computing steady state solutions of boundary value problems for nonlinear partial differential equations. Simple examples are used to illustrate that even theoretically convergent schemes can produce numerical steady state solutions that do not correspond to steady state solutions of the boundary value problem. This phenomenon must be considered in any computational study of nonunique solutions to partial differential equations that govern physical systems such as fluid flows. In particular, numerical calculations have been used to “suggest” that certain Euler equations do not have a unique solution. For Burgers' equation on a finite spatial interval with Neumann boundary conditions the only steady state solutions are constant (in space) functions. Moreover, according to recent theoretical results, for any initial condition the corresponding solution to Burgers' equation must converge to a constant as t → ∞. However, we present a convergent finite difference scheme that produces false nonconstant numerical steady state “solutions.” These erroneous solutions arise out of the necessary finite floating point arithmetic inherent in every digital computer. We suggest the resulting numerical steady state solution may be viewed as a solution to a “nearby” boundary value problem with high sensitivity to changes in the boundary conditions. Finally, we close with some comments on the relevance of this paper to some recent “numerical based proofs” of the existence of nonunique solutions to Euler equations and to aerodynamic design.     

n-sided polygonal hybrid finite elements with unified fundamental solution kernels for topology optimization

In topology optimization, the optimized design can be obtained based on spatial discretization of design domain using natural polygonal finite elements to reduce the influence of mesh geometry on topology optimization solutions. However, the natural polygonal finite elements require separate interpolants for each type of elements and involve troublesome domain integrals. In this study, an alternative n-sided polygonal hybrid finite element possessing multiple-node connection is formulated in a unified form to compress the checkerboard patterns caused by numerical instability in topology optimization. Different from the natural polygonal finite elements, the present polygonal hybrid finite elements involve two sets of independent displacement fields. The intra-element displacement field defined inside the element is approximated by the linear combination of the fundamental solution of the problem to achieve the purpose of the local satisfaction of the governing equations of the problem, but not the specific boundary conditions and the inter-element continuity conditions. To overcome such drawback, the inter-element displacement field defined over the entire element boundary is independently approximated by means of the conventional shape function interpolation. As a result, only line integrals along the element boundary are involved in the computation, whose dimension is reduced by one compared to the domain integrals in the natural polygonal finite elements, and more importantly, allowing us to flexibly construct any polygons from Voronoi tessellations in discretizing complex design domains using same fundamental solution kernels. Numerical results obtained indicate that the present n-sided polygonal hybrid finite elements can produce more accurate displacement solutions and smaller mean compliance, compared to the standard finite elements and the natural polygonal finite elements.     

Multiple solutions for a nonlinear and non-homogeneous problem in Orlicz-Sobolev spaces

We study a non-homogeneous boundary value problem in a smooth bounded domain in R^N. We prove the existence of at least two non-negative and non-trivial weak solutions. Our approach relies on Orlicz-Sobolev spaces theory combined with adequate variational methods and a variant of Mountain Pass Lemma.     

Nonlinear fourth-order elliptic equations with nonlocal boundary conditions

This paper is concerned with a class of fourth-order nonlinear elliptic equations with nonlocal boundary conditions, including a multi-point boundary condition in a bounded domain of Rn. Also considered is a second-order elliptic equation with nonlocal boundary condition, and the usual multi-point boundary problem in ordinary differential equations. The aim of the paper is to show the existence of maximal and minimal solutions, the uniqueness of a positive solution, and the method of construction for these solutions. Our approach to the above problems is by the method of upper and lower solutions and its associated monotone iterations. The monotone iterative schemes can be developed into computational algorithms for numerical solutions of the problem by either the finite difference method or the finite element method.     

Tangential Nevanlinna-Pick Interpolation and Its Connection with Hamburger Matrix Moment Problem

This paper is concerned with the solution of a certain tangential Nevanlinna-Pick interpolation for Nevanlinna functions. We use the so-called block Hankel vector method to establish two intrinsic connections between the tangential Nevanlinna-Pick interpolation in the Nevanlinna class and the truncated Hamburger matrix moment problem associated with the block Hankel vector under consideration: one is a congruent relationship between their information matrices, and the other is a divisor-remainder connection between their solutions. These investigations generalize our previous work on the Nevanlinna-Pick interpolation and power matrix moment problem.     

Convex interval interpolation with cubic splines,II

The problem of convex interval interpolation with cubicC ¹-splines has an infinite number of solutions, if it is solvable at all. For selecting one of the solutions a regularized mean curvature is minimized. The arising finite dimensional constrained program is solved numerically by means of a dualization approach.Dedicated to Professor Julius Albrecht on the occasion of his 65th birthday.     

Inhomogeneous Calderón reproducing formula on spaces of homogeneous type

In this paaper we use the Calderón-Zygmund operator theory to prove an inhomogenous Calderón reproducing formula on spaces of homogeneous type with finite or infinite measures. Our formula is new even for classical spaces of homogeneous type such as the surface of the unit ball and then-torus inR ⁿ, compact Lie groups,C ^∞-compact Riemannian manifolds, and the boundary of any bounded Lipschitz domain inR ⁿ.