On Factorization of Trigonometric Polynomials

We give a new proof of the operator version of the Fejér-Riesz Theorem using only ideas from elementary operator theory. As an outcome, an algorithm for computing the outer polynomials that appear in the Fejér-Riesz factorization is obtained. The extremal case, where the outer factorization is also *-outer, is examined in greater detail. The connection to Aglers model theory for families of operators is considered, and a set of families lying between the numerical radius contractions and ordinary contractions is introduced. The methods are also applied to the factorization of multivariate operator-valued trigonometric polynomials, where it is shown that the factorable polynomials are dense, and in particular, strictly positive polynomials are factorable. These results are used to give results about factorization of operator valued polynomials over , in terms of rational functions with fixed denominators.     

Strong convergence theorems for monotone mappings and relatively weak nonexpansive mappings

In this paper, we prove strong convergence theorems to a zero of monotone mapping and a fixed point of relatively weak nonexpansive mapping. Moreover, strong convergence theorems to a point which is a fixed point of relatively weak nonexpansive mapping and a solution of a certain variational problem are proved under appropriate conditions.     

Constructing three-dimensional mappings onto the unit sphere with the hypercomplex Szegö kernel

In classical complex analysis the Szegö kernel method provides an explicit way to construct conformal maps from a given simply-connected domain G⊂C onto the unit disc. In this paper we revisit this method in the three-dimensional case. We investigate whether it is possible to construct three-dimensional mappings from some elementary domains into the three-dimensional unit ball by using the hypercomplex Szegö kernel. In the cases of rectangular domains, L-shaped domains, cylinders and the symmetric double-cone the proposed method leads surprisingly to qualitatively very good results. In the case of the cylinder we get even better results than those obtained by the hypercomplex Bergman method that was very recently proposed by several authors.We round off with also giving an explicit example of a domain, namely the T-piece, where the method does not lead to the desired result. This shows that one has to adapt the methods in accordance with different classes of domains.     

Mathematical foundations of nonsmooth embedding methods

This paper establishes a mathematical foundation for application of the well known classical embedding approach to a class of nonsmooth functions, using a recently developed analogue of the derivative in cases where the functions involved fail to be differentiable in the usual sense. As part of this development we show how to obtain an extension to this case of the classical Hadamard theorem giving conditions for a map of ⁿ to itself to be a homeomorphism.The research reported here was sponsored by the National Science Foundation under Grant CCR-8801489, and by the Air Force Systems Command, USAF, under Grants AFOSR-88-0090 and AFOSR-89-0058. The US Government has certain rights in this material, and is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon.     

Strong convergence theorems for finitely many nonexpansive mappings

Under the assumption that E$E$ is a reflexive Banach space whose norm is uniformly Gêteaux differentiable and which has a weakly continuous duality mapping _Jφ$J_{\phi }$ with gauge function φ$\phi$, Ceng–Cubiotti–Yao [Strong convergence theorems for finitely many nonexpansive mappings and applications, Nonlinear Analysis 67 (2007) 1464–1473] introduced a new iterative scheme for a finite commuting family of nonexpansive mappings, and proved strong convergence theorems about this iteration. In this paper, only under the hypothesis that E$E$ is a reflexive Banach space which has a weakly continuous duality mapping _Jφ$J_{\phi }$ with gauge function φ$\phi$, and several control conditions about the iterative coefficient are removed, we present a short and simple proof of the above theorem.     

Conformal mappings onto prescribed regions via optimization techniques

Summary All well known extremal principles for conformal mappings of simply connected regionsR yield mappings onto disksD. It is shown here that given an arbitrary star shaped regionD as range a corresponding extremal principle is valid just by replacing the ordinary modulus in by a suitable positively homogeneous functional. If the star shaped regionD (bounded or not) is a convex polygon the extremal principle is equivalent to a linear (but infinite) programming problem, which can be solved approximately by passing to an ordinary (i.e. finite) linear programming problem. A numerical example whereR is a disk and the rangeD is an infinite strip is given.     

Viscosity approximation to common fixed points of families of nonexpansive mappings with generalized contractions mappings

Let X$X$ be a reflexive and smooth real Banach space which has a weakly sequentially continuous duality mapping. In this paper, we consider the following viscosity approximation scheme _xn+1=_λn+1f(_xn)+(1−_λn+1)_Tn+1xn$x_{n+1}={\lambda }_{n+1}f\left(x_n\right)+(1-{\lambda }_{n+1})T_{n+1}{x}_n$ (where f$f$ is a generalized contraction mapping) for infinitely many nonexpansive self-mappings _T1,_T2,_T3,…$T_1,T_2,T_3,\dots$ in X$X$. We establish a strong convergence result which generalizes some results in the literature.     

Trace and spectrum preserving linear mappings in Jordan-Banach algebras

In the first section we define the trace on the socle of a Jordan-Banach algebra in a purely spectral way and we prove that it satisfies several identities. In particular this trace defines the Faulkner bilinear form. In the second section, using analytic tools and the properties of the trace, we prove that a spectrum preserving linear mapping fromJ ontoJ ¹, whereJ andJ ¹ are semisimple Jordan-Banach algebras, is not far from being a Jordan isomorphism. It is in particular a Jordan isomorphism ifJ ¹ is primitive with non-zero socle.     

Fixed point theorems for the sum of two weakly sequentially continuous mappings

In the present paper, we prove some Krasnosel’skii-Leray-Schauder type fixed point theorems for weak topology. Some fixed point theorems for the sum of two weakly sequentially continuous mappings are also presented. Our results extend and improve on ones from several earlier works.     

Conjugate function method for numerical conformal mappings

We present a method for numerical computation of conformal mappings from simply or doubly connected domains onto so-called canonical domains, which in our case are rectangles or annuli. The method is based on conjugate harmonic functions and properties of quadrilaterals. Several numerical examples are given.     

A general iterative algorithm for nonexpansive mappings in Hilbert spaces

Let H be a real Hilbert space. Suppose that T is a nonexpansive mapping on H with a fixed point, f is a contraction on H with coefficient 0<α<1, and F:H→H is a k-Lipschitzian and η-strongly monotone operator with k>0,η>0. Let . We proved that the sequence {x_n} generated by the iterative method x_n+1=α_nγf(x_n)+(I−μα_nF)Tx_n converges strongly to a fixed point , which solves the variational inequality , for x∈F_ix(T).     

Strong convergence of an iterative method for nonexpansive mappings with new control conditions

The purpose of this paper is to study the convergence problem of an iterative method for nonexpansive mappings in Banach spaces under some new control conditions on parameters.     

On a Mann type implicit iteration process for continuous pseudo-contractive mappings

Let K$K$ be a nonempty closed convex subset of a Banach space E$E$, T:K→K$T:K\to K$ a continuous pseudo-contractive mapping. Suppose that {_αn}$\left\{{\alpha }_n\right\}$ is a real sequence in [0,1]$[0,1]$ satisfying appropriate conditions; then for arbitrary _x0∈K$x_0\in K$, the Mann type implicit iteration process {_xn}$\left\{x_n\right\}$ given by _xn=_αnxn−1+(1−_αn)T_xn,n≥0$x_n={\alpha }_n{x}_{n-1}+(1-{\alpha }_n)T{x}_n,n\ge 0$, strongly and weakly converges to a fixed point of T$T$, respectively.     

Compact embeddings and proper mappings

Sufficient conditions are given to assert that a C¹-proper mapping defined on a compact embedding between Banach spaces over K has a fixed point. The proof of the result is based upon continuation methods.     

A linear boundary value problem for weakly quasiregular mappings in space

Given a number a weakly L-quasiregular map on a domain in space is a map u in a Sobolev space that satisfies almost everywhere in In this paper, we study the problem concerning linear boundary values of weakly L-quasiregular mappings in space with dimension It turns out this problem depends on the power p of the underlying Sobolev space. For p not too far below the dimension n we show that a weakly quasiregular map in can only assume a quasiregular linear boundary value; however, for all and , we prove a rather surprising existence result that every linear map can be the boundary value of a weakly L-quasiregular map in Received July 20, 2000 / Accepted September 22, 2000 / Published online December 8, 2000     

Strong convergence theorems obtained by a generalized projections hybrid method for families of mappings in Banach spaces

Let C be a nonempty, closed and convex subset of a uniformly convex and smooth Banach space and let {T_n} be a family of mappings of C into itself such that the set of all common fixed points of {T_n} is nonempty. We consider a sequence {x_n} generated by the hybrid method by generalized projection in mathematical programming. We give conditions on {T_n} under which {x_n} converges strongly to a common fixed point of {T_n} and generalize the results given in [12], [14], [13] and [11].     

Bloch’s theorem for mappings of bounded and finite distortion

We study the covering properties of mappings of bounded and exponentially integrable distortion on the unit ball. We extend the results of Eremenko (Proc Am Math Soc 128:557–560, 2000) by proving Bloch-type theorems for mappings of exponentially integrable distortion. In the case of mappings of bounded distortion, we formulate and prove Bloch’s theorem in its most natural form. Research supported by the Academy of Finland, and by NSF grant DMS 0244421. Part of this research was done when the author was visiting at the University of Michigan. He wishes to thank the department for hospitality.     

Doubling measures,monotonicity, and quasiconformality

We construct quasiconformal mappings in Euclidean spaces by integration of a discontinuous kernel against doubling measures with suitable decay. The differentials of mappings that arise in this way satisfy an isotropic form of the doubling condition. We prove that this isotropic doubling condition is satisfied by the distance functions of certain fractal sets. Finally, we construct an isotropic doubling measure that is not absolutely continuous with respect to the Lebesgue measure. L.V.K. was supported by an NSF Young Investigator award under grant DMS 0601926. J.-M.W. was supported by the NSF grant DMS 0400810.