Perturbations of vectorial coverings and systems of equations in metric spaces

E. R. Avakov, A. V. Arutyunov, S. E. Zhukovski?, and E. S. Zhukovski? studied the problem of Lipschitz perturbations of conditional coverings of metric spaces. Here we propose some extension of the concept of conditional covering to vector-valued mappings; i.e., the mappings acting in products of metric spaces. The idea is that, to describe a mapping, we replace the covering constant by the matrix of covering coefficients of the components of the vector-valued mapping with respect to the corresponding arguments. We obtain a statement on the preservation of the property of conditional and unconditional vectorial coverings under Lipschitz perturbations; the main assumption is that the spectral radius of the product of the covering matrix and the Lipschitz matrix is less than one. In the scalar case this assumption is equivalent to the traditional requirement that the covering constant be greater than the Lipschitz constant. The statement can be used to study various simultaneous equations. As applications we consider: some statements on the solvability of simultaneous operator equations of a particular form arising in the problems on n-fold coincidence points and n-fold fixed points; as well as some conditions for the existence of periodic solutions to a concrete implicit difference equation.     

A characteristic operator function for the class of n-hypercontractions

We consider a class of bounded linear operators on Hilbert space called n-hypercontractions which relates naturally to adjoint shift operators on certain vector-valued standard weighted Bergman spaces on the unit disc. In the context of n-hypercontractions in the class C0⋅ we introduce a counterpart to the so-called characteristic operator function for a contraction operator. This generalized characteristic operator function Wn,T is an operator-valued analytic function in the unit disc whose values are operators between two Hilbert spaces of defect type. Using an operator-valued function of the form Wn,T, we parametrize the wandering subspace for a general shift invariant subspace of the corresponding vector-valued standard weighted Bergman space. The operator-valued analytic function Wn,T is shown to act as a contractive multiplier from the Hardy space into the associated standard weighted Bergman space.     

Elliptic grids, rational functions, and the Padé interpolation

We consider rational functions with n prescribed poles for which there exists a divided difference operator transforming them to rational functions with n−1 poles. The poles of such functions are shown to lie on the elliptic grids. There is a one-to-one correspondence between this problem of admissible grids and the Poncelet problem on two quadrics. Additionally, we outline an explicit scheme of the Padé interpolation with prescribed poles and zeros on the elliptic grids. Dedicated to Richard Askey on the occasion of his seventieth birthday. 2000 Mathematics Subject Classification Primary—42C05; Secondary—39A13, 41A05, 41A21.     

矩形网格上的二元切触有理插值

二元切触有理插值是有理插值的一个重要内容,而降低其函数的次数和解决其函数的存在性是有理插值的一个重要问题.二元切触有理插值算法的可行性大都是有条件的,且计算复杂度较大,有理函数的次数较高.利用二元Hermite（埃米特）插值基函数的方法和二元多项式插值误差性质,构造出了一种二元切触有理插值算法并将其推广到向量值情形.较之其它算法,有理插值函数的次数和计算量较低.最后通过数值实例说明该算法的可行性是无条件的,且计算量低.     

Unconditional Schauder decompositions and stopping times in the Lebesgue-Bochner spaces

We extend Troitsky's study of martingales in Banach lattices to include stopping times. Results from the theory of unconditional Schauder decompositions and multipliers are used to derive an optional stopping theorem for unbounded stopping times. We also apply these techniques to convergent nets of stopped processes, as well as to unconditional Schauder decompositions in vector-valued Lp-spaces (1<p<∞).     

Rational inversion of the Laplace transform

This paper studies new inversion methods for the Laplace transform of vector-valued functions arising from a combination of A-stable rational approximation schemes to the exponential and the shift operator semigroup. Each inversion method is provided in the form of a (finite) linear combination of the Laplace transform of the function and a finite amount of its derivatives. Seven explicit methods arising from A-stable schemes are provided, such as the Backward Euler, RadauIIA, Crank-Nicolson, and Calahan scheme. The main result shows that, if a function has an analytic extension to a sector containing the nonnegative real line, then the error estimate for each method is uniform in time.     

Resolvent growth and Birkhoff-regularity

We prove a long standing conjecture in the theory of two-point boundary value problems that unconditional basisness implies Birkhoff-regularity. It is a corollary of our two main results: minimal resolvent growth along a sequence of points implies nonvanishing of a regularity determinant, and sparseness of nth-order roots of eigenvalues in small sectors provided that eigen and associated functions of the boundary value problem form an unconditional basis.Considerations are based on a new direct method, exploiting almost orthogonality of Birkhoff's solutions of the equation l(y)=λy. This property was discovered earlier by the author.     

Vector-valued multiparameter singular integrals and pseudodifferential operators

We consider multiparameter singular integrals and pseudodifferential operators acting on mixed-norm Bochner spaces Lp₁,…,pN(Rn₁×?×RnN;X) where X is a UMD Banach space satisfying Pisier's property (α). These geometric conditions are shown to be necessary. We obtain a vector-valued version of a result by R. Fefferman and Stein, also providing a new, inductive proof of the original scalar-valued theorem. Then we extend a result of Bourgain on singular integrals in UMD spaces with an unconditional basis to a multiparameter situation. Finally we carry over a result of Yamazaki on pseudodifferential operators to the Bochner space setting, improving the known vector-valued results even in the one-parameter case.     

Vector-valued,rational interpolants III

In 1963, Wynn proposed a method for rational interpolation of vector-valued quantities given on a set of distinct interpolation points. He used continued fractions, and generalized inverses for the reciprocals of vector-valued quantities. In this paper, we present an axiomatic approach to vector-valued rational interpolation. Uniquely defined interpolants are constructed for vector-valued data so that the components of the resulting vector-valued rational interpolant share a common denominator polynomial. An explicit determinantal formula is given for the denominator polynomial for the cases of (i) vector-valued rational interpolation on distinct real or complex points and (ii) vector-valued Padé approximation. We derive the connection with theε-algorithm of Wynn and Claessens, and we establish a five-term recurrence relation for the denominator polynomials.     

Elliptic boundary value problems with λ-dependent boundary conditions

In this paper second order elliptic boundary value problems on bounded domains Ω⊂Rn with boundary conditions on ∂Ω depending nonlinearly on the spectral parameter are investigated in an operator theoretic framework. For a general class of locally meromorphic functions in the boundary condition a solution operator of the boundary value problem is constructed with the help of a linearization procedure. In the special case of rational Nevanlinna or Riesz-Herglotz functions on the boundary the solution operator is obtained in an explicit form in the product Hilbert space L²(Ω)⊕(L²m(∂Ω)), which is a natural generalization of known results on λ-linear elliptic boundary value problems and λ-rational boundary value problems for ordinary second order differential equations.     

A de Montessus type convergence study of a least-squares vector-valued rational interpolation procedure

In a recent paper of the author [A. Sidi, A new approach to vector-valued rational interpolation, J. Approx. Theory 130 (2004) 177–187], three new interpolation procedures for vector-valued functions F(z), where F:C→C^N, were proposed, and some of their algebraic properties were studied. One of these procedures, denoted IMPE, was defined via the solution of a linear least-squares problem. In the present work, we concentrate on IMPE, and study its convergence properties when it is applied to meromorphic functions with simple poles and orthogonal vector residues. We prove de Montessus and Koenig type theorems when the points of interpolation are chosen appropriately.     

On a boundary value problem of N. I. Ionkin type

We study the spectral properties of a second-order differential operator with regular but not strongly regular boundary conditions. We show that the system of root functions of this operator contains infinitely many associated functions. We prove that a specially chosen system of root functions of this operator forms a basis in the space L _p (0, 1), 1 < p < ∞, which is unconditional for p = 2.     

Convoluted C-cosine functions and semigroups. Relations with ultradistribution and hyperfunction sines

Convoluted C-cosine functions and semigroups in a Banach space setting extending the classes of fractionally integrated C-cosine functions and semigroups are systematically analyzed. Structural properties of such operator families are obtained. Relations between convoluted C-cosine functions and analytic convoluted C-semigroups, introduced and investigated in this paper are given through the convoluted version of the abstract Weierstrass formula which is also proved in the paper. Ultradistribution and hyperfunction sines are connected with analytic convoluted semigroups and ultradistribution semigroups. Several examples of operators generating convoluted cosine functions, (analytic) convoluted semigroups as well as hyperfunction and ultradistribution sines illustrate the abstract approach of the authors. As an application, it is proved that the polyharmonic operator Δn², n∈N, acting on L²[0,π] with appropriate boundary conditions, generates an exponentially bounded Kn-convoluted cosine function, and consequently, an exponentially bounded analytic Kn+1-convoluted semigroup of angle , for suitable exponentially bounded kernels Kn and Kn+1.     

Boundary behavior in Hilbert spaces of vector-valued analytic functions

In this paper we study the boundary behavior of functions in Hilbert spaces of vector-valued analytic functions on the unit disc D. More specifically, we give operator-theoretic conditions on Mz, where Mz denotes the operator of multiplication by the identity function on D, that imply that all functions in the space have non-tangential limits a.e., at least on some subset of the boundary. The main part of the article concerns the extension of a theorem by Aleman, Richter and Sundberg in [A. Aleman, S. Richter, C. Sundberg, Analytic contractions and non-tangential limits, Trans. Amer. Math. Soc. 359 (2007)] to the case of vector-valued functions.     

A study on compactly supported orthogonal vector-valued wavelets and wavelet packets

In this paper, vector-valued multiresolution analysis and orthogonal vector-valued wavelets are introduced. The definition for orthogonal vector-valued wavelet packets is proposed. A necessary and sufficient condition on the existence of orthogonal vector-valued wavelets is derived by means of paraunitary vector filter bank theory. An algorithm for constructing a class of compactly supported orthogonal vector-valued wavelets is presented. The properties of the vector-valued wavelet packets are investigated by using operator theory and algebra theory. In particular, it is shown how to construct various orthonormal bases of L²(R, C^s) from the orthogonal vector-valued wavelet packets.     

Interpolation on the hypersphere with Thiele type rational interpolants

In order to interpolate 2n?+?1 points on the unit hypersphere $ \mathcal{S}^{d-1}$ with a vector-valued rational function, we use the Generalised Inverse Rational Interpolants (GIRI) of Graves?CMorris. The construction process of these Thiele type rational interpolants is based on the Samelson??s inverse for vectors. We show that in general any GIRI of 2n?+?1 points of $ \mathcal{S}^{d-1}$ lies on $ \mathcal{S}^{d-1}$ . We also show that the stereographic projection induces a one-to-one correspondence between the set of vector-valued rational functions lying on $ \mathcal{S}^{d-1}$ and the set of Generalised Inverse Rational Fractions in the equator plane.     

Homogeneous differential-operator equations in locally convex spaces

We describe a general method that allows us to find solutions to homogeneous differential-operator equations with variable coefficients by means of continuous vector-valued functions. The “homogeneity” is not interpreted as the triviality of the right-hand side of an equation. It is understood in the sense that the left-hand side of an equation is a homogeneous function with respect to operators appearing in that equation. Solutions are represented as functional vector-valued series which are uniformly convergent and generated by solutions to a kth order ordinary differential equation, by the roots of the characteristic polynomial and by elements of a locally convex space. We find sufficient conditions for the continuous dependence of the solution on a generating set. We also solve the Cauchy problem for the considered equations and specify conditions for the existence and the uniqueness of the solution. Moreover, under certain hypotheses we find the general solution to the considered equations. It is a function which yields any particular solution. The investigation is realized by means of characteristics of operators such as the order and the type of an operator, as well as operator characteristics of vectors, namely, the operator order and the operator type of a vector relative to an operator. We also use a convergence of operator series with respect to an equicontinuous bornology.     

A de Montessus type convergence study for a vector-valued rational interpolation procedure

In a recent paper of the author [8], three new interpolation procedures for vector-valued functions F(z), where F: ℂ → ℂ^N, were proposed, and some of their algebraic properties were studied. In the present work, we concentrate on one of these procedures, denoted IMMPE, and study its convergence properties when it is applied to meromorphic functions. We prove de Montessus and Koenig type theorems in the presence of simple poles when the points of interpolation are chosen appropriately. We also provide simple closed-form expressions for the error in case the function F(z) in question is itself a vector-valued rational function whose denominator polynomial has degree greater than that of the interpolant.     

Restricted domains,arrow-social welfare functions and noncorruptible and non-manipulable social choice correspondences: The case of private alternatives

An n-person social choice problem is considered in which the alternatives are n dimensional vectors, with the ith component of such a vector being the part of the alternatives affecting individual i alone. Assuming that individuals are selfish (individual i must be indifferent between any two alternatives with the same components), that they may be indifferent among alternatives and that each individual may choose his preferences out of a different set of permissible preferences, we prove that any set of restricted domains of preferences admits an n person non-dictatorial Arrow-type social welfare function if and only if it admits a two-person Arrow-type social welfare function: we characterize all the sets of restricted domains of preferences which admit two-person Arrow-type social welfare functions (and therefore also admit n-person Arrow-type social welfare functions) and then we prove that we also characterized all the sets of restricted domains of preferences which admit nondictatorial, nonmanipulable, noncorruptible and rational social choice correspondences.