Interpolation Problems for Certain Classes of Slice Hyperholomorphic Functions

A general interpolation problem (which includes as particular cases the Nevanlinna–Pick and Carathéodory–Fejér interpolation problems) is considered in two classes of slice hyperholomorphic functions of the unit ball of the quaternions. In the Hardy space of the unit ball we present a Beurling–Lax type parametrization of all solutions, and the formula for the minimal norm solution. In the class of functions slice hyperholomorphic in the unit ball and bounded by one in modulus there (that is, in the class of Schur functions in the present framework) we present a necessary and sufficient condition for the problem to have a solution, and describe the set of all solutions in the indeterminate case.     

Bloch,Besov and Dirichlet Spaces of Slice Hyperholomorphic Functions

In this paper we begin the study of some important Banach spaces of slice hyperholomorphic functions, namely the Bloch, Besov and weighted Bergman spaces, and we also consider the Dirichlet space, which is a Hilbert space. The importance of these spaces is well known, and thus their study in the framework of slice hyperholomorphic functions is relevant, especially in view of the fact that this class of functions has recently found several applications in operator theory and in Schur analysis. We also discuss the property of invariance of these function spaces with respect to Möbius maps by using a suitable notion of composition.     

Bi-Inner Dilations and Bi-Stable Passive Scattering Realizations of Schur Class Operator-Valued Functions

Let S(U; Y) be the class of all Schur functions (analytic contractive functions) whose values are bounded linear operators mapping one separable Hilbert space U into another separable Hilbert space Y , and which are defined on a domain , which is either the open unit disk or the open right half-plane . In the development of the Darlington method for passive linear time-invariant input/state/output systems (by Arov, Dewilde, Douglas and Helton) the following question arose: do there exist simple necessary and sufficient conditions under which a function has a bi-inner dilation mapping into ; here U ₁ and Y ₁ are two more separable Hilbert spaces, and the requirement that Θ is bi-inner means that Θ is analytic and contractive on Ω and has unitary nontangential limits a.e. on ∂Ω. There is an obvious well-known necessary condition: there must exist two functions and (namely and ) satisfying and for almost all . We prove that this necessary condition is also sufficient. Our proof is based on the following facts. 1) A solution ψ _r of the first factorization problem mentioned above exists if and only if the minimal optimal passive realization of θ is strongly stable. 2) A solution ψ _l of the second factorization problem exists if and only if the minimal *-optimal passive realization of θ is strongly co-stable (the adjoint is strongly stable). 3) The full problem has a solution if and only if the balanced minimal passive realization of θ is strongly bi-stable (both strongly stable and strongly co-stable). This result seems to be new even in the case where θ is scalar-valued.      

The Schur Transformation for Generalized Nevanlinna Functions: Interpolation and Self-Adjoint Operator Realizations

The Schur transformation for generalized Nevanlinna functions has been defined and applied in [2]. In this paper we discuss its relation to a basic interpolation problem and study its effect on the minimal self-adjoint operator (or relation) realization of a generalized Nevanlinna function. D. Alpay acknowledges with thanks the Earl Katz family for endowing the chair which supported this research and the Netherlands Organization for Scientific Research, NWO (grant B 61-524). The research of A. Dijksma and H. Langer was partly supported by the Center for Advanced Studies in Mathematics, CASM, of the Department of Mathematics of Ben-Gurion University, that of H. Langer also by the Austrian Science Fund, Project P15540-N05. Received: September 25, 2006. Accepted: October 11, 2006.     

Generalized Schur Functions and Related de Branges-Rovnyak Spaces in the Banach Space Setting

We study the analogues of de Branges-Rovnyak spaces in the Banach space case. An important role is played by self-adjoint operators from the dual of a Banach space into the Banach space itself. A factorization theorem for such operators is proved in the case when they have a finite number of negative squares.     

Equality of Schur and Skew Schur Functions

We determine the precise conditions under which any skew Schur function is equal to a Schur function over both infinitely and finitely many variables. Received May 29, 2004     

Schur Convexity for Two Classes of Symmetric Functions and Their Applications

For x =(x1, x2, ···, xn) ∈ Rn+∪ Rn-, the symmetric functions Fn(x, r) and Gn(x, r) are defined by r1 + xFij n(x, r) = Fn(x1, x2, ···, xn; r) =x1≤iij1i2···ir ≤n j=1and r1- xGij n(x, r) = Gn(x1, x2, ···, xn; r) =,x1≤i1i2···ir ≤n j=1ij respectively, where r = 1, 2, ···, n, and i1, i2, ···, in are positive integers. In this paper,the Schur convexity of Fn(x, r) and Gn(x, r) are discussed. As applications, by a bijective transformation of independent variable for a Schur convex function, the authors obtain Schur convexity for some other symmetric functions, which subsumes the main results in recent literature; and by use of the theory of majorization establish some inequalities. In particular, the authors derive from the results of this paper the Weierstrass inequalities and the Ky Fan's inequality, and give a generalization of Safta's conjecture in the n-dimensional space and others.     

Multiplicity Free Schur, Skew Schur, and Quasisymmetric Schur Functions

In this paper we classify all Schur functions and skew Schur functions that are multiplicity free when expanded in the basis of fundamental quasisymmetric functions, termed F-multiplicity free. Combinatorially, this is equivalent to classifying all skew shapes whose standard Young tableaux have distinct descent sets. We then generalize our setting, and classify all F-multiplicity free quasisymmetric Schur functions with one or two terms in the expansion, or one or two parts in the indexing composition. This identifies composition shapes such that all standard composition tableaux of that shape have distinct descent sets. We conclude by providing such a classification for quasisymmetric Schur function families, giving a classification of Schur functions that are in some sense almost F-multiplicity free.     

Lagrange Inversion and Schur Functions

Macdonald defined an involution on symmetric functions by considering the Lagrange inverse of the generating function of the complete homogeneous symmetric functions. The main result we prove in this note is that the images of skew Schur functions under this involution are either Schur positive or Schur negative symmetric functions. The proof relies on the combinatorics of Lagrange inversion. We also present a q-analogue of this result, which is related to the q-Lagrange inversion formula of Andrews, Garsia, and Gessel, as well as the operator of Bergeron and Garsia.     

The Schur Algorithm for Generalized Schur Functions II: Jordan Chains and Transformations of Characteristic Functions

 In the first paper of this series (Daniel Alpay, Tomas Azizov, Aad Dijksma, and Heinz Langer: The Schur algorithm for generalized Schur functions I: coisometric realizations, Operator Theory: Advances and Applications 129 (2001), pp. 1–36) it was shown that for a generalized Schur function s(z), which is the characteristic function of a coisometric colligation V with state space being a Pontryagin space, the Schur transformation corresponds to a finite-dimensional reduction of the state space, and a finite-dimensional perturbation and compression of its main operator. In the present paper we show that these formulas can be explained using simple relations between V and the colligation of the reciprocal s(z)⁻¹ of the characteristic function s(z) and general factorization results for characteristic functions. Received October 31, 2001; in revised form August 21, 2002 RID="a" ID="a" Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday     

Interpolation, Schur Functions and Moment Problems

No Abstract.  .     

完全对称函数的Schur凸性

定义了一完全对称函数并研究该称函数的Schur凸性,Schur乘性凸性及Schur调和凸性,作为应用探讨了与其相关的一些不等式.     

Kernel Decompositions for Schur Functions on the Polydisk

A certain kernel (sometimes called the Pick kernel) associated to Schur functions on the disk is always positive semi-definite. A generalization of this fact is well-known for Schur functions on the polydisk. In this article, we show that the “Pick kernel” on the polydisk has a great deal of structure beyond being positive semi-definite. It can always be split into two kernels possessing certain shift invariance properties.     

Hypergeometric Functions Related to Schur Q-Polynomials and the BKP Equation

We introduce hypergeometric functions related to projective Schur functions Q and describe their properties. Linear equations, integral representations, and Pfaffian representations are obtained. These hypergeometric functions are vacuum expectations of free fermion fields and are therefore tau functions of the so-called BKP hierarchy of integrable equations.     

Distributions and the Global Operator of Slice Monogenic Functions

In this paper we consider functions \(f\) defined on an open set \(U\) of the Euclidean space \(\mathbb{R }^{n+1}\) and with values in the Clifford Algebra \(\mathbb{R }_n\) . Slice monogenic functions \(f: U \subseteq \mathbb{R }^{n+1} \rightarrow \mathbb{R }_n\) belong to the kernel of the global differential operator with non constant coefficients given by \( \mathcal{G }=|{\underline{x}}|^2\frac{\partial }{\partial x_0} \ + \ {\underline{x}} \ \sum _{j=1}^n x_j\frac{\partial }{\partial x_j}. \) Since the operator \(\mathcal{G }\) is not elliptic and there is a degeneracy in \( {\underline{x}}=0\) , its kernel contains also less smooth functions that have to be interpreted as distributions. We study the distributional solutions of the differential equation \(\mathcal{G }F(x_0,{\underline{x}})=G(x_0,{\underline{x}})\) and some of its variations. In particular, we focus our attention on the solutions of the differential equation \( ({\underline{x}}\frac{\partial }{\partial x_0} \ - E)F(x_0,{\underline{x}})=G(x_0,{\underline{x}}), \) where \(E= \sum _{j=1}^n x_j\frac{\partial }{\partial x_j}\) is the Euler operator, from which we deduce properties of the solutions of the equation \( \mathcal{G }F(x_0,{\underline{x}})=G(x_0,{\underline{x}})\) .