Scattered hermite interpolation by ridge functions

In this paper we study the problem of Hermite interpolation on scattered data using ridge functions, and give a wide class of ridge functions that may be used in practice to interpolate Hermite data.     

On the Nevanlinna-Pick interpolation problem for generalized stieltjes functions

The solutions of the Nevanlinna-Pick interpolation problem for generalized Stieltjes matrix functions are parametrized via a fractional linear transformation over a subset of the class of classical Stieltjes functions. The fractional linear transformation of some of these functions may have a pole in one or more of the interpolation points, hence not all Stieltjes functions can serve as a parameter. The set of excluded parameters is characterized in terms of the two related Pick matrices.Dedicated to the memory of M. G. Kreîn     

Step-by-step solving of ordered interpolation problem for stieltjes functions

We investigate a step-by-step solving of ordered generalized interpolation problems for Stieltjes matrix functions and obtain a multiplicative representation for the sequence of resolvent matrices. Thematrix factors inmultiplicative representations of the resolventmatrices are expressed through the Schur–Stieltjes parameters, for which we obtain explicit formulas and give an algorithm of step-by-step solving of Stieltjes type interpolation problems. As examples, we consider step-by-step solutions of the Stieltjes matrix moment problem and the problems by Nevanlinna–Pick and Caratheodory.     

Gleason's problem and tangential homogeneous interpolation for hyperholomorphic quaternionic functions

We study a version of Gleason's problem in the setting of functions of class C¹ in the unit ball of ?². We use the setting of hyperholomorphic functions to define and solve the problem. Finally, we briefly discuss a tangential interpolation problem for hyperholomorphic functions.     

The inverse interpolation problem for operators

The inverse problem for a real interpolation functor in the class of ideal spaces is discussed. The most complete solution is obtained for consistent interpolation couples. It follows from the theorems proved in this article and from the concepts introduced that the Marcinkiewicz theorem on interpolation of weak type operators cannot essentially be strengthened.Translated fromMatematicheskie Zametki, Vol. 59, No. 3, pp. 323–333, May, 1996.     

Indefinite metric in Schur's interpolation problem for analytic functions. VI

The first five parts have been published in issues 37, 38, 41, 42, and 45 of this collection.     

Indefinite metric in Schur's interpolation problem for analytic functions. V

A theorem on the factorization of the radii of the Weyl limit circle, arising in the Schur interpolation problem, is proved by the methods of the J-theory.The first four parts have been published in the issues 37, 38, 41, and 42 of this collection.Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 45, pp. 16–26, 1986.     

The Appell interpolation problem

A general linear interpolation problem is considered. We will call it the Appell interpolation problem because the solution can be expressed by a basis of Appell polynomials. Some classical and non-classical examples are also considered. Finally, numerical calculations are given.     

The Appell interpolation problem

A general linear interpolation problem is considered. We will call it the Appell interpolation problem because the solution can be expressed by a basis of Appell polynomials. Some classical and non-classical examples are also considered. Finally, numerical calculations are given.     

The least solution for the polynomial interpolation problem

Supported by the National Science Foundation under Grant No. DMS-8701275     

The exact error of trigonometric interpolation for differentiable functions

In [1], G. Halász gives some properties of the order of trigonometric approximation as a function of the Lipschitz parameter. Here we show that these properties completely characterize this function.     

The fundamentality of sets of ridge functions

Summary We investigate the fundamentality of the set of all continuous ridge functions in the spaceC( ⁿ ) as well as inC(X) for a general Banach space,X. Both positive and negative results are obtained. Necessary and sufficient conditions for the fundamentality are given for certain sets of ridge functions inC( ⁿ ).     

J-Inner matrix functions, interpolation and inverse problems for canonical systems, II: The inverse monodromy problem

This is the second of a planned sequence of papers on inverse problems for canonical systems of differential equations. It is devoted to the inverse monodromy problem for canonical integral and differential systems. In this part, which focuses on the case of a diagonal signature matrixJ, a parametrization is obtained of the set of all solutionsM (t) for the inverse problem for integral systems in terms of two chains of entire matrix valued inner functions. Special classes of solutions correspond to special choices of these chains. This theme will be elaborated upon further in a third part of this paper which will be published in a subsequent issue of this journal. There the emphasis will be on symmetries and growth conditions all of which serve to specify or restrict the chains alluded to above, from the outside, so to speak.