Rational matrix functions associated with the Nevanlinna-Pick problem in the class <Emphasis Type="Italic">S</Emphasis>[<Emphasis Type="Italic">a,b</Emphasis>] and orthogonal on a compact interval

We consider the interpolation Nevanlinna-Pick problem with infinitely many interpolation nodes in the class S[a, b] and rational matrix functions associated with this problem and orthogonal on the segment [a, b]. We obtain a criterion of complete indeterminacy of the Nevanlinna-Pick problem in terms of orthogonal rational matrix functions. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 6, pp. 764–770, June, 2007.     

Error estimates for 3- narrow finite elements

We obtain error estimates for finite element approximations of the lowest degree valid uniformly for a class of three-dimensional narrow elements. First, for the Lagrange interpolation we prove optimal error estimates, both in order and regularity, in for . For it is known that this result is not true. Applying extrapolation results we obtain an optimal order error estimate for functions sligthly more regular than . These results are valid both for tetrahedral and rectangular elements. Second, for the case of rectangular elements, we obtain optimal, in order and regularity, error estimates for an average interpolation valid for functions in with and .

     

and interpolation between Banach lattices

We study the possibility of obtaining the -norm by an interpolation method starting from a couple of Banach lattice norms. We describe all couples of Banach lattice norms in such that the -norm is a strict interpolation norm between them. Further we consider the possibility of obtaining the -norm by any method which guarantees interpolation of not only linear operators (= bilinear forms on but also of all polylinear forms. Here we show that either one of the initial norms has to be proportional to the -norm, or both have to be weighted -norms.

     

Free Interpolation in the Spaces of Analytic Functions With Derivative of Order <Emphasis Type="Italic">s</Emphasis> From the Hardy Space

We consider the problem of free interpolation for the spaces of analytic functions with derivative of order s in the Hardy space H^p. For the sets that satisfy the Stolz condition, we obtain a condition necessary for interpolation: if 1 ≤ p < ∞, then the set must be a union of s sparse sets. For p = ∞, we obtain a necessary and sufficient condition for interpolation: the set must be a union of s + 1 sparse sets. In this case, we construct an extension operator. Bibliography 11 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 303, 2003, pp. 169–202.     

Weak interpolation in extensions of the logics <Emphasis Type="Italic">S</Emphasis><Subscript>4</Subscript> and <Emphasis Type="Italic">K</Emphasis><Subscript>4</Subscript>

Conditions are specified which are necessary and sufficient for a logic over K4 to possess the weak interpolation property. For this goal to be met, simple transitive modal algebras are described, and we establish a criterion for the class of such algebras to be amalgamable. For extensions of K4, the weak interpolation property is proved decidable. Supported by RFBR (project No. 06-01-00358) and by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (grant NSh-335.2008.1). __________ Translated from Algebra i Logika, Vol. 47, No. 6, pp. 705–722, November–December, 2008.     

Optimal individual stability estimates for -semigroups in Banach spaces

In a previous paper we proved that the asymptotic behavior of a -semigroup is completely determined by growth properties of the resolvent of its generator and geometric properties of the underlying Banach space as described by its Fourier type. The given estimates turned out to be optimal. The method of proof uses complex interpolation theory and reflects the full semigroup structure. In the present paper we show that these uniform estimates have to be replaced by weaker ones, if individual initial value problems and local resolvents are considered because the full semigroup structure is lacking. In a different approach this problem has also been studied by Huang and van Neerven, and a part of our straightforward estimates can be inferred from their results. We mainly stress upon the surprising fact that these estimates turn out to be optimal. Therefore it is not possible to obtain the optimal uniform estimates mentioned above from individual ones. Concerning Hardy-abscissas, individual orbits and their local resolvents behave as badly as general vector valued functions and their Laplace-transforms. This is in strict contrast to the uniform situation of a -semigroup itself and the resolvent of its generator where a simple dichotomy holds true.

     

Absence of the interpolation property in the calculi <Emphasis Type="Italic">Lα</Emphasis> and <Emphasis Type="Italic">Lf</Emphasis>

We prove the absence of the Craig interpolation property in the polymodal calculi related to some Ershov spaces.     

Interpolation of Ideal Measures by Abstract K and J Spaces

We work with the abstract K and J interpolation method generated by a sequence lattice Г. We investigate the deviation of an interpolated operator from a given operator ideal by establishing formulae for the ideal measure of the interpolated operator in terms of the ideal measures of restrictions of the operator. Formulae are given in terms of the norms of the shift operators on Г.     

Interpolation spaces between<Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript> and<Emphasis Type="Italic">L</Emphasis><Superscript>∞</Superscript> on spaces of homogeneous type

Using the maximal function characterization of Hardy spaces, we study the interpolation spaces be-tween H1 and L∞ on spaces of homogeneous type.     

Trianguläre interpolation

Triangular interpolation problems (problems with biorthogonal polynomial sequence) are studied. Convergence of the corresponding interpolation series is investigated. It is shown that the interpolation polynomials form a basis in appropriate nuclear Fréchet spaces. The results are applied to the interpolation of weighted remainders and the Abel-Gončarov interpolation problem.      

矩形域上分形插值研究

该文给出了矩形域上分形插值数学模型, 分形插值曲面的计算公式, 证明了分形插值曲面迭代函数系唯一性定理, 导出了分形插值曲面的维数定理,并应用实际数据进行了分形插值曲面的实例研究. 为工程中长期寻求的粗糙表面模拟提供了理论基础和实用方法.     

On multivariate Hermite interpolation

We study the problem of Hermite interpolation by polynomials in several variables. A very general definition of Hermite interpolation is adopted which consists of interpolation of consecutive chains of directional derivatives. We discuss the structure and some aspects of poisedness of the Hermite interpolation problem; using the notion of blockwise structure which we introduced in [10], we establish an interpolation formula analogous to that of Newton in one variable and use it to derive an integral remainder formula for a regular Hermite interpolation problem. For Hermite interpolation of degreen of a functionf, the remainder formula is a sum of integrals of certain (n + 1)st directional derivatives off multiplied by simplex spline functions.     

Interpolation Functors In Weak-Type Interpolation

For interpolation in the diagonal case, i.e. with respect to the two couples (X, X) and (Y, Y), there exists a natural relation between weak-type and strong-type interpolation. Indeed, weak-type interpolation is related to the “M-couples” (ΛX, MX) and (ΛY, MY) of the Lorentz spaces of X and Y. Since ΛZ ? MZ for any space Z, any weak-type interpolation space also has the (strong-type) interpolation property for the “Λ-couples” (ΛX, ΛX) and (ΛY, ΛY). In this paper a scale ${cal G}, c>0$ , of interpolation functors with respect to the Λ-couples is introduced such that all generated interpolation spaces (also) have the weak-type interpolation property. Moreover, we will show that a space is a weak interpolation space if and only if it is generated by one of these functors.     

Block based Newton-like blending osculatory rational interpolation

With Newton’s interpolating formula, we construct a kind of block based Newton-like blending osculatory interpolation.The interpolation provides us many flexible interpolation schemes for choices which include the expansive Newton’s polynomial interpolation as its special case. A bivariate analogy is also discussed and numerical examples are given to show the effectiveness of the interpolation.     

The scalar Nevanlinna-Pick interpolation problem with boundary conditions

We show that if the Nevanlinna-Pick interpolation problem is solvable by a function mapping into a compact subset of the unit disc, then the problem remains solvable with the addition of any number of boundary interpolation conditions, provided the boundary interpolation values have modulus less than unity. We give new, inductive proofs of the Nevanlinna-Pick interpolation problem with any finite number of interpolation points in the interior and on the boundary of the domain of interpolation (the right half plane or unit disc), with function values and any finite number of derivatives specified. Our solutions are analytic on the closure of the domain of interpolation. Our proofs only require a minimum of matrix theory and operator theory. We also give new, straightforward algorithms for obtaining minimal H_∞ norm solutions. Finally, some numerical examples are given.     

Multivariate Interpolation Using Carlitz's Inversion Formulas

Multivariate rational exponential Lagrange interpolation formulas, Hermite interpolation formulas, and Hermite–Fejér interpolation formulas of the Newton type are established by using Carlitz's inversion formulas. The recurrence relation for constructing Lagrange interpolation is also given. In addition, by setting q1 in the obtained formulas, we obtain the corresponding polynomial interpolation formulas with combinatorial form.     

On general Hermite trigonometric interpolation

Summary A sequence of general Hermite trigonometric interpolation polynomials with equidistant interpolation points is given. Integrating these interpolation formulae a sequence of quadrature formulae for the integration of periodic functions is obtained. Derivative-free remainders are stated for these interpolation and quadrature formulae.This work was done at the Max-Planck-Institut für Physik und Astrophysik, München.     

Interpolation properties for provability logics GL and GLP

We study interpolation properties of provability logics. We prove the Lyndon interpolation for GL and the uniform interpolation for GLP.     

On interpolation families of wavelet sets

The question of which groups are isomorphic to groups of interpolation maps for interpolation families of wavelet sets was raised by Dai and Larson. In this article it is shown that any finite group is isomorphic to a group of interpolation maps for some interpolation family of wavelet sets.

     

About measures and nodal systems for which the Hermite interpolants uniformly converge to continuous functions on the circle and interval

We obtain the Laurent polynomial of Hermite interpolation on the unit circle for nodal systems more general than those formed by the n-roots of complex numbers with modulus one. Under suitable assumptions for the nodal system, that is, when it is constituted by the zeros of para-orthogonal polynomials with respect to appropriate measures or when it satisfies certain properties, we prove the convergence of the polynomial of Hermite-Fejér interpolation for continuous functions. Moreover, we also study the general Hermite interpolation problem on the unit circle and we obtain a sufficient condition on the interpolation conditions for the derivatives, in order to have uniform convergence for continuous functions.Finally, we obtain some improvements on the Hermite interpolation problems on the interval and for the Hermite trigonometric interpolation.