Shepard-Bernoulli operators

We introduce the Shepard-Bernoulli operator as a combination of the Shepard operator with a new univariate interpolation operator: the generalized Taylor polynomial. Some properties and the rate of convergence of the new combined operator are studied and compared with those given for classical combined Shepard operators. An application to the interpolation of discrete solutions of initial value problems is given.

     

Anisotropic weak Hardy spaces and interpolation theorems

In this paper, the authors establish the anisotropic weak Hardy spaces associated with very general discrete groups of dilations. Moreover, the atomic decomposition theorem of the anisotropic weak Hardy spaces is also given. As some applications of the above results, the authors prove some interpolation theorems and obtain the boundedness of the singular integral operators on these Hardy spaces.     

齐型空间上$L^1$ 与{\rm BMO}的内插空间

本文讨论齐型空间上$L^1$ 与{\rm BMO}的内插空间, 得到下列结果:对于本文讨论齐型空间上$L^1$ 与{\rm BMO}的内插空间, 得到下列结果:对于本文讨论齐型空间上$L^1$ 与{\rm BMO}的内插空间, 得到下列结果:对于摘要：本文讨论齐型空间上L^1与BMO的内插空间，得到下列结果：对于0〈θ〈1，1≤q≤∞，有（L^1，BMO）θ，q=Lpq，其中θ=1-1/p。     

Interpolation theorem for Marcinkiewicz spaces with applications to Lorentz gamma spaces

This paper is devoted to the interpolation principle between spaces of weak type. We characterise interpolation spaces between two Marcinkiewicz spaces in terms of Hardy type operators involving suprema. We study general properties of such operators and their behavior on Lorentz gamma spaces. A particular emphasis is placed on elementary and comprehensive proofs.     

Limit theorems of the theory of discrete periodic splines

It is known from the discrete harmonic analysis that the interpolation problem with equidistant interpolation points has a unique solution. If the right-hand sides in the interpolation problem are fixed, the spline depends on two parameters: the spline order and the number of points located between neighboring interpolation points. We find explicit expressions for the limits of interpolation spllines with respect to each parameter separately and show that both repeated limits exist. We also prove that these repeated limits are equal and their value is an interpolation trigonometric polynomial. Bibliography: 10 titles. Illustrations: 2 figures.     

Estimate of the norm of the Lagrange interpolation operator in a multidimensional Sobolev space

We obtain an estimate of the norm of the Lagrange interpolation operator in a multidimensional Sobolev space. It is shown that, under a suitable choice of the sequence of multi-indices, interpolation polynomials converge to the interpolated function and their rate of convergence is of the order of the best approximation of this function.     

Equations of optimal nonlinear filtration and interpolation for partially observed Markov processes

Recurrence relations are obtained for problems of optimal filtration and interpolation of partially observed discrete Markov chains. We present the system of differential equations for problems of optimal nonlinear filtration for Markov processes with continuous time and the system of inverse differential equations for problems of optimal nonlinear interpolation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 8, pp. 971–976, August, 1994.     

Interpolation of Classical Lorentz Spaces1

We describe the K-functional and identify the real interpolated spaces of general quasi–Banach couples of classical Lorentz spaces. Applications are given which include interpolation of spaces of Lorentz–Zygmund type.     

A Banach couple is determined by the collection of its interpolation spaces

We prove that an arbitrary Banach couple is uniquely determined by the collection of all its interpolation spaces, which extends a result by N. Aronszajn and E. Gagliardo.

     

Polynomial interpolation of cryptographic functions related to Diffie-Hellman and discrete logarithm problem

Recently, the first author introduced some cryptographic functions closely related to the Diffie-Hellman problem called P-Diffie-Hellman functions. We show that the existence of a low-degree polynomial representing a P-Diffie-Hellman function on a large set would lead to an efficient algorithm for solving the Diffie-Hellman problem. Motivated by this result we prove lower bounds on the degree of such interpolation polynomials. Analogously, we introduce a class of functions related to the discrete logarithm and show similar reduction and interpolation results.     

The correction operator for the canonical interpolation operator of the Adini element and the lower bounds of eigenvalues

In this paper,we develop a correction operator for the canonical interpolation operator of the Adini element.We use this new correction operator to analyze the discrete eigenvalues of the Adini element method for the fourth order elliptic eigenvalue problem in the three dimensions.We prove that the discrete eigenvalues are smaller than the exact ones.     

Some Interpolating Operators of de la Vallée-Poussin Type

We consider discrete versions of the de la Vallée-Poussin algebraic operator. We give a simple sufficient condition in order that such discrete operators interpolate, and in particular we study the case of the Bernstein-Szeg weights. Furthermore we obtain good error estimates in the cases of the sup-norm and L¹-norm, which are critical cases for the classical Lagrange interpolation.     

On the Interpolation by Discrete Splines with Equidistant Nodes

In this paper we consider equidistant discrete splines S(j), j , which may grow as O(|j|^s) as |j|→∞. Such splines are relevant for the purposes of digital signal processing. We give the definition of the discrete B-splines and describe their properties. Discrete splines are defined as linear combinations of shifts of the B-splines. We present a solution to the problem of discrete spline cardinal interpolation of the sequences of power growth and prove that the solution is unique within the class of discrete splines of a given order.     

Commutators of maximal functions on spaces of homogeneous type and their weighted,local versions

We obtain the characterizations of commutators of several versions of maximal functions on spaces of homogeneous type. In addition, with the aid of interpolation theory, we provide weighted version of the commutator theorems by establishing new characterizations of the weighted BMO space. Finally, a concrete example shows that the local version of commutators also has an independent interest.     

Interpolation properties of scales of Banach spaces

It is shown that any interpolation scales joining weight spaces L _p or similar spaces have many remarkable properties. Not only are such scales intrinsically interpolation scales, but an analog of the Arazy-Cwikel theorem describing interpolation spaces between the spaces from the scale is valid.     

Revisiting Landauʼs density theorems for Paley–Wiener spaces

We present a surprisingly simple approach to Landau?s density theorems for sampling and interpolation, which provides stronger versions of these results. In particular, we extend the interpolation theorem to unbounded spectra.     

The functional calculus of full operators with discrete spectrum

We construct the functional calculus for full operators with discrete spectrum over Banach spaces in the interpolation classes of symbols associated with given operators. We describe new classes of full operators in Banach spaces. Translated fromMatematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 1, 1998 pp. 127–135.     

一个无限可微函数类的最优Lagrange 插值

本文研究\,$[-1,1]$上的一个无限可微函数类$F_\infty$在空间$L_\infty[-1,1]$及加权空间$L_{p,\omega}[-1,1]$, $1\le p< \infty$ ($\omega$是$(-1,1)$上的非负连续可积函数)的最优Lagrange插值.我们证明了基于首项系数为1且于$L_{p,\omega}[-1,1]$上有最小范数的多项式零点的Lagrange插值对$1\le p< \infty$是最优的. 同时我们给出了当结点组包含端点时的最优结点组.     

Entropy function spaces and interpolation

We associate to every function space, and to every entropy function E, a scale of spaces Λp,q(E) similar to the classical Lorentz spaces Lp,q. Necessary and sufficient conditions for they to be normed spaces are proved, their role in real interpolation theory is analyzed, and a number of applications to functional and interpolation properties of several variants of Lorentz spaces and entropy spaces are given.