On One class of Hermite projectors

We conjecture that every ideal projector on \({\mathbb {C}}\left[ x_1,\ldots ,x_d\right] \) whose kernel is generated by precisely d polynomials is Hermite (i.e., the limit of Lagrange interpolation projectors). We validate this conjecture in case the d generators of the kernel have no roots at infinity.     

On the Limits of Lagrange Projectors

This article addresses a question of Carl de Boor (In: Constructive theory of functions, Varna 2005, pp. 51–63, Marin Drinov Academic, Sofia, [2006]): What ideal projectors are the limits of Lagrange projectors? The results of this paper answer the question in the sense that for every ideal projector P, we prescribe finitely many computations that determine whether the projector P is a limit of Lagrange projectors.     

Konvergenz in verallgemeinerten topologischen Räumen

Spline – projectors in BANACH spaces. A class of BANACH spaces is studied in which it is possible to construct explicitly spline – projectors. These projectors do not depend on the special HILBERT structure that is used for construction; therefore they are unique determined.     

Representation of projectors involving Minkowski inverse in Minkowski space

A certain class of results about the different representations of Oblique projectors is present in the literature. These results represent Oblique projectors as the functions of orthogonal projectors with given onto and along spaces. But these results are valid under the restriction that the functions of orthogonal projectors involved are invertible. In this paper we extend and generalize these results. The extension lies in making a transition from Euclidean space to Minkowski space M and the generalization is obtain by voiding the invertibility condition and use of the Minkowski inverse. Furthermore, the nobility lies in utilizing the m-projectors instead of the regular orthogonal projectors.     

On simultaneous block-diagonalization of cyclic sequences of commuting matrices

We study simultaneous block-diagonalization of cyclic d-tuples of commuting matrices. Some application to ideal projectors are also presented. In particular, we extend Hans Stetter's theorem characterizing Lagrange projectors.     

On the pointwise limits of bivariate lagrange projectors

A linear algebra proof is given of the fact that the nullspace of a finite-rank linear projector, on polynomials in two complex variables, is an ideal if and only if the projector is the bounded pointwise limit of Lagrange projectors, i.e., projectors whose nullspace is a radical ideal, i.e., the set of all polynomials that vanish on a certain given finite set. A characterization of such projectors is also given in the real case. More generally, a characterization is given of those finite-rank linear projectors, on polynomials in d complex variables, with nullspace an ideal that are the bounded pointwise limit of Lagrange projectors. The characterization is in terms of a certain sequence of d commuting linear maps and so focuses attention on the algebra generated by such sequences.     

Spline interpolation at knot averages

It is well known that when interpolation points coincide with knots, the knot sequence must obey some restriction in order to guarantee the existence and boundedness of the interpolation projector. But, when the interpolation points are chosen to be the knot averages, the corresponding quadratic or cubic spline interpolation projectors are bounded independently of the knot sequence. Based on this fact, de Boor in 1975 made a conjecture that interpolation by splines of orderk at knot averages is bounded for anyk. In this paper we disprove de Boor's conjecture fork 20.Communicated by Wolfgang Dahmen.     

On spaces of fractal functions

In this paper we Ointroduce linear-spaces consisting of continuous functions whose graphs are the attactors of a special class of iterated function systems. We show that such spaces are finite dimensional and give the bases of these spaces in an implicit way. Given such a space, we discuss how to obtain a set of knots for which the Lagrange interpolation problem by the space is uniquely solvable.     

Haar expansions of a class of fractal interpolation functions and their logical derivatives

In this paper, we study a special class of fractal interpolation functions, and give their Haar-wavelet expansions. On the basis of the expansions, we investigate the Hölder smoothness of such funstions and their logical derivatives of order α.     

Convergence Analysis of Processes with Valiant Projection Operators in Hilbert Space

Convex feasibility problems require to find a point in the intersection of a finite family of convex sets. We propose to solve such problems by performing set-enlargements and applying a new kind of projection operators called valiant projectors. A valiant projector onto a convex set implements a special relaxation strategy, proposed by Goffin in 1971, that dictates the move toward the projection according to the distance from the set. Contrary to past realizations of this strategy, our valiant projection operator implements the strategy in a continuous fashion. We study properties of valiant projectors and prove convergence of our new valiant projections method. These results include as a special case and extend the 1985 automatic relaxation method of Censor.     

Some properties of fractal interpolation functions on Sierpinski gasket

In this paper, we discuss some basic properties of uniform fractal interpolation functions (FIFs), which is a special class of FIFs, on Sierpinski gasket. We firstly study the min-max property of uniform FIFs. Then we present a necessary and sufficient condition such that uniform FIFs have finite energy. Normal derivative and Laplacian of uniform FIFs are also discussed.     

A Lie theoretic categorification of the coloured Jones polynomial

We use the machinery of categorified Jones-Wenzl projectors to construct a categorification of a type A Reshetikhin-Turaev invariant of oriented framed tangles where each strand is labelled by an arbitrary finite-dimensional representation. As a special case, we obtain a categorification of the coloured Jones polynomial of links.     

A note on generalized and hypergeneralized projectors

Groß and Trenkler [Generalized and hypergeneralized projectors, Linear Algebra Appl. 264 (1997) 463–474] have introduced two generalizations of orthogonal projectors called generalized projectors and hypergeneralized projectors. In this note we characterize these generalizations by their spectral decompositions.     

Orthogonal Helmholtz decomposition in arbitrary dimension using divergence-free and curl-free wavelets

We present tensor-product divergence-free and curl-free wavelets, and define associated projectors. These projectors enable the construction of an iterative algorithm to compute the Helmholtz decomposition of any vector field, in wavelet domain. This decomposition is localized in space, in contrast to the Helmholtz decomposition calculated by Fourier transform. Then we prove the convergence of the algorithm in dimension two for any kind of wavelets, and in larger dimension for the particular case of Shannon wavelets. We also present a modification of the algorithm by using quasi-isotropic divergence-free and curl-free wavelets. Finally, numerical tests show the validity of this approach for a large class of wavelets.     

On a Conjecture of Carl de Boor Regarding the Limits of Lagrange Interpolants

The purpose of this paper is to provide a counterexample to a conjecture of Carl de Boor [2], that every ideal projector is a limit of Lagrange projectors. The counterexample is based on a construction of A. Iarrobino [9] pointed to in this context by G. Ellingsrud (as mentioned in de Boor's paper [2]). We also show that the conjecture is true for polynomials in two variables.     

Fast multipole methods for approximating a function from sampling values

Both barycentric Lagrange interpolation and barycentric rational interpolation are thought to be stable and effective methods for approximating a given function on some special point sets. A direct evaluation of these interpolants due to N interpolation points at M sampling points requires \(\mathcal {O}(NM)\) arithmetic operations. In this paper, we introduce two fast multipole methods to reduce the complexity to \(\mathcal {O}(\max \left \{N,M\right \})\). The convergence analysis is also presented in this paper.     

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.     

Critical Point in the Problem of Maximizing the Transition Probability Using Measurements in an <Emphasis Type="Italic">n</Emphasis>-Level Quantum System

We consider the problem of maximizing the transition probability in an n-level quantum system from a given initial state to a given final state using nonselective quantum measurements. We find a sequence of measurements that is a critical point of the transition probability and, moreover, a local maximum in each variable on the set of one-dimensional projectors. We consider the class of one-dimensional projectors because these projectors describe the measurements of populations of pure states of the system.     

The global existence of solutions for two classes of chemotaxis models

This paper is concerned with the global existence and uniform boundedness of solutions for two classes of chemotaxis models in two or three dimensional spaces. Firstly, by using detailed energy estimates, special interpolation relation and uniform Gronwall inequality, we prove the global existence of uniformly bounded solutions for a class of chemotaztic systems with linear chemotactic-sensitivity terms and logistic reaction terms. Secondly, by applying detailed analytic semigroup estimates and special iteration techniques, we obtain the global existence of uniformly bounded solutions for a class of chemotactic systems with nonlinear chemotacticsensitivity terms, which extends the global existence results of [6] to other general cases.