The convergence of three spline methods for scattered data interpolation and fitting

The convergences of three L₁ spline methods for scattered data interpolation and fitting using bivariate spline spaces are studied in this paper. That is, L₁ interpolatory splines, splines of least absolute deviation, and L₁ smoothing splines are shown to converge to the given data function under some conditions and hence, the surfaces from these three methods will resemble the given data values.     

The Koszul dual of a weakly Koszul module

We study the so-called weakly Koszul modules and characterise their Koszul duals. We show that the (adjusted) associated graded module of a weakly Koszul module exactly determines the homology modules of the Koszul dual. We give an example of a quasi-Koszul module which is not weakly Koszul.     

The behavior of best -approximations as . A counterexample of convergence

In the space of summable sequences we give an example of a one-dimensional affine subspace C such that the best L_p-approximations of 0 from C fail to converge as p↓1. We thus give an answer to this problem of convergence in infinite measure spaces.     

On holomorphic functions of slow growth on coverings of strongly pseudoconvex manifolds

In this paper we extend some results of the paper [M. Gromov, G. Henkin, M. Shubin, Holomorphic L²-functions on coverings of pseudoconvex manifolds, Geom. Funct. Anal. 8 (1998) 552-585].     

Spectrum and analytical indices of the C-algebra of Wiener-Hopf operators

We study multivariate generalisations of the classical Wiener-Hopf algebra, which is the C^∗-algebra generated by the Wiener-Hopf operators, given by convolutions restricted to convex cones. By the work of Muhly and Renault, this C^∗-algebra is known to be isomorphic to the reduced C^∗-algebra of a certain restricted action groupoid, given by the action of Euclidean space on a certain compactification. Using groupoid methods, we construct composition series for the Wiener-Hopf C^∗-algebra by a detailed study of this compactification. We compute the spectrum, and express homomorphisms in K-theory induced by the symbol maps which arise by the subquotients of the composition series in analytical terms. Namely, these symbols maps turn out to be given by an analytical family index of a continuous family of Fredholm operators. In a subsequent paper, we also obtain a topological expression of these indices.     

The algebra of harmonic functions for a matrix-valued transfer operator

We analyze matrix-valued transfer operators. We prove that the fixed points of transfer operators form a finite-dimensional C^∗-algebra. For matrix weights satisfying a low-pass condition we identify the minimal projections in this algebra as correlations of scaling functions, i.e., limits of cascade algorithms.     

A unique decomposition result for HT factors with torsion free core

We prove that II₁ factors M have a unique (up to unitary conjugacy) cross-product type decomposition around “core subfactors” N⊂M satisfying the property HT of [S. Popa, On a class of type II₁ factors with Betti numbers invariants, Ann. of Math. (2) 163 (2006) 809-899] and a certain “torsion freeness” condition. In particular, this shows that isomorphism of factors of the form Lαi(Z²)?Fni, i=1,2, for Fni⊂SL(2,Z) free groups of rank ni and αj=e2πitj, tj∉Q, implies n₁=n₂.     

Direct limits of infinite-dimensional Lie groups compared to direct limits in related categories

Let G be a Lie group which is the union of an ascending sequence G₁⊆G₂⊆? of Lie groups (all of which may be infinite-dimensional). We study the question when in the category of Lie groups, topological groups, smooth manifolds, respectively, topological spaces. Full answers are obtained for G the group Diffc(M) of compactly supported C^∞-diffeomorphisms of a σ-compact smooth manifold M; and for test function groups of compactly supported smooth maps with values in a finite-dimensional Lie group H. We also discuss the cases where G is a direct limit of unit groups of Banach algebras, a Lie group of germs of Lie group-valued analytic maps, or a weak direct product of Lie groups.     

Functional spectrum of contractions

In this paper, we introduce a new kind of spectrum for the C⋅0-class contractions. Since elements in this spectrum are functions, rather than numbers, we shall call it functional spectrum. Functional spectrum is a “large” closed subset of the Hardy space over the unit disk, and in many cases there is a canonical embedding of classical spectrum into functional spectrum. The study is carried out in the setting of the Hardy space over the bidisk H²(D²), on which every C⋅0-class contraction has a representation. A key tool is reduction operator. The reduction operator also gives rise to an equivalent statement of the Invariant Subspace Problem.     

Eventual differentiability of functional differential equations in Banach spaces

This paper concerns the regularity of a functional differential equation in the form: , t>0, where A is the generator of an analytic semigroup on a Banach space X, and B₁,B₂ are α(γ−A)-bounded linear operator for 0<α<1. By spectral analysis, it is shown that the associated solution semigroup of this equation is eventually differentiable.