Ordinary differential inequalities and quasimonotonicity in ordered topological vector spaces

A well known comparison theorem on ordinary differential inequalities with quasimonotone right-hand side was carried over by
Volkmann (1972) to (pre)ordered topological vector spaces. We prove that the quasimonotonicity of is a necessary condition here if is continuous. Then it is shown that quasimonotonicity can be verified by considering only a few positive continuous linear functionals in the definition (for instance in by taking coordinate functionals).

     

Exact iterative reconstruction algorithm for multivariate irregularly sampled functions in spline-like spaces: The -Theory

We prove that the exact reconstruction of a function from its samples on any ``sufficiently dense" sampling set can be obtained, as long as is known to belong to a large class of spline-like spaces in . Moreover, the reconstruction can be implemented using fast algorithms. Since a limiting case is the space of bandlimited functions, our result generalizes the classical Shannon-Whittaker sampling theorem on regular sampling and the Paley-Wiener theorem on non-uniform sampling.

     

RUC-Bases in Orlicz and Lorentz Operator Spaces

If is an RUC-basis in somecouple of non-commutative L _p-spaces, then is an RUC-basic sequence in any non-commutative Orlicz or Lorentz space which is an interpolation space for this couple.     

Nonsmooth Characterizations of Asplund Spaces and Smooth Variational Principles

We show that the Asplund property of Banach spaces is not only sufficient but also a necessary condition for the fulfillment of some basic results in nonsmooth analysis involving Fréchet-like normals and subdifferentials as well as their sequential limits. In this way we obtain new characterizations of Asplund spaces within the framework of nonsmooth analysis. Then we study several versions of smooth variational principles in Asplund spaces, provide necessary and sufficient conditions for the validity of such principles, and establish their relationships with certain subdifferential properties of lower semicontinuous functions.     

Generalized sampling in shift-invariant spaces with multiple stable generators

The aim of this paper is to derive stable generalized sampling in a shift-invariant space with ? stable generators. This is done in the light of the theory of frames in the product Hilbert space (? times). The generalized samples are expressed as the frame coefficients of an appropriate function in with respect to some particular frame in . Since any multiply stable generated shift-invariant space is the image of by means of a bounded invertible operator, the generalized sampling is obtained from some dual frame expansions in . An example in the setting of the Hermite cubic splines is exhibited.     

Pointwise bounds for L 2 eigenfunctions on locally symmetric spaces

We prove pointwise bounds for L ² eigenfunctions of the Laplace-Beltrami operator on locally symmetric spaces with -rank one if the corresponding eigenvalues lie below the continuous part of the L ² spectrum. Furthermore, we use these bounds in order to obtain some results concerning the L ^p spectrum.     

On some iterated weighted spaces

It is proved that the Hörmander and spaces (Ω₁⊂Rn, Ω₂⊂Rm open sets, 1?p<∞, ki Beurling-Björck weights, k=k₁⊗k₂) are isomorphic whereas the iterated spaces and are not if 1<p≠q<∞. A similar result for weighted Lp-spaces of entire analytic functions is also obtained. Finally a result on iterated Besov spaces is given: and are not isomorphic when 1<q≠2<∞.     

Riemannian three dimensional molecular spaces

A simple manipulation of the first order density function permits to define a curved 3D Riemannian coordinate set, which can substitute the usual flat 3D Cartesian space, where atoms and molecules are supposed to exist. Several simple models are discussed. Gaussian type orbitals generate a space division with positive and negative curvatures, the later one being near the centre of the functions; contrarily Slater type orbitals provide a positive curvature everywhere.     

A counterexample for boundedness of pseudo-differential operators on modulation spaces

We prove that pseudo-differential operators with symbols in the class ( ) are not always bounded on the modulation space ().

     

Learning and approximation by Gaussians on Riemannian manifolds

Learning function relations or understanding structures of data lying in manifolds embedded in huge dimensional Euclidean spaces is an important topic in learning theory. In this paper we study the approximation and learning by Gaussians of functions defined on a d-dimensional connected compact C ^∞ Riemannian submanifold of which is isometrically embedded. We show that the convolution with the Gaussian kernel with variance σ provides the uniform approximation order of O(σ ^s ) when the approximated function is Lipschitz s ∈(0, 1]. The uniform normal neighborhoods of a compact Riemannian manifold play a central role in deriving the approximation order. This approximation result is used to investigate the regression learning algorithm generated by the multi-kernel least square regularization scheme associated with Gaussian kernels with flexible variances. When the regression function is Lipschitz s, our learning rate is (log² m)/m) ^{s/(8 s + 4 d)} where m is the sample size. When the manifold dimension d is smaller than the dimension n of the underlying Euclidean space, this rate is much faster compared with those in the literature. By comparing approximation orders, we also show the essential difference between approximation schemes with flexible variances and those with a single variance. Supported partially by the Research Grants Council of Hong Kong [Project No. CityU 103405], City University of Hong Kong [Project No. 7001983], National Science Fund for Distinguished Young Scholars of China [Project No. 10529101], and National Basic Research Program of China [Project No. 973-2006CB303102].