Positive Bases for Linear Spaces

New Bases for Polynomial-Based Spaces

Since it is well-known that the Vandermonde matrix is ill-conditioned, while the interpolation itself is not unstable in function space, this paper surveys the choices of other new bases. These bases are data-dependent and are categorized into discretely l2-orthonormal and continuously L2-orthonormal bases. The first one construct a unitary Gramian matrix in the space l2(X) while the late...

Dual Generalized Bases in Linear Topological Spaces

Diagrams for Positive Bases

The theory of positive bases is a useful tool in the study of convexity. In this paper we describe a representation of a positive basis of a finite-dimensional linear space, which is analogous to that of a poly tope by a Gale diagram (see [4, §5.4 and §6.3; 7, Chapter 3]). The basic idea is implicit in the work of Chandler Davis [2] but, as we now have the advantage of experience in the use of ...

Greedy Bases for Besov Spaces

We prove that the Banach spaces (⊕n=1`p )`q , which are isomorphic to the Besov spaces on [0, 1], have greedy bases, whenever 1 ≤ p ≤ ∞ and 1 < q < ∞. Furthermore, the Banach spaces (⊕n=1`p )`1 , with 1 < p ≤ ∞, and (⊕n=1`p )c0 , with 1 ≤ p < ∞ do not have a greedy bases. We prove as well that the space (⊕n=1`p )`q has a 1-greedy basis if and only if 1 ≤ p = q ≤ ∞.

Local Bases for Refinable Spaces

We provide a new representation of a refinable shift invariant space with a compactly supported generator, in terms of functions with a special property of homogeneity. In particular, these functions include all the homogeneous polynomials that are reproducible by the generator, which links this representation to the accuracy of the space. We completely characterize the class of homogeneous fun...