spline wavelets on triangulations

In this paper we investigate spline wavelets on general triangulations. In particular, we are interested in wavelets generated from piecewise quadratic polynomials. By using the Powell-Sabin elements, we set up a nested family of spaces of quadratic splines, which are suitable for multiresolution analysis of Besov spaces. Consequently, we construct wavelet bases on general triangulations and give explicit expressions for the wavelets on the three-direction mesh. A general theory is developed so as to verify the global stability of these wavelets in Besov spaces. The wavelet bases constructed in this paper will be useful for numerical solutions of partial differential equations.

     

Cesàro function spaces fail the fixed point property

The Cesàro sequence spaces , are reflexive but they have the fixed point property. In this paper we prove that in contrast to these sequence spaces the corresponding Cesàro function spaces on both and for are not reflexive and they fail to have the fixed point property.

     

Baire and Volterra spaces

In this paper we describe broad classes of spaces for which the Baire space property is equivalent to the assertion that any two dense -sets have dense intersection. We also provide examples of spaces where the equivalence does not hold. Finally, our techniques provide an easy proof of a new internal characterization of perfectly meager subspaces of and characterize metric spaces that are always of first category.

     

Spaces of rational loops on a real projective space

We show that the loop spaces on real projective spaces are topologically approximated by the spaces of rational maps . As a byproduct of our constructions we obtain an interpretation of the Kronecker characteristic (degree) of an ornament via particle spaces.

     

Spaces of type and the Hankel convolution

In this paper we introduce new function spaces that are denoted by , -1/2$"> and and that are spaces of type where the Hankel convolution and the Hankel transformation are defined. The spaces will play the same role in the Hankel setting that the spaces play in the theory of Fourier transformation.     

Perfectly normal non-metrizable non-Archimedean spaces are generalized Souslin lines

In this paper we prove the equivalence between the existence of perfectly normal, non-metrizable, non-archimedean spaces and the existence of ``generalized Souslin lines", i.e., linearly ordered spaces in which every collection of disjoint open intervals is -discrete, but which do not have a -discrete dense set. The key ingredient is the observation that every first countable linearly ordered space has a dense non-archimedean subspace.

     

Composition operators on spaces of entire functions

In this paper we study composition operators on spaces of entire functions. We determine which entire functions induce bounded composition operators on the Paley-Wiener space, , and on the spaces. In addition, we characterize compact composition operators on these spaces. We also study the cyclic properties of composition operators acting on .

     

Pseudonormality and starcompactness of -products

In this paper we shall prove the following: For every non-trivial -product , of uncountable number of spaces, having at least two points, is not pseudonormal. And every non-trivial -product is not strongly starcompact.

     

Strong paracompactness and w- -refinability of inverse limits

In this paper we construct inverse systems of which each space is strongly paracompact and each bonding map and each projection are open and onto maps, and the limit space is paracompact and not strongly paracompact and we investigate -refinability-like properties of limit spaces of inverse systems.

     

A geometric parametrization for the virtual Euler characteristics of the moduli spaces of real and complex algebraic curves

We determine an expression for the virtual Euler characteristics of the moduli spaces of -pointed real ) and complex () algebraic curves. In particular, for the space of real curves of genus with a fixed point free involution, we find that the Euler characteristic is where is the th Bernoulli number. This complements the result of Harer and Zagier that the Euler characteristic of the moduli space of complex algebraic curves is

The proof uses Strebel differentials to triangulate the moduli spaces and some recent techniques for map enumeration to count cells. The approach involves a parameter that permits specialization of the formula to the real and complex cases. This suggests that itself may describe the Euler characteristics of some related moduli spaces, although we do not yet know what these spaces might be.

     

Composition operators on Banach function spaces

We study the boundedness and the compactness of composition operators on some Banach function spaces such as absolutely continuous Banach function spaces on a -finite measure space, Lorentz function spaces on a -finite measure space and rearrangement invariant spaces on a resonant measure space. In addition, we study some properties of the spectra of a composition operator on the general Banach function spaces.

     

Classes of Hardy spaces associated with operators, duality theorem and applications

Let be the infinitesimal generator of an analytic semigroup on with suitable upper bounds on its heat kernels. In Auscher, Duong, and McIntosh (2005) and Duong and Yan (2005), a Hardy space and a space associated with the operator were introduced and studied. In this paper we define a class of spaces associated with the operator for a range of acting on certain spaces of Morrey-Campanato functions defined in New Morrey-Campanato spaces associated with operators and applications by Duong and Yan (2005), and they generalize the classical spaces. We then establish a duality theorem between the spaces and the Morrey-Campanato spaces in that same paper. As applications, we obtain the boundedness of fractional integrals on and give the inclusion between the classical spaces and the spaces associated with operators.

     

Isometries for the Carathéodory metric

Given two open unit balls and in complex Banach spaces, we consider a holomorphic mapping such that and is an isometry. Under some additional hypotheses on the Banach spaces involved, we prove that is a complex closed analytic submanifold of .

     

Coarse embeddings of metric spaces into Banach spaces

There are several characterizations of coarse embeddability of locally finite metric spaces into a Hilbert space. In this note we give such characterizations for general metric spaces. By applying these results to the spaces , we get their coarse embeddability into a Hilbert space for . This together with a theorem by Banach and Mazur yields that coarse embeddability into and into are equivalent when . A theorem by G.Yu and the above allow us to extend to , , the range of spaces, coarse embeddings into which is guaranteed for a finitely generated group to satisfy the Novikov Conjecture.

     

Alternate signs Banach-Saks property and real interpolation of operators

In the space of bounded linear operators acting between Banach spaces we define a seminorm vanishing on the subspace of operators having the alternate signs Banach-Saks property. We obtain logarithmically convex-type estimates of the seminorm for operators interpolated by the Lions-Peetre real method. In particular, the estimates show that the alternate signs Banach-Saks property is inherited from a space of an interpolation pair to the real interpolation spaces with respect to for all and .

     

A new approximation technique for div-curl systems

In this paper, we describe an approximation technique for div-curl systems based in where is a domain in . We formulate this problem as a general variational problem with different test and trial spaces. The analysis requires the verification of an appropriate inf-sup condition. This results in a very weak formulation where the solution space is and the data reside in various negative norm spaces. Subsequently, we consider finite element approximations based on this weak formulation. The main approach of this paper involves the development of ``stable pairs" of discrete test and trial spaces. With this approach, we enlarge the test space so that the discrete inf-sup condition holds and we use a negative-norm least-squares formulation to reduce to a uniquely solvable linear system. This leads to optimal order estimates for problems with minimal regularity which is important since it is possible to construct magnetostatic field problems whose solutions have low Sobolev regularity (e.g., with ). The resulting algebraic equations are symmetric, positive definite and well conditioned. A second approach using a smaller test space which adds terms to the form for stabilization will also be mentioned. Some numerical results are also presented.

     

Stabilization of Tsirelson-type norms on spaces

We consider classical Tsirelson-type norms of and their modified versions on spaces, . We show that the modified Tsirelson-type norms do not distort any of the subspaces of the spaces. We prove that Tsirelson-type norms, being equivalent to their modified versions, may at most 2-distort spaces.

     

Block diagonal polynomials

In this paper we introduce and study a certain class of polynomials in spaces with unconditional finite dimensional decomposition. Some applications to the existence of copies of in spaces of polynomials and to the stabilization of polynomial algebras are given.