On characterizations of commutativity of C-algebras

We give three kinds of characterizations of the commutativity of C- algebras. The first is the one from operator monotone property of functions regarded as the nonlinear version of Stinespring theorem, the second one is the characterization of commutativity of local type from expansion formulae of related functions and the third one is of global type from multiple positivity of those nonlinear positive maps induced from functions.

     

Refinable bivariate quartic -splines for multi-level data representation and surface display

In this paper, a second-order Hermite basis of the space of -quartic splines on the six-directional mesh is constructed and the refinable mask of the basis functions is derived. In addition, the extra parameters of this basis are modified to extend the Hermite interpolating property at the integer lattices by including Lagrange interpolation at the half integers as well. We also formulate a compactly supported super function in terms of the basis functions to facilitate the construction of quasi-interpolants to achieve the highest (i.e., fifth) order of approximation in an efficient way. Due to the small (minimum) support of the basis functions, the refinable mask immediately yields (up to) four-point matrix-valued coefficient stencils of a vector subdivision scheme for efficient display of -quartic spline surfaces. Finally, this vector subdivision approach is further modified to reduce the size of the coefficient stencils to two-point templates while maintaining the second-order Hermite interpolating property.

     

On the density of Banach spaces with the Grothendieck property

Using the method of forcing we prove that consistently there is a Banach space of continuous functions on a compact Hausdorff space with the Grothendieck property and with density less than the continuum. It follows that the classical result stating that ``no nontrivial complemented subspace of a Grothendieck space is separable' cannot be strengthened by replacing ``is separable' by ``has density less than that of ', without using an additional set-theoretic assumption. Such a strengthening was proved by Haydon, Levy and Odell, assuming Martin's axiom and the negation of the continuum hypothesis. Moreover, our example shows that certain separation properties of Boolean algebras are quite far from the Grothendieck property.

     

Generalized spherical functions on reductive -adic groups

Let be the group of rational points of a connected reductive -adic group and let be a maximal compact subgroup satisfying conditions of Theorem 5 from Harish-Chandra (1970). Generalized spherical functions on are eigenfunctions for the action of the Bernstein center, which satisfy a transformation property for the action of . In this paper we show that spaces of generalized spherical functions are finite dimensional. We compute dimensions of spaces of generalized spherical functions on a Zariski open dense set of infinitesimal characters. As a consequence, we get that on that Zariski open dense set of infinitesimal characters, the dimension of the space of generalized spherical functions is constant on each connected component of infinitesimal characters. We also obtain the formula for the generalized spherical functions by integrals of Eisenstein type. On the Zariski open dense set of infinitesimal characters that we have mentioned above, these integrals then give the formula for all the generalized spherical functions. At the end, let as mention that among others we prove that there exists a Zariski open dense subset of infinitesimal characters such that the category of smooth representations of with fixed infinitesimal character belonging to this subset is semi-simple.

     

Completely monotonic functions involving the gamma and -gamma functions

We give an infinite family of functions involving the gamma function whose logarithmic derivatives are completely monotonic. Each such function gives rise to an infinitely divisible probability distribution. Other similar results are also obtained for specific combinations of the gamma and -gamma functions.

     

Error estimates on anisotropic elements for functions in weighted Sobolev spaces

In this paper we prove error estimates for a piecewise average interpolation on anisotropic rectangular elements, i.e., rectangles with sides of different orders, in two and three dimensions.

Our error estimates are valid under the condition that neighboring elements have comparable size. This is a very mild assumption that includes more general meshes than those allowed in previous papers. In particular, strong anisotropic meshes arising naturally in the approximation of problems with boundary layers fall under our hypotheses.

Moreover, we generalize the error estimates allowing on the right-hand side some weighted Sobolev norms. This extension is of interest in singularly perturbed problems.

Finally, we consider the approximation of functions vanishing on the boundary by finite element functions with the same property, a point that was not considered in previous papers on average interpolations for anisotropic elements.

As an application we consider the approximation of a singularly perturbed reaction-diffusion equation and show that, as a consequence of our results, almost optimal order error estimates in the energy norm, valid uniformly in the perturbation parameter, can be obtained.

     

A -sampling theorem and product formula for continuous -Jacobi functions

In this paper we derive a q-analogue of the sampling theorem for Jacobi functions. We also establish a product formula for the nonterminating version of the q-Jacobi polynomials. The proof uses recent results in the theory of q-orthogonal polynomials and basic hypergeometric functions.

     

A hereditarily subspace of without the Schur property

Let p_1 > p_2 > \cdots > 1$">. We construct an easily determined -symmetric basic sequence in , which spans a hereditarily subspace without the Schur property. An immediate consequence is the existence of hereditarily subspaces of without the Schur property.

     

Quantum cohomology and the -Schur basis

We prove that structure constants related to Hecke algebras at roots of unity are special cases of -Littlewood-Richardson coefficients associated to a product of -Schur functions. As a consequence, both the 3-point Gromov-Witten invariants appearing in the quantum cohomology of the Grassmannian, and the fusion coefficients for the WZW conformal field theories associated to are shown to be -Littlewood-Richardson coefficients. From this, Mark Shimozono conjectured that the -Schur functions form the Schubert basis for the homology of the loop Grassmannian, whereas -Schur coproducts correspond to the integral cohomology of the loop Grassmannian. We introduce dual -Schur functions defined on weights of -tableaux that, given Shimozono's conjecture, form the Schubert basis for the cohomology of the loop Grassmannian. We derive several properties of these functions that extend those of skew Schur functions.

     

-bounds for spectral multipliers on graphs

We prove that certain spectral multipliers associated with the discrete Laplacian on graphs satisfying the doubling volume property and the Poincaré inequality are bounded on the Hardy space .

     

Weak compactness is equivalent to the fixed point property in

A nonempty, closed, bounded, convex subset of has the fixed point property if and only if it is weakly compact.

     

Analytic -adic cell decomposition and integrals

We prove a conjecture of Denef on parameterized -adic analytic integrals using an analytic cell decomposition theorem, which we also prove in this paper. This cell decomposition theorem describes piecewise the valuation of analytic functions (and more generally of subanalytic functions), the pieces being geometrically simple sets, called cells. We also classify subanalytic sets up to subanalytic bijection.

     

On the theory of elliptic functions based on

Based on properties of the hypergeometric series , we develop a theory of elliptic functions that shares many striking similarities with the classical Jacobian elliptic functions.

     

Subspaces of

The relationship of the Hardy space and the space of integrable functions is examined in terms of intermediate spaces of functions that are described as sums of atoms. It is proved that these spaces have dual spaces that lie between the space of functions of bounded mean oscillation, , and . Furthermore, the spaces intermediate to and are shown to be dual to spaces similar to the space of functions of vanishing mean oscillation. The proofs are extensions of classical proofs.

     

Homotopy groups of -contact toric manifolds

Contact toric manifolds of Reeb type are a subclass of contact toric manifolds which have the property that they are classified by the images of the associated moment maps. We compute their first and second homotopy group terms of the images of the moment map. We also explain why they are -contact.

     

Inverse-type estimates on -finite element spaces and applications

This work is concerned with the development of inverse-type inequalities for piecewise polynomial functions and, in particular, functions belonging to -finite element spaces. The cases of positive and negative Sobolev norms are considered for both continuous and discontinuous finite element functions.The inequalities are explicit both in the local polynomial degree and the local mesh size.The assumptions on the -finite element spaces are very weak, allowing anisotropic (shape-irregular) elements and varying polynomial degree across elements. Finally, the new inverse-type inequalities are used to derive bounds for the condition number of symmetric stiffness matrices of -boundary element method discretisations of integral equations, with element-wise discontinuous basis functions constructed via scaled tensor products of Legendre polynomials.

     

An ultrafilter with property

Let be a -supercompact cardinal. We show that carries a normal ultrafilter with a property introduced by Menas. With it we give a transparent proof of Kamo's theorem that carries a normal ultrafilter with the partition property.

     

A -analogue of the Whittaker-Shannon-Kotel'nikov sampling theorem

The Whittaker-Shannon-Kotel'nikov (WSK) sampling theorem plays an important role not only in harmonic analysis and approximation theory, but also in communication engineering since it enables engineers to reconstruct analog signals from their samples at a discrete set of data points. The main aim of this paper is to derive a -analogue of the Whittaker-Shannon-Kotel'nikov sampling theorem. The proof uses recent results in the theory of -orthogonal polynomials and basic hypergeometric functions, in particular, new results on the addition theorems for -exponential functions.

     

Rational Landen transformations on

The Landen transformation preserves the value of an elliptic integral, and its iteration produces the classical arithmetic-geometric mean AGM. We present analogous transformations for rational functions integrated over the whole real line.

     

A reproducing kernel space model for -functions

A new model for generalized Nevanlinna functions will be presented. It involves Bezoutians and companion operators associated with certain polynomials determined by the generalized zeros and poles of . The model is obtained by coupling two operator models and expressed by means of abstract boundary mappings and the corresponding Weyl functions.