The bestL 2-approximation by finite sums of functions with separable variables

Summary We consider the problem of the best approximation of a given functionh L ² (X × Y) by sums _k=1 ⁿ f _k f _k, with a prescribed numbern of products of arbitrary functionsf _k L ² (X) andg _k L ² (Y). As a co-product we develop a new proof of the Hilbert—Schmidt decomposition theorem for functions lying inL ² (X × Y).     

Local error estimation for singly-implicit formulas by two-step Runge-Kutta methods

In this paper it is shown that the local discretization error ofs-stage singly-implicit methods of orderp can be estimated by embedding these methods intos-stage two-step Runge-Kutta methods of orderp+1, wherep=s orp=s+1. These error estimates do not require any extra evaluations of the right hand side of the differential equations. This is in contrast with the error estimation schemes based on embedded pairs of two singly-implicit methods proposed by Burrage.The work of A. Bellen and M. Zennaro was supported by the CNR and MPI. The work of Z. Jackiewicz was supported by the CNR and by the NSF under grant DMS-8520900.     

Well posed cauchy problems for complex nonlinear equations must be semilinear

The first author was supported in part by CNPq     

Quasi-Free Resolutions Of Hilbert Modules

The notion of a quasi-free Hilbert module over a function algebra $\mathcal{A}$ consisting of holomorphic functions on a bounded domain $\Omega$ in complex m space is introduced. It is shown that quasi-free Hilbert modules correspond to the completion of the direct sum of a certain number of copies of the algebra $\mathcal{A}$. A Hilbert module is said to be weakly regular (respectively, regular) if there exists a module map from a quasi-free module with dense range (respectively, onto). A Hilbert module $\mathcal{M}$ is said to be compactly supported if there exists a constant $\beta$ satisfying $\|\varphi f\| \leq \beta \ |\varphi \| \textsl{X} \|f \|$ for some compact subset X of $\Omega$ and $\varphi$ in $\mathcal{A}$, f in $\mathcal{M}$. It is shown that if a Hilbert module is compactly supported then it is weakly regular. The paper identifies several other classes of Hilbert modules which are weakly regular. In addition, this result is extended to yield topologically exact resolutions of such modules by quasi-free ones.     

A remark on compact matrix quantum groups

A simpler set of axioms of the theory of compact matrix quantum groups (pseudogroups) is found.Supported by Japan Society for Promoting Science. On leave from the Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, Hoa 74, 00-682 Warsaw, Poland.     

On coherent states for the simplest quantum groups

The coherent states for the simplest quantum groups (q-Heisenberg-Weyl, SU _q (2) and the discrete series of representations of SU _q (1, 1)) are introduced and their properties investigated. The corresponding analytic representations, path integrals, and q-deformation of Berezin's quantization on , a sphere, and the Lobatchevsky plane are discussed.     

A New Inertia Theorem for Stein Equations,Inertia of Invertible Hermitian Block Toeplitz Matrices and Matrix Orthogonal Polynomials

In this paper we present an inertia result for Stein equations with an indefinite right hand side. This result is applied to establish connnections between the inertia of invertible hermitian block Toeplitz matrices and associated orthogonal polynomials.     

Presentations of the first homotopy groups of the unitary groups

We describe explicit presentations of all stable and the first nonstable homotopy groups of the unitary groups. In particular, for each n 2 we supply n homotopic maps that each represent the (n - 1)!-th power of a suitable generator of _2n SU(n) _n!. The product of these n commuting maps is the constant map to the identity matrix.