Showing papers in "Mathematische Zeitschrift in 1992"

The least solution for the polynomial interpolation problem

Abstract: We consider the following problem: given a subspace Λ of the dual Π′ of the space Π of svariate polynomials, find a space P ⊂ Π which is correct for Λ in the sense that each continuous linear functional on Λ can be interpolated by a unique p ∈ P . We provide a map, Λ 7→ Λ↓ ⊂ Π, which we call the least map, that solves this interpolation problem and give a comprehensive discussion of its properties. This least solution, Λ↓, is a homogeneous space and is shown to have minimal degree among all possible solutions. It is the unique minimal degree solution which is dual (in a natural sense) to all minimal degree solutions. It also interacts nicely with various maps applied to Λ, such as convolution, translation, change of variables, and, particularly, differentiation. Our approach is illustrated by detailed examples, concerning finite-dimensional Λ’s spanned by point-evaluations or line integrals. Methods which facilitate the identification of the least solution are established. The paper is complemented by [BR3], in which an algorithmic approach for obtaining Λ↓ is presented whose computational aspects are detailed. AMS (MOS) Subject Classifications: primary 41A05, 41A63, 41A10; secondary 13F20, 13F25, 13A15

The Dirichlet series of Koecher and Maaß and modular forms of weight 3/2

Abstract: The basis problem for modular forms is the question which modular forms can be expressed as linear combinations of theta series, in order that this question makes sense one has of course to specify which type of theta series should be taken into account. We want here to follow (in a special case) the approach taken e.g. in [B61, W a l l which (in its general form) asks" Which (cuspidal) modular forms F of degree n and weight k for the group FoC")(N) can be expressed as linear combinations of the theta series of degree n of integral quadratic forms of rank m = 2 k and the same level N? Moreover, one wants to derive an explicit expression for such a linear combination in terms of the Petersson inner products of F with the theta series involved. The representation theoretic version of this question is to ask: Which (cuspidat) automorphic representations of the adelic symplectic (or metaplectic) group

A variational approach in weighted Sobolev spaces to the operator - ... + .../... X1 in exterior domains of IR3.

Abstract: Consider the Dirichlet problem −vΔu+k∂ 1 u = f withv, k>0 in ℝ3 or in an exterior domain of ℝ3 where the skew-symmetric differential operator −1=∂/∂x1 is a singular perturbation of the Laplacian. Because of the inhomogeneity of the fundamental solution we study existence, uniqueness and regularity in Sobolev spaces with anisotropic weights. In these spaces the operator ∂1 yields an additional positive definite term giving better results than in Sobolev spaces with radial weights. The elliptic equation −vΔu +k∂1 u=f can be taken as a model problem for the Oseen equations, a linearized form of the Navier-Stokes equations.

Involutions of compact Klein surfaces

Abstract: Let X be a compact Klein surface of topological genus g and k bounda ry components and let tp: X-+X be a dianalytic involution. We are interested in determining q0 up to topological conjugacy by a finite number of invariants mainly connected with Fix(q~), the fixed point set of qo. We shall also investigate corresponding inclusions between non-Eucl idean crystal lographic groups and use these to consider the subspaces of Teichmfiller space of Klein surfaces admitting involutions. As we shall see, Fix(cp) consists of (a) a finite number of isolated fixed points, (b) a finite number of simple closed curves. By analogy with the case of plane algebraic curves we shall call a fixed simple closed curve an oval. Ovals will be called twisted or untwisted according to whether they have M6bius band or annular ne ighbourhoods respectively. Of course, twisted ovals can only occur on non-orientable surfaces.