Lie-group interpolation and variational recovery for internal variables

We propose a variational procedure for the recovery of internal variables, in effect extending them from integration points to the entire domain. The objective is to perform the recovery with minimum error and at the same time guarantee that the internal variables remain in their admissible spaces. The minimization of the error is achieved by a three-field finite element formulation. The fields in the formulation are the deformation mapping, the target or mapped internal variables and a Lagrange multiplier that enforces the equality between the source and target internal variables. This formulation leads to an L2 projection that minimizes the distance between the source and target internal variables as measured in the L2 norm of the internal variable space. To ensure that the target internal variables remain in their original space, their interpolation is performed by recourse to Lie groups, which allows for direct polynomial interpolation of the corresponding Lie algebras by means of the logarithmic map. Once the Lie algebras are interpolated, the mapped variables are recovered by the exponential map, thus guaranteeing that they remain in the appropriate space

Cubature formulae for orthogonal polynomials in terms of elements of finite order of compact simple Lie groups

AbstractThe paper contains a generalization of known properties of Chebyshev polynomials of the second kind in one variable to polynomials of n variables based on the root lattices of compact simple Lie groups G of any type and of any rank n. The results, inspired by work of H. Li and Y. Xu where they derived cubature formulae from A-type lattices, yield Gaussian cubature formulae for each simple Lie group G based on nodes (interpolation points) that arise from regular elements of finite order in G. The polynomials arise from the irreducible characters of G and the nodes as common zeros of certain finite subsets of these characters. The consistent use of Lie theoretical methods reveals the central ideas clearly and allows for a simple uniform development of the subject. Furthermore it points to genuine and perhaps far reaching Lie theoretical connections

Besov and Triebel--Lizorkin spaces on Lie groups

In this paper we develop a theory of Besov and Triebel--Lizorkin spaces on general noncompact Lie groups endowed with a sub-Riemannian structure. Such spaces are defined by means of hypoelliptic sub-Laplacians with drift, and endowed with a measure whose density with respect to a right Haar measure is a continuous positive character of the group. We prove several equivalent characterizations of their norms, we establish comparison results also involving Sobolev spaces of recent introduction, and investigate their complex interpolation and algebra properties.Comment: 35 page