where L is the left regular representation of G. From this it can be deduced that for any v in V and f in C c (G) the vector π(f) is smooth, and more precisely that if X lies in U(g) then π(X)π(f)v = π(LXf)v. This implies that V ∞ is dense in V , since if {fn} is a Dirac sequence on G then π(fn)v → v. The subspace of V ∞ spanned by the π(f)v with f in C c (G) is called the Gårding subspace of V...

and K(s) = [ AK BK CK DK ] be an n κ order controller with transfer function K(s) = CK(sI − AK)−1BK + DK . We seek a reduced-order controller Kr(s) of order r with r nκ to replace K(s). Assume that both G(s) and K(s) are single-input single-output (SIS0), and that K(s) is a stable stabilizing controller. (The general case follows simply and will be presented in the full paper.) Requiring Kr(s) ...

Recent results on residual smoothing are reviewed, and it is observed that certain of these are equivalent to results obtained by different means that relate “peaks” and “plateaus” in residual norm sequences produced by certain pairs of Krylov subspace methods.

In this paper, structure preserving order reduction of proportionally damped and undamped second order systems is presented. The discussion is based on Second Order Krylov Subspace method and it is shown that for systems with a proportional damping, the damping matrix does not contribute to the projection matrices and the reduction can be carried out using the classical Krylov subspaces. As a r...

We study the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian. We prove that if u is a solution of (−∆)u = g in Ω, u ≡ 0 in R\Ω, for some s ∈ (0, 1) and g ∈ L∞(Ω), then u is C(R) and u/δ|Ω is C up to the boundary ∂Ω for some α ∈ (0, 1), where δ(x) = dist(x, ∂Ω). For this, we develop a fractional analog of the Krylov boundary Harnack method. Moreov...

Many problems in scientific computing involving a large sparse matrix A are solved by Krylov subspace methods. This includes methods for the solution of large linear systems of equations with A, for the computation of a few eigenvalues and associated eigenvectors of A, and for the approximation of nonlinear matrix functions of A. When the matrix A is non-Hermitian, the Arnoldi process commonly ...

The R-Function Method (RFM) solution structure is a functional expression that satisfies all given boundary conditions exactly and contains some undetermined functional components. It is complete if there exists a choice of undetermined component that transform the solution structure into an exact solution. Such a structure was used by Kantorovich (Kantorovich and Krylov, 1958) and his students...

Xavier Cabré Abstra t. We consider a class of second order linear elliptic operators intrinsically defined on Riemannian manifolds, and which correspond to nondivergent operators in Euclidean space. Under the assumption that the sectional curvature is nonnegative, we prove a global Krylov-Safonov Harnack inequality and, as a consequence, a Liouville theorem for solutions of such equations. From...