It is well known that for multipliers f of the Drury-Arveson space H n, ‖f‖∞ does not dominate the operator norm of Mf . We show that in general ‖f‖∞ does not even dominate the essential norm of Mf . A consequence of this is that there exist multipliers f of H n for which Mf fails to be essentially hyponormal, i.e., if K is any compact, self-adjoint operator, then the inequality M∗ f Mf −MfM f ...

The supremum over all knot sequences of the max-norm of the orthogonal spline projector is studied with respect to the order k of the splines and their smoothness. It is first bounded from below in terms of the max-norm of the orthogonal projector onto a space of incomplete polynomials. Then, for continuous and for differentiable splines, its order of growth is shown to be √ k.

Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal publications. In this lecture we describe applications of low rank approximation in optimization. Firstly, let us give a short overview of the last lecture. We defined the operator norm of a matrix ‖.‖2 and the Frobenius norm ‖.‖F and we showed that the best rank k approximation of a given matrix M is the ...

In this paper, we investigate the accuracy-enhancement for the discontinuous Galerkin (DG) method for solving one-dimensional nonlinear symmetric systems of hyperbolic conservation laws. For nonlinear equations, the divided difference estimate is an important tool that allows for superconvergence of the post-processed solutions in the local L2 norm. Therefore, we first prove that the L2 norm of...

Some new classes of compacta K are considered for which C(K) endowed with the pointwise topology has a countable cover by sets of small local norm–diameter.

The anchored hyperplane location problem is to locate a hyperplane passing through some given points P ⊆ IR and minimizing either the sum of weighted distances (median problem), or the maximum weighted distance (center problem) to some other points Q ⊆ IR. This problem of computational geometry is analyzed by using nonlinear programming techniques. If the distances are measured by a norm, it wi...

We investigate the convergence of GMRES for an n by n Jordan block J . For each k that divides n we derive the exact form of the kth ideal GMRES polynomial and prove the equality max ‖v‖=1 min p∈πk ‖p(J)v‖ = min p∈πk max ‖v‖=1 ‖p(J)v‖, where πk denotes the set of polynomials of degree at most k and with value one at the origin, and ‖ · ‖ denotes the Euclidean norm. In other words, we show that ...

Let K be the field of real or complex numbers. In this note we characterize all inner product norms p1, . . . , pm on K for which the norm x 7−→ max{p1(x), . . . , pm(x)} on K is monotonic.

Given a polarization of an even unimodular lattice and integer k ≥ 1, we define a family of unimodular lattices L(M, N, k). Of special interest are certain L(M,N, 3) of rank 72. Their minimum norms lie in {4, 6, 8}. Norms 4 and 6 do occur. Consequently, 6 becomes the highest known minimum norm for rank 72 even unimodular lattices. We discuss how norm 8 might occur for such a L(M,N, 3). We note ...

Let E be a real uniformly convex Banach space which has the Fréchet differentiable norm, and K a nonempty, closed, and convex subset of E. Let T : K ® K be an asymptotically -strictly pseudocontractive mapping with a nonempty fixed point set. We prove that (I T) is demiclosed at 0 and obtain a weak convergence theorem of the modified Mann’s algorithm for T under suitable control conditions. Mor...