Weak Norm Research Articles

Interior maximum norm estimates for finite element methods

Interior a priori error estimates in the maximum norm are derived from interior Ritz-Galerkin equations which are common to a class of methods used in approximating solutions of second order elliptic boundary value problems. The estimates are valid for a large class of piecewise polynomial subspaces used in practice, which are defined on quasi-uniform meshes. It is shown that the error in an interior domain Ω 1 {\Omega _1} can be estimated with the best order of accuracy that is possible locally for the subspaces used plus the error in a weaker norm over a slightly larger domain which measures the effects from outside of the domain Ω 1 {\Omega _1} .

On the residual correlation of finite-dimensional discrete Fourier transforms of stationary signals (Corresp.)

The covariance matrix of the Fourier coefficients of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> - sampled stationary random signals is studied. Three theorems are established. 1) If the covariance sequence is summable, the magnitude of every off-diagonal covariance element converges to zero as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N \rightarrow \infty</tex> . 2) If the covariance sequence is only square summable, the magnitude of the covariance elements sufficiently far from the diagonal converges to zero as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N \rightarrow \infty</tex> . 3) If the covariance sequence is square summable, the weak norm of the matrix containing only the off-diagonal elements converges to zero as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N \rightarrow \infty</tex> . The rates of convergence are also determined when the covariance sequence satisfies additional conditions.

An extension of a theorem of Bessaga

In [1], C. Bessaga has shown that if $X$ is a linear topological space admitting a weak incomplete norm $w$ and $A \subset X$ is closed in the $w$-completion of $X$, then $A$ is negligible in $X$. The present paper establishes this result in a space admitting a weak incomplete linear metric.

Interior estimates for Ritz-Galerkin methods

Interior a priori error estimates in Sobolev norms are derived from interior Ritz-Galerkin equations which are common to a class of methods used in approximating solutions of second order elliptic boundary value problems. The estimates are valid for a large class of piecewise polynomial subspaces used in practice, which are defined on both uniform and nonuniform meshes. It is shown that the error in an interior domain Ω \Omega can be estimated with the best order of accuracy that is possible locally for the subspaces used plus the error in a weaker norm over a slightly larger domain which measures the effects from outside of the domain Ω \Omega . Additional results are given in the case when the subspaces are defined on a uniform mesh. Applications to specific boundary value problems are given.

Capacitary differentiability of potentials of finite Radon measures

We study differentiability properties of a potential of the type $K\star \mu$, where $\mu$ is a finite Radon measure in $\mathbb{R}^N$ and the kernel $K$ satisfies $|\nabla^j K(x)| \le C\, |x|^{-(N-1+j)}, \quad j=0,1,2.$ We introduce a notion of differentiability in the capacity sense, where capacity is classical capacity in the de la Vallee Poussin sense associated with the kernel $|x|^{-(N-1)}.$ We require that the first order remainder at a point is small when measured by means of a normalized weak capacity norm in balls of small radii centered at the point. This implies weak $L^{N/(N-1)}$ differentiability and thus $L^{p}$ differentiability in the Calderon--Zygmund sense for $1\le p < N/(N-1)$. We show that $K\star \mu$ is a.e. differentiable in the capacity sense, thus strengthening a recent result by Ambrosio, Ponce and Rodiac. We also present an alternative proof of a quantitative theorem of the authors just mentioned, giving pointwise Lipschitz estimates for $K\star \mu.$ As an application, we study level sets of newtonian potentials of finite Radon measures.