Quantitative bounds on rates of approximation by linear combinations of Heaviside plane waves are obtained for sufficiently differentiable functions f which vanish rapidly enough at infinity: for d odd and f ∈ Cd(R), with lower-order partials vanishing at infinity and dth-order partials vanishing as ‖x‖−(d+1+ε), ε > 0, on any domain ⊂ R with unit Lebesgue measure, the L2( )-error in approximati...

To address the classic interior tomography problem where projections at each view extend only to the shadow of a circular region completely interior to the subject being scanned, previously we showed that the exact recovery of twoand three-dimensional piecewise smooth images is guaranteed using a one-dimensional generalized total variation seminorm penalty which allows a much faster reconstruct...

Exploiting sparsity in the image gradient magnitude has proved to be an effective means for reducing the sampling rate in the projection view angle in computed tomography (CT). Most of the image reconstruction algorithms, developed for this purpose, solve a nonsmooth convex optimization problem involving the image total variation (TV). The TV seminorm is the ℓ1 norm of the image gradient magnit...

It has been proved that algebraic polynomials P are dense in the space Lp(R, dμ), p∈ (0,∞), iff the measure μ is representable as dμ = wp dν with a finite non-negative Borel measure ν and an upper semi-continuous function w : R → R+ : = [0,∞) such that P is a dense subset of the space C0 w : = {f ∈ C(R) : w(x)f(x) → 0 as |x| → ∞} equipped with the seminorm ‖f‖w := supx∈R w(x)|f(x)|. The similar...

The title above is wrong, because the strong dual of a Banach space is too strong to assert that the natural correspondence between a space and its bidual is an isomorphism. This, from a categorical point of view, is indeed the right duality concept because it yields a self adjoint dualisation functor. However, for many applications the non–reflexiveness problem can be solved by replacing the n...

This paper considers the numerical solution of elliptic boundary value problems with a complicated (nonperiodic) diffusion matrix which is smooth but highly oscillating on very different scales. We study the influence of the scales and amplitudes of these oscillations to the regularity of the solution. We introduce weighted Sobolev norms of integer order, where the (p + 1)st seminorm is weighte...

This paper studies finite element approximations of the stochastic Allen–Cahn equation with gradient-type multiplicative noise that is white in time and correlated in space. The sharp interface limit—as the diffuse interface thickness vanishes—of the stochastic Allen–Cahn equation is formally a stochastic mean curvature flow which is described by a stochastically perturbed geometric law of the ...

Abstract. The main purpose of this paper is to present a closer look at how the H1and L2-errors for quadratic interpolation on a triangle are determined by the triangle geometry and the anisotropic behavior of the third order derivatives of interpolated functions. We characterize quantitatively the anisotropic behavior of a third order derivative tensor by its orientation and anisotropic ratio....

Exploiting sparsity in the image gradient magnitude has proved to be an effective means for reducing the sampling rate in the projection view angle in computed tomography (CT). Most of the image reconstruction algorithms, developed for this purpose, solve a nonsmooth convex optimization problem involving the image total variation (TV). The TV seminorm is the `1 norm of the image gradient magnit...

We introduce a novel generic energy functional that we employ to solve inverse imaging problems within a variational framework. The proposed regularization family, termed as structure tensor total variation (STV), penalizes the eigenvalues of the structure tensor and is suitable for both grayscale and vector-valued images. It generalizes several existing variational penalties, including the tot...