In this article, we present a sufficient condition for the exponential exp⁡(−f) to have tail decay stronger than any Gaussian, where f is defined on locally convex space X and grows faster squared seminorm X. particular, our result proves that exp⁡(−p(x)2+ε+αq(x)2) integrable all α,ε>0 w.r.t. Radon Gaussian measure nuclear X, if p q are continuous seminorms with compatible kernels. This can be ...

In 2014, Ludwig showed the limiting behavior of anisotropic Gagliardo s-seminorm a function f as s→1− and s→0+, which extend results due to Bourgain-Brezis-Mironescu (BBM) Maz'ya-Shaposhnikova (MS) respectively. Recently, Brezis, Van Schaftingen Yung provided different approach by replacing strong Lp norm in weak quasinorm. They characterized case for s=1 that complements BBM formula. The corre...

This paper considers a wavelet analogue of the classical Ginzburg-Landau energy, where the Hseminorm is replaced by the Besov seminorm defined via an arbitrary regular wavelet. We prove that functionals of this type converge, in the Γ-sense, to a weighted analogue of the TV functional on characteristic functions of finite-perimeter sets. The Γ-limiting functional is defined explicitly, in terms...

We study arbitrary-order Hermite difference methods for the numerical solution of initial-boundary value problems for symmetric hyperbolic systems. These differ from standard difference methods in that derivative data (or equivalently local polynomial expansions) are carried at each grid point. Time-stepping is achieved using staggered grids and Taylor series. We prove that methods using deriva...

Let l be a length function on a group G, and let Ml denote the operator of pointwise multiplication by l on l(G). Following Connes, Ml can be used as a “Dirac” operator for C ∗ r (G). It defines a Lipschitz seminorm on C∗ r (G), which defines a metric on the state space of C∗ r (G). We investigate whether the topology from this metric coincides with the weak-∗ topology (our definition of a “com...

Intuitively, the more regular a problem, the easier it should be to solve. Examples drawn from ordinary and partial differential equations, as well as from approximation, support the intuition. Traub and Wozniakowski conjectured that this is always the case. In this paper, we study linear problems. We prove a weak form of the conjecture, and show that this weak form cannot be strengthened. To d...

Strong stability preserving (SSP) high order time discretizations were developed to ensure nonlinear stability properties necessary in the numerical solution of hyperbolic partial differential equations with discontinuous solutions. SSP methods preserve the strong stability properties – in any norm, seminorm or convex functional – of the spatial discretization coupled with first order Euler tim...

Motivated by the interior tomography problem, we propose a method for exact reconstruction of a region of interest of a function from its local Radon transform in any number of dimensions. Our aim is to verify the feasibility of a one-dimensional reconstruction procedure that can provide the foundation for an efficient algorithm. For a broad class of functions, including piecewise polynomials a...

In contrast to the usual Lipschitz seminorms associated to ordinary metrics on compact spaces, we show by examples that Lipschitz seminorms on possibly non-commutative compact spaces are usually not determined by the restriction of the metric they define on the state space, to the extreme points of the state space. We characterize the Lipschitz norms which are determined by their metric on the ...

The integral fractional Laplacian of order $s \in (0,1)$ is a nonlocal operator. It known that solutions to the Dirichlet problem involving such an operator exhibit algebraic boundary singularity regardless domain regularity. This, in turn, deteriorates global regularity and as result convergence rate numerical solutions. For finite element discretizations, we derive local error estimates $H^s$...