On the Lattice Properties of Almost L-Weakly and Almost M-Weakly Compact Operators

We establish the domination property and some lattice approximation properties for almost L-weakly and almost M-weakly compact operators. Then, we consider the linear span of positive almost L-weakly (resp., almost M-weakly) compact operators and give results about when they form a Banach lattice and have an order continuous norm.

Efficient multivariate approximation on the cube

We combine a periodization strategy for weighted $$L_{2}$$ L 2 -integrands with efficient approximation methods in order to approximate multivariate non-periodic functions on the high-dimensional cube $$\left[ -\frac{1}{2},\frac{1}{2}\right] ^{d}$$ - 1 2 , 1 2 d . Our concept allows to determine conditions on the d-variate torus-to-cube transformations $${\psi :\left[ -\frac{1}{2},\frac{1}{2}\right] ^{d}\rightarrow \left[ -\frac{1}{2},\frac{1}{2}\right] ^{d}}$$ ψ : - 1 2 , 1 2 d → - 1 2 , 1 2 d such that a non-periodic function is transformed into a smooth function in the Sobolev space $${\mathcal {H}}^{m}(\mathbb {T}^{d})$$ H m ( T d ) when applying $$\psi $$ ψ . We adapt $$L_{\infty }(\mathbb {T}^{d})$$ L ∞ ( T d ) - and $$L_{2}(\mathbb {T}^{d})$$ L 2 ( T d ) -approximation error estimates for single rank-1 lattice approximation methods and adjust algorithms for the fast evaluation and fast reconstruction of multivariate trigonometric polynomials on the torus in order to apply these methods to the non-periodic setting. We illustrate the theoretical findings by means of numerical tests in up to $$d=5$$ d = 5 dimensions.

Generalized Darcy’s Law in Filtration Theory

We study the hydrodynamics of flow in a porous medium modeling the grain filling in filters. Using the lattice approximation, we derive the structure of the current in porous media and obtain the transverse diffusion coefficient D which proves to be proportional to the diameter d of the grains as constituents of the medium. We consider the axially-symmetric stationary flow in a cylindrical filter and show that the vertical velocity takes its maximal value at the wall, this effect being known as the “near-wall” one. We analyze the solution to the Euler equation with the modified Darcy force, which depends not only on the velocity but also on the gradient of the pressure included in the Darcy coefficient. Finally, within the scope of the perturbation method, we derive the main filtration equation and discuss the influence of modifying the Darcy’s law on the efficiency of the filtration process.

NONCOMMUTATIVE DE LEEUW THEOREMS

Let $\text{H}$ be a subgroup of some locally compact group $\text{G}$. Assume that $\text{H}$ is approximable by discrete subgroups and that $\text{G}$ admits neighborhood bases which are almost invariant under conjugation by finite subsets of $\text{H}$. Let $m:\text{G}\rightarrow \mathbb{C}$ be a bounded continuous symbol giving rise to an $L_{p}$-bounded Fourier multiplier (not necessarily completely bounded) on the group von Neumann algebra of $\text{G}$ for some $1\leqslant p\leqslant \infty$. Then, $m_{\mid _{\text{H}}}$ yields an $L_{p}$-bounded Fourier multiplier on the group von Neumann algebra of $\text{H}$ provided that the modular function ${\rm\Delta}_{\text{G}}$ is equal to 1 over $\text{H}$. This is a noncommutative form of de Leeuw’s restriction theorem for a large class of pairs $(\text{G},\text{H})$. Our assumptions on $\text{H}$ are quite natural, and they recover the classical result. The main difference with de Leeuw’s original proof is that we replace dilations of Gaussians by other approximations of the identity for which certain new estimates on almost-multiplicative maps are crucial. Compactification via lattice approximation and periodization theorems are also investigated.