We now discuss convergence of the Fourier series on compact intervals I. ‘Convergence’ depends on the notion of convergence we use, such as (i) L: uj → u in L if ‖uj − u‖L2 → 0 as j →∞. (ii) uniform, or C: uj → u uniformly if ‖uj−u‖C0 = supx∈I |uj(x)−u(x)| → 0. (iii) uniform with all derivatives, or C∞: uj → u in C∞ if for all non-negative integers k, supx∈I |∂uj(x)− ∂u(x)| → 0. (iv) pointwise:...

In this paper we study the convergence of a centred finite difference scheme on a non-uniform mesh for a 1D elliptic problem subject to general boundary conditions. On a non-uniform mesh, the scheme is, in general, only first-order consistent. Nevertheless, we prove for s ∈ (1/2, 2] order O(hs)-convergence of solution and gradient if the exact solution is in the Sobolev space H1+s(0, L), i.e. t...

The aim of this paper is to set up appropriate uniform convergence spaces in which to reformulate and enrich the Order Completion Method [25] for nonlinear PDEs. In this regard, we consider an appropriate space ML (X) of normal lower semi-continuous functions. The space ML (X) appears in the ring theory of C (X) and its various extensions [15], as well as in the theory of nonlinear, PDEs [25] a...

We are concerned with the inviscid limit of the Navier-Stokes equations to the Euler equations in R. We first observe that a pathwise Kolmogorov hypothesis implies the uniform boundedness of the α-order fractional derivatives of the velocity for some α > 0 in the space variables in L, which is independent of the viscosity μ > 0. Then it is shown that this key observation yields the Lequicontinu...

We prove that the additive group (E∗, τk(E)) of an L∞-Banach space E, with the topology τk(E) of uniform convergence on compact subsets of E, is topologically isomorphic to a subgroup of the unitary group of some Hilbert space (is unitarily representable). This is the same as proving that the topological group (E∗, τk(E)) is uniformly homeomorphic to a subset of ` κ 2 for some κ. As an immediat...

A net (xγ)γ∈Γ in a locally solid Riesz space (X,τ) is said to be unbounded τ-convergent x if |xγ−x|∧u⟶τ0 for all u∈X+. We recall that there linear topology uτ on X such τ-convergence coincides with uτ-convergence. It turns out characterised as the weakest which τ order bounded subsets. this motivation we introduce, uniform lattice (L,u), uniformity u⁎ L u subsets of L. shown induced by (X,τ), t...

A uniform space is a topological space together with some additional structure which allows one to make sense of uniform properties such as completeness or uniform convergence. Motivated by previous work of J. Rivera-Letelier, we give a new construction of the Berkovich analytic space associated to an affinoid algebra as the completion of a canonical uniform structure on the associated rigid-an...

We show that the category of convergence approach spaces is a simultaneously reective and coreective subcategory of the category of latticevalued limit spaces. Further we study the preservation of diagonal conditions, which characterize approach spaces. It is shown that the category of preapproach spaces is a simultaneously reective and coreective subcategory of the category of lattice-valued p...

We characterize all geometric perturbations of an open set, for which the solution of a nonlinear elliptic PDE of p-Laplacian type with Dirichlet boundary condition is stable in the L∞-norm. The necessary and sufficient conditions are jointly expressed by a geometric property associated to the γp-convergence. If the dimension of the space N satisfies N − 1 < p ≤ N and if the number of the conne...