We prove a non conventional pointwise convergence theorem for a nilsystem, and give an explicit formula for the limit.

Every clone of functions comes naturally equipped with a topology—the topology of pointwise convergence. A clone C is said to have automatic homeomorphicity with respect to a class C of clones, if every clone-isomorphism of C to a member of C is already a homeomorphism (with respect to the topology of pointwise convergence). In this paper we study automatic homeomorphicity-properties for polymo...

This paper analyzes the convergence of discrete approximations to the linearized equations arising from an unsteady one-dimensional hyperbolic equation with a convex flux function. A simple modified Lax–Friedrichs discretization is used on a uniform grid, and a key point is that the numerical smoothing increases the number of points across the nonlinear discontinuity as the grid is refined. It ...

We approximate functionals depending on the gradient of u and on the behaviour of u near the discontinuity points, by families of non-local functionals where the gradient is replaced by finite differences. We prove pointwise convergence, Γ-convergence, and a compactness result which implies, in particular, the convergence of minima and minimizers. AMS (MOS) subject classifications: 49J45, 65K10.

We show that if X is a Σ 1 separable metrizable space which is not σ-compact then C p (X), the space of bounded real-valued continuous functions on X with the topology of pointwise convergence, is Borel-Π 1 -complete. Assuming projective determinacy we show that if X is projective not σ-compact and n is least such that X is Σ1n then Cp(X), the space of real-valued continuous functions on X with...

The cascade algorithm plays an important role in computer graphics and wavelet analysis. For an initial function φ0, a cascade sequence (φn)n=0 is constructed by the iteration φn = Caφn−1, n = 1, 2, . . . , where Ca is defined by Cag = ∑ α∈Z a(α)g(2 · −α), g ∈ Lp(R). In this paper, under a condition that the sequence (φn)n=0 is bounded in L∞(R), we prove that the following three statements are ...

We establish weak and strong convergence of Ishikawa type iterates of two pointwise asymptotic nonexpansive maps in a Hadamard space. For weak and strong convergence results, we drop “rate of convergence condition”, namely , to answer in the affirmative to the open question posed by Tan and Xu [1] even in a general setup.     1 1 n n c x      

We approximate functionals depending on the gradient of u and on the behaviour of u near the discontinuity points, by families of non-local function-als where the gradient is replaced by nite diierences. We prove pointwise convergence, ?-convergence, and a compactness result which implies, in particular , the convergence of minima and minimizers.