The regularization functional induced by the graph Laplacian of a random neighborhood graph based on the data is adaptive in two ways. First it adapts to an underlying manifold structure and second to the density of the data-generating probability measure. We identify in this paper the limit of the regularizer and show uniform convergence over the space of Hölder functions. As an intermediate s...

We characterize convex cocompact subgroups of the mapping class group of a surface in terms of uniform convergence actions on the zero locus of the limit set. We also construct subgroups that act as uniform convergence groups on their limit sets, but are not convex cocompact.

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:...

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:...

It is a classical problem in Fourier analysis to give conditions for a single sine or cosine series to be uniformly convergent. Several authors gave conditions for this problem supposing that the coefficients are monotone, non-negative or more recently, general monotone. There are also results for the regular convergence of double sine series to be uniform in case the coefficients are monotone ...

The roots of Bernoulli polynomials, Bn(z), when plotted in the complex plane, accumulate around a peculiar H-shaped curve. Karl Dilcher proved in 1987 that, on compact subsets of C, the Bernoulli polynomials asymptotically behave like sine or cosine. Here we establish the asmptotic behavior of Bn(nz), compute the distribution of real roots of Bernoulli polynomials and show that, properly rescal...

The spectral shift function ξL(E) for a Schrödinger operator restricted to a finite cube of length L in multi-dimensional Euclidean space, with Dirichlet boundary conditions, counts the number of eigenvalues less than or equal to E ∈ R created by a perturbation potential V . We study the behavior of this function ξL(E) as L→∞ for the case of a compactly-supported and bounded potential V . After...

Kernel density estimation (KDE) is a popular nonparametric density estimation method. We (1) derive finite-sample high-probability density estimation bounds for multivariate KDE under mild density assumptions which hold uniformly in x ∈ R and bandwidth matrices. We apply these results to (2) mode, (3) density level set, and (4) class probability estimation and attain optimal rates up to logarit...

For n particles diiusing throughout R (or R d), let n;t (A), A 2 B, t 0, be the random measure that counts the number of particles in A at time t. It is shown that for some basic models (Brownian particles with or without branching and diiusion with a simple interaction) the processes f(n;t () ? E n;t ())= p n : t 2 0; M]; 2 C L (R)g, n 2 N, converge in law uniformly in (t;). Previous results c...

We study the continuity of the correspondence of interim ε-rationalizable actions in incomplete information games. We introduce a topology on types, called uniformweak topology, under which two types of a player are close if they have similar firstorder beliefs, attach similar probabilities to other players having similar first-order beliefs, and so on, where the degree of similarity is uniform...