Pointwise smoothness of space-filling functions

Pointwise smoothness of space-filling functions

We study irregularity properties of generic Peano functions; we apply these results to the determination of the pointwise smoothness of a Peano function introduced by Lebesgue and of some related functions, showing that they are either monohölder or multifractal functions. We test on these examples several numerical variants of the multifractal formalism, and we show how a change of topology on...

Pointwise Multipliers of Besov Spaces of Smoothness Zero and Spaces of Continuous Functions

Recurrence for Pandimensional Space-Filling Functions

A space-filling function is a bijection from the unit line segment to the unit square, cube, or hypercube. The function from the unit line segment is continuous. The inverse function, while welldefined, is not continuous. Space-filling curves, the finite approximations to space-filling functions, have found application in global optimization, database indexing, and dimension reduction among oth...

Graph-based Semi-supervised Learning: Realizing Pointwise Smoothness Probabilistically

As the central notion in semi-supervised learning, smoothness is often realized on a graph representation of the data. In this paper, we study two complementary dimensions of smoothness: its pointwise nature and probabilistic modeling. While no existing graph-based work exploits them in conjunction, we encompass both in a novel framework of Probabilistic Graph-based Pointwise Smoothness (PGP), ...

Smoothness of Multiple Refinable Functions

We consider the smoothness of solutions of a system of refinement equations written in the form φ = ∑ α∈Z a(α)φ(2 · − α), where the vector of functions φ = (φ1, . . . , φr) is in (Lp(R)) and a is a finitely supported sequence of r× r matrices called the refinement mask. We use the generalized Lipschitz space Lip∗(ν, Lp(R)), ν > 0, to measure smoothness of a given function. Our method is to rela...