Luca Brandolini

Localization and convergence of eigenfunction expansions

We state a localization principle for expansions in eigenfunctions of a self-adjoint second order elliptic operator and we prove an equiconvergence result between eigenfunction expansions and trigonometric expansions. We then study the Gibbs phenomenon for eigenfunction expansions of piecewise smooth functions on two-dimensional manifolds.

Non-divergence equations structured on Hörmander vector fields: heat kernels and Harnack inequalities

In this work the authors deal with linear second order partial differential operators of the following type

Average decay of Fourier transforms and geometry of convex sets

H=\partial_{t}-L=\partial_{t}-\sum_{i,j=1}^{q}a_{ij}(t,x) X_{i}X_{j}-\sum_{k=1}^{q}a_{k}(t,x)X_{k}-a_{0}(t,x)

On the Koksma-Hlawka inequality

where

Quadrature rules and distribution of points on manifolds

X_{1},X_{2},\ldots,X_{q}

Basic properties of nonsmooth Hörmander's vector fields and Poincaré's inequality

is a system of real Hormanders vector fields in some bounded domain

Fourier Analysis and Convexity

\Omega\subseteq\mathbb{R}^{n}

A Koksma-Hlawka inequality for simplices

,

AVERAGE DECAY ESTIMATES FOR FOURIER TRANSFORMS OF MEASURES SUPPORTED ON CURVES

A=\left\{ a_{ij}\left( t,x\right) \right\} _{i,j=1}^{q}

Planar convex bodies, Fourier transform, lattice points, and irregularities of distribution

is a real symmetric uniformly positive definite matrix such that