arXiv subjects

Giacomo Gigante

Publications and source records attributed to Giacomo Gigante.

27 records · Page 2Linked to original sources

arXiv2017

Mixed $L^p(L^2)$ norms of the lattice point discrepancy

We estimate some mixed $L^{p}\left( L^{2}\right) $ norms of the discrepancy between the volume and the number of integer points in $rΩ-x$, a dilated by a factor $r$ and translated by a vector $x$ of a convex body $Ω$ in $\mathbb{R}^{d}$, $ \left\{ {\int_{\mathbb{T}^{d}}}\left( \frac{1}{H} {\int_{R}^{R+H}}\left\vert \sum_{k\in\mathbb{Z}^{d}}χ_{rΩ-x}(k)-r^{d}\left\vert Ω\right\vert \right\vert^{2}dr\right)^{p/2}dx\right\} ^{1/p}. $ We obtain estimates for fixed values of $H$ and $R\to\infty$, and also asymptotic estimates when $H\to\infty$.

math.NT↗

arXiv2016

Diameter bounded equal measure partitions of Ahlfors regular metric measure spaces

The algorithm devised by Feige and Schechtman for partitioning higher dimensional spheres into regions of equal measure and small diameter is combined with David and Christ's construction of dyadic cubes to yield a partition algorithm suitable to any connected Ahlfors regular metric measure space of finite measure.

math.NA↗

arXiv2015

Low-discrepancy sequences for piecewise smooth functions on the two-dimensional torus

We produce explicit low-discrepancy infinite sequences which can be used to approximate the integral of a smooth periodic function restricted to a convex domain with positive curvature in R^2. The proof depends on simultaneous diophantine approximation and a general version of the Erdos-Turan inequality.

math.NT↗

arXiv2015

$L^{p}$ and Weak-$L^{p}$ estimates for the number of integer points in translated domains

Revisiting and extending a recent result of M.Huxley, we estimate the $L^{p}\left( \mathbb{T}^{d}\right) $ and Weak-$L^{p}\left( \mathbb{T}^{d}\right) $ norms of the discrepancy between the volume and the number of integer points in translated domains.

math.NT↗

arXiv2010

Quadrature rules and distribution of points on manifolds

We study the error in quadrature rules on a compact manifold. As in the Koksma-Hlawka inequality, we consider a discrepancy of the sampling points and a generalized variation of the function. In particular, we give sharp quantitative estimates for quadrature rules of functions in Sobolev classes.

math.NT↗

arXiv2010

Trigonometric approximation and a general form of the Erdős Turán inequality

There exists a positive function $ψ(t)${on}$t\geq0${, with fast decay at infinity, such that for every measurable set}$Ω${in the Euclidean space and}$R>0${, there exist entire functions}$A(x) ${and}$B(x) ${of exponential type}$R${, satisfying\}$A(x)\leq χ_Ω(x)\leq B(x)${and}$| B(x)-A(x)| \leqslantψ(R\operatorname*{dist}(x,\partialΩ)) $. This leads to Erdős Turán estimates for discrepancy of point set distributions in the multi dimensional torus. Analogous results hold for approximations by eigenfunctions of differential operators and discrepancy on compact manifolds.

math.NT↗

arXiv2010

Convolution operators defined by singular measures on the motion group

This paper contains an $L^{p}$ improving result for convolution operators defined by singular measures associated to hypersurfaces on the motion group. This needs only mild geometric properties of the surfaces, and it extends earlier results on Radon type transforms on $\mathbb{R}^{n}$. The proof relies on the harmonic analysis on the motion group.

math.FA↗

arXiv2006

Average decay estimates for Fourier transforms of measures supported on curves

We consider Fourier transforms of densities supported on curves in R^d. We obtain sharp lower and close to sharp upper bounds for the L^q decay rates.

math.CA↗

arXiv2003

On the Boundedness in $H^{1/4}$ of the Maximal Square Function Associated with the Schroedinger Equation

A long standing conjecture for the linear Schroedinger equation states that 1/4 of derivative in $L^2$, in the sense of Sobolev spaces, suffices in any dimension for the solution to that equation to converge almost everywhere to the initial datum as the time goes to 0. This is only known to be true in dimension 1 by work of Carleson. In this paper we show that the conjecture is true on spherical averages. To be more precise, we prove the $L^2$ boundedness of the associated maximal square function on the Sobolev class $H^{1/4}(R^n)$ in any dimension n.

math.CA↗