arXiv subjects
Publications and source records attributed to Leonardo Colzani.
W. Schmidt, H. Montgomery, and J. Beck proved a result on irregularities of distribution with respect to $d$-dimensional balls. In this paper, we extend their result to any $d$-dimensional convex body with a smooth boundary and finite order of contact. As an intermediate step, we prove a geometric inequality for the Fourier transform of the characteristic function of a convex body.
Given a probability space $(X,\mu)$, a square integrable function $f$ on such space and a (unilateral or bilateral) shift operator $T$, we prove under suitable assumptions that the ergodic means $N^{-1}\sum_{n=0}^{N-1} T^nf$ converge pointwise almost everywhere to zero with a speed of convergence which, up to a small logarithmic transgression, is essentially of the order of $N^{-1/2}$. We also provide a few applications of our results, especially in the case of shifts associated with toral endomorphisms.
We prove several results concerning the discrepancy, tested on balls in the $d$-dimensional torus $\mathbb{T}^{d}$, between absolutely continuous measures and finite atomic measures.
Given an irrational vector $α$ in $\mathbb{R}^{d}$, a continuous function $f(x)$ on the torus $\mathbb{T}^{d}$ and suitable weights $Φ(N,n)$ such that $\sum_{n=-\infty}^{+\infty}Φ(N,n)=1$, we estimate the speed of convergence to the integral $\int_{\mathbb{T}^{d}}f(y)dy$ of the weighted sum $\sum_{n=-\infty}^{+\infty}Φ(N,n) f(x+nα)$ as $N\rightarrow +\infty$. Whereas for the arithmetic means $N^{-1}\sum_{n=1}^{N}f(x+nα)$ the speed of convergence is never faster than $cN^{-1}$, for other means such speed can be accelerated. We estimate the speed of convergence in two theorems with different flavor. The first result is a metric one, and it provides an estimate of the speed of convergence in terms of the Fourier transform of the weights $Φ(N,n)$ and the smoothness of the function $f(x)$ which holds for almost every $α$. The second result is a deterministic one, and the speed of convergence is estimated also in terms of the Diophantine properties of the given irrational vector $α\in\mathbb R^d$.
We prove a general result on irregularities of distribution for Borel sets intersected with bounded measurable sets or affine half-spaces.
We provide a multidimensional weighted Euler--MacLaurin summation formula on polytopes and a multidimensional generalization of a result due to L. J. Mordell on the series expansion in Bernoulli polynomials. These results are consequences of a more general series expansion; namely, if $χ_{τ\mathcal{P}}$ denotes the characteristic function of a dilated integer convex polytope $\mathcal{P}$ and $q$ is a function with suitable regularity, we prove that the periodization of $qχ_{τ\mathcal{P}}$ admits an expansion in terms of multivariate Bernoulli polynomials. These multivariate polynomials are related to the Lerch Zeta function. In order to prove our results we need to carefully study the asymptotic expansion of $\widehat{qχ_{τ\mathcal{P}}}$, the Fourier transform of $qχ_{τ\mathcal{P}}$.
We prove an Euler-Maclaurin formula for double polygonal sums and, as a corollary, we obtain approximate quadrature formulas for integrals of smooth functions over polygons with integer vertices. Our Euler-Maclaurin formula is in the spirit of Pick's theorem on the number of integer points in an integer polygon and involves weighted Riemann sums, using tools from Harmonic analysis. Finally, we also exhibit a classical trick, dating back to Huygens and Newton, to accelerate convergence of these Riemann sums.
We add another brick to the large building comprising proofs of Pick's theorem. Although our proof is not the most elementary, it is short and reveals a connection between Pick's theorem and the pointwise convergence of multiple Fourier series of piecewise smooth functions.
We consider the discrepancy of the integer lattice with respect to the collection of all translated copies of a dilated convex body having a finite number of flat, possibly non-smooth, points in its boundary. We estimate the $L^{p}$ norm of the discrepancy with respect to the translation variable as the dilation parameter goes to infinity. If there is a single flat point with normal in a rational direction we obtain an asymptotic expansion for this norm. Anomalies may appear when two flat points have opposite normals. When all the flat points have normals in generic irrational directions, we obtain a smaller discrepancy. Our proofs depend on careful estimates for the Fourier transform of the characteristic function of the convex body.
We give asymptotic estimates of the variance of the number of integer points in translated thin annuli in any dimension.
We estimate the $L^{p}$ 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 $\mathbb{R}^{d}$ with smooth boundary with strictly positive curvature, \[ \left\{ {\displaystyle\int_{\mathbb R}}{\displaystyle\int_{\mathbb{T}^{d}}}\left\vert \sum_{k\in\mathbb{Z}^{d}}χ_{rΩ-x}(k)-r^{d}\left\vert Ω\right\vert \right\vert ^{p}dxdμ(r-R) \right\} ^{1/p}, \] where $μ$ is a Borel measure compactly supported on the positive real axis and $R\to+\infty$.
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 $H\to\infty$.
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.
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.
We study here the error of numerical integration on metric measure spaces adapted to a decomposition of the space into disjoint subsets. We consider both the error for a single given function, and the worst case error for all functions in a given class of potentials. The main tools are the classical Marcinkiewicz-Zygmund inequality and ad hoc definitions of function spaces on metric measure spaces. The same techniques are used to prove the existence of point distributions in metric measure spaces with small $L^p$ discrepancy with respect to certain classes of subsets, for example metric balls.
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.
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.