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Bianca Gariboldi

Publications and source records attributed to Bianca Gariboldi.

15 recordsLinked to original sources
arXiv2025

Sampling theorems for inverse problems on Riemannian manifolds

We consider inverse problems consisting of the reconstruction of an unknown signal $f$ from noisy measurements $y=Ff+\text{noise}$, where $Ff$ is a function on a Riemannian manifold without boundary $\mathcal M$. We consider the case when only pointwise samples are available, namely $y_j = (Ff)(x_j)+\eta_j$, where $\{x_j\}_{j=1}^n\subseteq\mathcal M$ is a Marcinkiewicz-Zygmund family. We derive sampling theorems providing explicit bounds on the reconstruction error depending on $n$, the smoothness of $f$ and the properties of $F$. We study in detail the case when $F$ is a convolution on a compact two-point homogeneous space. As a corollary, we state a sampling theorem for convolutions on the two-dimensional sphere, and discuss four relevant examples related to terrestrial and celestial measurements.

math.FA↗
arXiv2025

On a discrete approach to lower bounds in discrepancy theory

In this paper, we prove that some renowned lower bounds in discrepancy theory admit a discrete analogue. Namely, we prove that the lower bound of the discrepancy for corners in the unit cube due to Roth holds true also for a suitable finite family of corners. We also prove two analogous results for the discrepancy on the torus with respect to squares and balls.

math.CA↗
arXiv2024

Single radius spherical cap discrepancy on compact two-point homogeneous spaces

In this note we study estimates from below of the single radius spherical discrepancy in the setting of compact two-point homogeneous spaces. Namely, given a $d$-dimensional manifold $\mathcal M$ endowed with a distance $ρ$ so that $(\mathcal M, ρ)$ is a two-point homogeneous space and with the Riemannian measure $μ$, we provide conditions on $r$ such that if $D_r$ denotes the discrepancy of the ball of radius $r$, then, for an absolute constant $C>0$ and for every set of points $\{x_j\}_{j=1}^N$, one has $\int_{\mathcal M} |D_{r}(x)|^2\, dμ(x)\geqslant C N^{-1-\frac1d}$. The conditions on $r$ that we have depend on the dimension $d$ of the manifold and cannot be achieved when $d \equiv 1 \ ( \operatorname{mod}4)$. Nonetheless, we prove a weaker estimate for such dimensions as well.

math.CA↗
arXiv2023

On the speed of convergence in the ergodic theorem for shift operators

Given a probability space $(X,μ)$, 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.

math.CA↗
arXiv2022

Irregularities of distribution on two point homogeneous spaces

We study the irregularities of distribution on two-point homogeneous spaces. Our main result is the following: let $d$ be the real dimension of a two point homogeneous space $\mathcal{M}$, let $\left( \{ a_{j}\} _{j=1}^{N},\{ x_{j}\} _{j=1}^{N}\right) $ be a system of positive weights and points on $\mathcal{M}$ and let \[ D_{r}( x) =\sum_{j=1}^{N}a_{j}χ_{B_{r}(x)}(x_{j})-μ(B_{r}(x)) \] be the discrepancy associated with the ball $B_{r}( x) $. Then, if $d\not \equiv 1(\operatorname{mod}4)$, for any radius $0<r<π/2$, we obtain the sharp estimate \[ \int_{\mathcal{M}}\left( \left\vert D_{r}( x) \right\vert ^{2}+\left\vert D_{2r}( x) \right\vert ^{2}\right) dμ( x) \geqslant cN^{-1-\frac{1}{d}}. \]

math.AP↗
arXiv2022

Euler-MacLaurin summation formula on polytopes and expansions in multivariate Bernoulli polynomials

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 $qχ_{τ\mathcal{P}}$.

math.CA↗
arXiv2022

Almost positive kernels on compact Riemannian manifolds

We show how to build a kernel \[ K_X(x,y)=\sum_{m=0}^Xh(λ_m/{λ_X})φ_m(x)\overline{φ_m(y)} \] on a compact Riemannian manifold $M$, which is positive up to a negligible error and such that $K_X(x,x)\approx X$. Here $0=λ_0^2\leλ_1^2\le\ldots$ are the eigenvalues of the Laplace-Beltrami operator on $M$, listed with repetitions, and $φ_0,\,φ_1,\ldots$ an associated system of eigenfunctions, forming an orthonormal basis of $L^2(M)$. The function $h$ is smooth up to a certain minimal degree, even, compactly supported in $[-1,1]$ with $h(0)=1$, and $K_X(x,y)$ turns out to be an approximation to the identity.

math.AP↗
arXiv2020

On a sharp lemma of Cassels and Montgomery on manifolds

Let $\left( \mathcal{M},g\right) $ be a $d$-dimensional compact connected Riemannian manifold and let $\left\{ φ_{m}\right\}_{m=0}^{+\infty}$ be a complete sequence of orthonormal eigenfunctions of the Laplace-Beltrami operator on $\mathcal{M}$. We show that there exists a positive constant $C$ such that for all integers $N$ and $X$ and for all finite sequences of $N$ points in $\mathcal{M}$, $\left\{ x\left( j\right) \right\}_{j=1}^{N}$, and positive weights $\left\{ a_{j}\right\}_{j=1}^{N}$ we have \[ \sum_{m=0}^{X} | \sum_{j=1}^{N} a_{j} φ_{m} ( x( j) ) | ^{2}\geq \max \{ CX\sum_{j=1}^{N}a_{j}^{2},( \sum_{j=1}^{N}a_{j}) ^{2}\}.\]

math.AP↗
arXiv2019

Discrepancy of a convex set with zero curvature at one point

Let $Ω\subset \mathbb{R}^{d}$ be a convex body with everywhere positive curvature except at the origin and with the boundary $\partial Ω$ as the graph of the function $y=|x|^γ$ in a neighborhood of the origin with $γ\geq 2$. We consider the $L^{p}$ norm of the discrepancy with respect to translations and rotations of a dilated copy of the set $Ω$.

math.NT↗
arXiv2018

Optimal asymptotic bounds for designs on manifolds

We extend to the case of a $d$-dimensional compact connected oriented Riemannian manifold $\mathcal M$ the theorem of A. Bondarenko, D. Radchenko and M. Viazovska on the existence of $L$-designs consisting of $N$ nodes, for any $N\ge C_{\mathcal M} L^d$. For this, we need to prove a version of the Marcinkiewicz-Zygmund inequality for the gradient of diffusion polynomials.

math.AP↗
arXiv2018

Discrepancy for convex bodies with isolated flat points

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.

math.FA↗
arXiv2018

$L^p$ norms of the lattice point discrepancy

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 $R\to+\infty$.

math.CA↗
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↗