arXiv subjects

Stefan Steinerberger

Publications and source records attributed to Stefan Steinerberger.

215 records · Page 12Linked to original sources

arXiv2015

A Rigidity Phenomenon for the Hardy-Littlewood Maximal Function

The Hardy-Littlewood maximal function $\mathcal{M}$ and the trigonometric function $\sin{x}$ are two central objects in harmonic analysis. We prove that $\mathcal{M}$ characterizes $\sin{x}$ in the following way: let $f \in C^α(\mathbb{R}, \mathbb{R})$ be a periodic function and $α> 1/2$. If there exists a real number $0 < γ< \infty$ such that the averaging operator $$ (A_xf)(r) = \frac{1}{2r}\int_{x-r}^{x+r}{f(z)dz}$$ has a critical point in $r = γ$ for every $x \in \mathbb{R}$, then $$f(x) = a+b\sin{(cx + d)} \qquad \mbox{for some}~a,b,c,d \in \mathbb{R}.$$ This statement can be used to derive a characterization of trigonometric functions as those nonconstant functions for which the computation of the maximal function $\mathcal{M}$ is as simple as possible. The proof uses the Lindemann-Weierstrass theorem from transcendental number theory.

math.CA↗

arXiv2015

Localization of Quantum States and Landscape Functions

Eigenfunctions in inhomogeneous media can have strong localization properties. Filoche \& Mayboroda showed that the function $u$ solving $(-Δ+ V)u = 1$ controls the behavior of eigenfunctions $(-Δ+ V)ϕ= λϕ$ via the inequality $$|ϕ(x)| \leq λu(x) \|ϕ\|_{L^{\infty}}.$$ This inequality has proven to be remarkably effective in predicting localization and recently Arnold, David, Jerison, Mayboroda \& Filoche connected $1/u$ to decay properties of eigenfunctions. We aim to clarify properties of the landscape: the main ingredient is a localized variation estimate obtained from writing $ϕ(x)$ as an average over Brownian motion $ω(\cdot)$ in started in $x$ $$ϕ(x) = \mathbb{E}_{x}\left(ϕ(ω(t)) e^{λt-\int_{0}^{t}{V(ω(z))dz}} \right).$$ This variation estimate will guarantee that $ϕ$ has to change at least by a factor of 2 in a small ball, which implicitly creates a landscape whose relationship with $1/u$ we discuss.

math.SP↗

arXiv2015

On the Discrepancy of Jittered Sampling

We study the discrepancy of jittered sampling sets: such a set $\mathcal{P} \subset [0,1]^d$ is generated for fixed $m \in \mathbb{N}$ by partitioning $[0,1]^d$ into $m^d$ axis aligned cubes of equal measure and placing a random point inside each of the $N = m^d$ cubes. We prove that, for $N$ sufficiently large, $$ \frac{1}{10}\frac{d}{N^{\frac{1}{2} + \frac{1}{2d}}} \leq \mathbb{E} D_N^*(\mathcal{P}) \leq \frac{\sqrt{d} (\log{N})^{\frac{1}{2}}}{N^{\frac{1}{2} + \frac{1}{2d}}},$$ where the upper bound with an unspecified constant $C_d$ was proven earlier by Beck. Our proof makes crucial use of the sharp Dvoretzky-Kiefer-Wolfowitz inequality and a suitably taylored Bernstein inequality; we have reasons to believe that the upper bound has the sharp scaling in $N$. Additional heuristics suggest that jittered sampling should be able to improve known bounds on the inverse of the star-discrepancy in the regime $N \gtrsim d^d$. We also prove a partition principle showing that every partition of $[0,1]^d$ combined with a jittered sampling construction gives rise to a set whose expected squared $L^2-$discrepancy is smaller than that of purely random points.

math.NA↗

arXiv2015

Lower bounds on nodal sets of eigenfunctions via the heat flow

We study the size of nodal sets of Laplacian eigenfunctions on compact Riemannian manifolds without boundary and recover the currently optimal lower bound by comparing the heat flow of the eigenfunction with that of an artifically constructed diffusion process. The same method should apply to a number of other questions; for example, we prove a sharp result saying that a nodal domain cannot be entirely contained in a small neighbourhood of a 'reasonably flat' surface. We expect the arising concepts to have more connections to classical theory and pose some conjectures in that direction.

math.AP↗

arXiv2015

Sharp L^1 Poincare inequalities correspond to optimal hypersurface cuts

Let $Ω\subset \mathbb{R}^n$ be a convex. If $u: Ω\rightarrow \mathbb{R}$ has mean 0, then we have the classical Poincaré inequality $$ \|u \|_{L^p} \leq c_p \mbox{diam}(Ω) \| \nabla u \|_{L^p}$$ with sharp constants $c_2 = 1/π$ (Payne \& Weinberger, 1960) and $c_1 = 1/2$ (Acosta \& Duran, 2005) independent of the dimension. The sharp constants $c_p$ for $1 < p < 2$ have recently been found by Ferone, Nitsch \& Trombetti (2012). The purpose of this short paper is to prove a much stronger inequality in the endpoint $L^1$: we combine results of Cianchi and Kannan, Lovász \& Simonovits to show that $$\left\|u\right\|_{L^{1}(Ω)} \leq \frac{2}{\log{2}} M_{}(Ω) \left\|\nabla u\right\|_{L^{1}(Ω)}$$ where $M_{}(Ω)$ is the average distance between a point in $Ω$ and the center of gravity of $Ω$. If $Ω$ is a simplex, this yields an improvement by a factor of $\sim \sqrt{n}$ in $n$ dimensions. By interpolation, this implies that that for every convex $Ω\subset \mathbb{R}^n$ and every $u:Ω\rightarrow \mathbb{R}$ with mean 0 $$ \left\|u\right\|_{L^{p}(Ω)}\leq \left(\frac{2}{\log{2}} M_{}(Ω) \right)^{\frac{1}{p}}\mbox{diam}(Ω)^{1-\frac{1}{p}}\left\|\nabla u\right\|_{L^{p}(Ω)}. $$

math.CA↗

arXiv2015

Lower bounds for the truncated Hilbert transform

Given two intervals $I, J \subset \mathbb{R}$, we ask whether it is possible to reconstruct a real-valued function $f \in L^2(I)$ from knowing its Hilbert transform $Hf$ on $J$. When neither interval is fully contained in the other, this problem has a unique answer (the nullspace is trivial) but is severely ill-posed. We isolate the difficulty and show that by restricting $f$ to functions with controlled total variation, reconstruction becomes stable. In particular, for functions $f \in H^1(I)$, we show that $$ \|Hf\|_{L^2(J)} \geq c_1 \exp{\left(-c_2 \frac{\|f_x\|_{L^2(I)}}{\|f\|_{L^2(I)}}\right)} \| f \|_{L^2(I)} ,$$ for some constants $c_1, c_2 > 0$ depending only on $I, J$. This inequality is sharp, but we conjecture that $\|f_x\|_{L^2(I)}$ can be replaced by $\|f_x\|_{L^1(I)}$.

math.CA↗

arXiv2015

Prescribing the nodal set of the first eigenfunction in each conformal class

We consider the problem of prescribing the nodal set of the first nontrivial eigenfunction of the Laplacian in a conformal class. Our main result is that, given a separating closed hypersurface $Σ$ in a compact Riemannian manifold $(M,g_0)$ of dimension $d \geq 3$, there is a metric $g$ on $M$ conformally equivalent to $g_0$ and with the same volume such that the nodal set of its first nontrivial eigenfunction is a $C^0$-small deformation of $Σ$ (i.e., $Φ(Σ)$ with $Φ: M \to M$ a diffeomorphism arbitrarily close to the identity in the $C^0$ norm).

math.DG↗

arXiv2014

A Remark on Disk Packings and Numerical Integration of Harmonic Functions

We are interested in the following problem: given an open, bounded domain $Ω\subset \mathbb{R}^2$, what is the largest constant $α= α(Ω) > 0$ such that there exist an infinite sequence of disks $B_1, B_2, \dots, B_N, \dots \subset \mathbb{R}^2$ and a sequence $(n_i)$ with $n_i \in \left\{1,2\right\}$ such that $$ \sup_{N \in \mathbb{N}}{N^α\left\| χ_Ω - \sum_{i=1}^{N}{(-1)^{n_i}χ_{B_i}}\right\|_{L^1(\mathbb{R}^2)}} < \infty,$$ where $χ$ denotes the characteristic function? We prove that certain (somewhat peculiar) domains $Ω\subset \mathbb{R}^2$ satisfy the property with $α= 0.53$. For these domains there exists a sequence of points $(x_i)_{i=1}^{\infty}$ in $Ω$ with weights $(a_i)_{i=1}^{\infty}$ such that for all harmonic functions $u:\mathbb{R}^2 \rightarrow \mathbb{R}$ $$ \left|\int_Ω{u(x)dx} - \sum_{i=1}^{N}{a_i u(x_i)}\right| \leq C_Ω\frac{\|u\|_{L^{\infty}(Ω)}}{N^{0.53}},$$ where $C_Ω$ depends only on $Ω$. This gives a Quasi-Monte-Carlo method for harmonic functions which improves on the probabilistic Monte-Carlo bound $\|u\|_{L^{2}(Ω)}/N^{0.5}$ \textit{without} introducing a dependence on the total variation. We do not know which decay rates are optimal.

math.NA↗

arXiv2014

Local Extrema in Quantum Chaos

We numerically investigate the distribution of extrema of 'chaotic' Laplacian eigenfunctions on two-dimensional manifolds. Our contribution is two-fold: (a) we count extrema on grid graphs with a small number of randomly added edges and show the behavior to coincide with the 1957 prediction of Longuet-Higgins for the continuous case and (b) compute the regularity of their spatial distribution using \textit{discrepancy}, which is a classical measure from the theory of Monte Carlo integration. The first part suggests that grid graphs with randomly added edges should behave like two-dimensional surfaces with ergodic geodesic flow; in the second part we show that the extrema are more regularly distributed in space than the grid $\mathbb{Z}^2$.

nlin.CD↗

arXiv2014

An uncertainty principle on compact manifolds

Breitenberger's uncertainty principle on the torus $\mathbb{T}$ and its higher-dimensional analogue on $\mathbb{S}^{d-1}$ are well understood. We give describe an entire family of uncertainty principles on compact manifolds $(M,g)$, which includes the classical Heisenberg-Weyl uncertainty principle (for $M=B(0,1) \subset \mathbb{R}^d$ the unit ball with the flat metric) and the Goh-Goodman uncertainty principle (for $M=\mathbb{S}^{d-1}$ with the canonical metric) as special cases. This raises a new geometric problem related to small-curvature low-distortion embeddings: given a function $f:M \rightarrow \mathbb{R}$, which uncertainty principle in our family yields the best result? We give a (far from optimal) answer for the torus, discuss disconnected manifolds and state a variety of other open problems.

math.CA↗

arXiv2014

A filtering technique for Markov chains with applications to spectral embedding

Spectral methods have proven to be a highly effective tool in understanding the intrinsic geometry of a high-dimensional data set $\left\{x_i \right\}_{i=1}^{n} \subset \mathbb{R}^d$. The key ingredient is the construction of a Markov chain on the set, where transition probabilities depend on the distance between elements, for example where for every $1 \leq j \leq n$ the probability of going from $x_j$ to $x_i$ is proportional to $$ p_{ij} \sim \exp \left( -\frac{1}{\varepsilon}\|x_i -x_j\|^2_{\ell^2(\mathbb{R}^d)}\right) \qquad \mbox{where}~\varepsilon>0~\mbox{is a free parameter}.$$ We propose a method which increases the self-consistency of such Markov chains before spectral methods are applied. Instead of directly using a Markov transition matrix $P$, we set $p_{ii} = 0$ and rescale, thereby obtaining a transition matrix $P^*$ modeling a non-lazy random walk. We then create a new transition matrix $Q = (q_{ij})_{i,j=1}^{n}$ by demanding that for fixed $j$ the quantity $q_{ij}$ be proportional to $$ q_{ij} \sim \min((P^*)_{ij}, ((P^*)^2)_{ij}, \dots, ((P^*)^k)_{ij}) \qquad \mbox{where usually}~ k=2.$$ We consider several classical data sets, show that this simple method can increase the efficiency of spectral methods and prove that it can correct randomly introduced errors in the kernel.

cs.DM↗

arXiv2014

Convolution Estimates for Singular Measures and Some Global Nonlinear Brascamp-Lieb Inequalities

We give a $L^2\times L^2 \rightarrow L^2$ convolution estimate for singular measures supported on transversal hypersurfaces in $\mathbb{R}^n$, which improves earlier results of Bejenaru, Herr & Tataru as well as Bejenaru & Herr. The arising quantities are relevant in the study of the validity of bilinear estimates for dispersive partial differential equations. We also prove a class of global, nonlinear Brascamp-Lieb inequalities with explicit constants in the same spirit.

math.CA↗

arXiv2014

On the curvature of level sets of harmonic functions

If a real harmonic function inside the open unit disk $B(0,1) \subset \mathbb{R}^2$ has its level set $\left\{x: u(x) = u(0)\right\}$ diffeomorphic to an interval, then we prove the sharp bound $κ\leq 8$ on the curvature of the level set $\left\{x: u(x) = u(0)\right\}$ in the origin. The bound is sharp and we give the unique (up to symmetries) extremizer.

math.CA↗

arXiv2014

New Bounds for the Traveling Salesman Constant

Let $X_1, X_2, \dots, X_n$ be independent and uniformly distributed random variables in the unit square $[0,1]^2$ and let $L(X_1, \dots, X_n)$ be the length of the shortest traveling salesman path through these points. In 1959, Beardwood, Halton $\&$ Hammersley proved the existence of a universal constant $β$ such that $$ \lim_{n \rightarrow \infty}{n^{-1/2}L(X_1, \dots, X_n)} = β\qquad \mbox{almost surely.}$$ The best bounds for $β$ are still the ones originally established by Beardwood, Halton $\&$ Hammersley $0.625 \leq β\leq 0.922$. We slightly improve both upper and lower bounds.

math.PR↗

arXiv2013

A geometric uncertainty principle with an application to Pleijel's estimate

Consider partitions of an open, bounded domain in $\mathbb{R}^n$. Then an average element of the partition has either its Fraenkel asymmetry or its deviation from the smallest element in the partition bounded away from 0 by a universal constant. As an application, we give an (unspecified) improvement of Pleijel's estimate on the number of nodal domains of a Laplacian eigenfunction similar to recent work of Bourgain and improve a bound coming from spectral partition problems.

math.MG↗

arXiv2013

Minimal Periods for Ordinary Differential Equations in Strictly Convex Banach Spaces and Explicit Bounds for some l^p-Spaces

Let x(t) be a non-constant T-periodic solution to the ordinary differential equation x'= f(x) in a Banach space X where f is assumed to be Lipschitz continuous with constant L. Then there exists a constant c such that T L >= c, with c only depending on X. It is known that c >= 6 in any Banach space and that c = 2π in any Hilbert space, but whereas the bound of c = 2 pi is sharp in any Hilbert space, there exists only one known example of a Banach space such that c = 6 is optimal. In this paper, we show that the inequality is in fact strict in any strictly convex Banach space. Moreover, we improve the lower bound for l^p(R^n) and L^p(M, μ) for a range of p close to p = 2 by using a form of Wirtinger's inequality for functions in W^{1,p}([0, T ], L^p(M, μ)).

math.CA↗

arXiv2013

Dispersion dynamics for the defocusing generalized Korteweg-de Vries equation

We study dispersion for the defocusing gKdV equation. It is expected that it is not possible for the bulk of the $L^2-$mass to concentrate in a small interval for a long time. We study a variance-type functional exploiting Tao's monotonicity formula in the spirit of earlier work by Tao as well as Kwon & Shao and quantify its growth in terms of sublevel estimates.

math.AP↗