arXiv subjects

Arie Israel

Publications and source records attributed to Arie Israel.

24 records · Page 2Linked to original sources

arXiv2015

An arithmetic-geometric mean inequality for products of three matrices

Consider the following noncommutative arithmetic-geometric mean inequality: given positive-semidefinite matrices $\mathbf{A}_1, \dots, \mathbf{A}_n$, the following holds for each integer $m \leq n$: $$ \frac{1}{n^m}\sum_{j_1, j_2, \dots, j_m = 1}^{n} ||| \mathbf{A}_{j_1} \mathbf{A}_{j_2} \dots \mathbf{A}_{j_m} ||| \geq \frac{(n-m)!}{n!} \sum_{\substack{j_1, j_2, \dots, j_m = 1 \\ \text{all distinct}}}^{n} ||| \mathbf{A}_{j_1} \mathbf{A}_{j_2} \dots \mathbf{A}_{j_m} |||,$$ where $||| \cdot |||$ denotes a unitarily invariant norm, including the operator norm and Schatten p-norms as special cases. While this inequality in full generality remains a conjecture, we prove that the inequality holds for products of up to three matrices, $m \leq 3$. The proofs for $m = 1,2$ are straightforward; to derive the proof for $m=3$, we appeal to a variant of the classic Araki-Lieb-Thirring inequality for permutations of matrix products.

math.SP↗

arXiv2015

The Eigenvalue Distribution of Time-Frequency Localization Operators

We estimate the distribution of the eigenvalues of a family of time-frequency localization operators whose eigenfunctions are the well-known Prolate Spheroidal Wave Functions from mathematical physics. These operators are fundamental to the theory of bandlimited functions and have applications in signal processing. Unlike previous approaches which rely on complicated formulas for the eigenvalues, our approach is simple: We build an orthonormal basis of modulated bump functions (known as wave packets in time-frequency analysis) which approximately diagonalizes the operator of interest.

math.CA↗

arXiv2014

Fitting a Sobolev function to data

We exhibit an algorithm to solve the following extension problem: Given a finite set $E \subset \mathbb{R}^n$ and a function $f: E \rightarrow \mathbb{R}$, compute an extension $F$ in the Sobolev space $L^{m,p}(\mathbb{R}^n)$, $p>n$, with norm having the smallest possible order of magnitude, and secondly, compute the order of magnitude of the norm of $F$. Here, $L^{m,p}(\mathbb{R}^n)$ denotes the Sobolev space consisting of functions on $\mathbb{R}^n$ whose $m$th order partial derivatives belong to $L^p(\mathbb{R}^n)$. The running time of our algorithm is at most $C N \log N$, where $N$ denotes the cardinality of $E$, and $C$ is a constant depending only on $m$,$n$, and $p$.

math.CA↗

arXiv2012

The Structure of Sobolev Extension Operators

Let $L^{m,p}(\R^n)$ denote the Sobolev space of functions whose $m$-th derivatives lie in $L^p(\R^n)$, and assume that $p>n$. For $E \subset \R^n$, denote by $L^{m,p}(E)$ the space of restrictions to $E$ of functions $F \in L^{m,p}(\R^n)$. It is known that there exist bounded linear maps $T : L^{m,p}(E) \rightarrow L^{m,p}(\R^n)$ such that $Tf = f$ on $E$ for any $f \in L^{m,p}(E)$. We show that $T$ cannot have a simple form called "bounded depth."

math.CA↗

arXiv2012

Sobolev Extension By Linear Operators

Let $L^{m,p}(\R^n)$ be the Sobolev space of functions with $m^{th}$ derivatives lying in $L^p(\R^n)$. Assume that $n< p < \infty$. For $E \subset \R^n$, let $L^{m,p}(E)$ denote the space of restrictions to $E$ of functions in $L^{m,p}(\R^n)$. We show that there exists a bounded linear map $T : L^{m,p}(E) \rightarrow L^{m,p}(\R^n)$ such that, for any $f \in L^{m,p}(E)$, we have $Tf = f$ on $E$. We also give a formula for the order of magnitude of $\|f\|_{L^{m,p}(E)}$ for a given $f : E \rightarrow \R$ when $E$ is finite.

math.CA↗

arXiv2010

A Bounded Linear Extension Operator for $L^{2,p}(\R^2)$

For a finite $E \subset \R^2$, $f:E \rightarrow \R$, and $p>2$, we produce a continuous $F:\R^2 \rightarrow \R$ depending linearly on $f$, taking the same values as $f$ on $E$, and with $L^{2,p}(\R^2)$ semi-norm minimal up to a factor $C=C(p)$. This solves the Whitney extension problem for the Sobolev space $L^{2,p}(\R^2)$. A standard method for solving extension problems is to find a collection of local extensions, each defined on a small square, which if chosen to be mutually consistent can be patched together to form a global extension defined on the entire plane. For Sobolev spaces the standard form of consistency is not applicable due to the (generically) non-local structure of the trace norm. In this paper, we define a new notion of consistency among local Sobolev extensions and apply it toward constructing a bounded linear extension operator. Our methods generalize to produce similar results for the $n$-dimensional case, and may be applicable toward understanding higher smoothness Sobolev extension problems.

math.CA↗