Smoothed analysis of symmetric random matrices with continuous distributions

We study invertibility of matrices of the form $D+R$ where $D$ is an arbitrary symmetric deterministic matrix, and $R$ is a symmetric random matrix whose independent entries have continuous distributions with bounded densities. We show that $|(D+R)^{-1}| = O(n^2)$ with high probability. The bound is completely independent of $D$. No moment assumptions are placed on $R$; in particular the entries of $R$ can be arbitrarily heavy-tailed.Comment: Several very small revisions mad

Inverse-Closedness of a Banach Algebra of Integral Operators on the Heisenberg Group

Let $\mathbb{H}$ be the general, reduced Heisenberg group. Our main result establishes the inverse-closedness of a class of integral operators acting on $L^{p}(\mathbb{H})$, given by the off-diagonal decay of the kernel. As a consequence of this result, we show that if $\alpha_{1}I+S_{f}$, where $S_{f}$ is the operator given by convolution with $f$, $f\in L^{1}_{v}(\mathbb{H})$, is invertible in \B(L^{p}(\mathbb{H})), then (\alpha_{1}I+S_{f})^{-1}=\alpha_{2}I+S_{g}$, and$g\in L^{1}_{v}(\mathbb{H})$. We prove analogous results for twisted convolution operators and apply the latter results to a class of Weyl pseudodifferential operators. We briefly discuss relevance to mobile communications.Comment: This version corrects two mistakes and recognizes the work of other authors related to a corollary of our main theore

Strong divergence of reconstruction procedures for the Paley–Wiener space PW(1_π) and the Hardy space H^1

Previous results on certain sampling series have left open if divergence only occurs for certain subsequences or, in fact, in the limit. Here we prove that divergence occurs in the limit. We consider three canonical reconstruction methods for functions in the Paley–Wiener space PW^1_π. For each of these we prove an instance when the reconstruction diverges in the limit. This is a much stronger statement than previous results that provide only lim sup divergence. We also address reconstruction for functions in the Hardy space H^1 and show that for any subsequence of the natural numbers there exists a function in H^1 for which reconstruction diverges in lim sup. For two of these sampling series we show that when divergence occurs, the sampling series has strong oscillations so that the maximum and the minimum tend to positive and negative infinity. Our results are of interest in functional analysis because they go beyond the type of result that can be obtained using the Banach–Steinhaus Theorem. We discuss practical implications of this work; in particular the work shows that methods using specially chosen subsequences of reconstructions cannot yield convergence for the Paley–Wiener Space PW^1_π

Eigenvalue Estimates and Mutual Information for the Linear Time-Varying Channel

We consider linear time-varying channels with additive white Gaussian noise. For a large class of such channels we derive rigorous estimates of the eigenvalues of the correlation matrix of the effective channel in terms of the sampled time-varying transfer function and, thus, provide a theoretical justification for a relationship that has been frequently observed in the literature. We then use this eigenvalue estimate to derive an estimate of the mutual information of the channel. Our approach is constructive and is based on a careful balance of the trade-off between approximate operator diagonalization, signal dimension loss, and accuracy of eigenvalue estimates.Comment: Submitted to IEEE Transactions on Information Theory This version is a substantial revision of the earlier versio

Local spectrum of truncations of Kronecker products of Haar distributed unitary matrices

We address the local spectral behavior of the random matrix Π_1U^(⊗k)Π_2U^(⊗k∗)Π_1 where U is a Haar distributed unitary matrix of size n × n, the factor k is at most c0lgn for a small constant c_0 > 0, and Π_1, Π_2 are arbitrary projections on ℓ^n^k_2of ranks proportional to n^k. We prove that in this setting the k-fold Kronecker product behaves similarly to the well-studied case when k = 1

Expected Supremum of a Random Linear Combination of Shifted Kernels

We address the expected supremum of a linear combination of shifts of the sinc kernel with random coefficients. When the coefficients are Gaussian, the expected supremum is of order \sqrt{\log n}, where n is the number of shifts. When the coefficients are uniformly bounded, the expected supremum is of order \log\log n. This is a noteworthy difference to orthonormal functions on the unit interval, where the expected supremum is of order \sqrt{n\log n} for all reasonable coefficient statistics.Comment: To appear in the Journal of Fourier Analysis and Application