An Exercise(?) in Fourier Analysis on the Heisenberg Group

Bump, Daniel; Diaconis, Persi; Hicks, Angela; Miclo, Laurent; Widom, Harold

doi:10.5802/afst.1533

Cited by 13 publications

(21 citation statements)

References 37 publications

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“…We hope this can be used to give sharp results for the constant. Finally we note that the approach to bound β via an associated absorbing Markov chain was used in [5]. There, a geometric path argument was used to complete the analysis.…”

Section: This Givesmentioning

confidence: 99%

Useful Bounds on the Extreme Eigenvalues and Vectors of Matrices for Harper’s Operators

Bump

Diaconis

Hicks

et al. 2017

Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics

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Abstract. In analyzing a simple random walk on the Heisenberg group we encounter the problem of bounding the extreme eigenvalues of an n × n matrix of the form M = C + D where C is a circulant and D a diagonal matrix. The discrete Schrödinger operators are an interesting special case. The Weyl and Horn bounds are not useful here. This paper develops three different approaches to getting good bounds. The first uses the geometry of the eigenspaces of C and D, applying a discrete version of the uncertainty principle. The second shows that, in a useful limit, the matrix M tends to the harmonic oscillator on L 2 (R) and the known eigenstructure can be transferred back. The third approach is purely probabilistic, extending M to an absorbing Markov chain and using hitting time arguments to bound the Dirichlet eigenvalues. The approaches allow generalization to other walks on other groups.

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Section: This Givesmentioning

confidence: 99%

Useful Bounds on the Extreme Eigenvalues and Vectors of Matrices for Harper’s Operators

Bump

Diaconis

Hicks

et al. 2017

Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics

Self Cite

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“…The reader interested in learning more about the connections between random walks on groups and topics such as solid-state physics and ergodic theory is referred to [1]. While the literature contains a wealth of asymptotic estimates for quite general classes of groups (see, for instance, [7]), the main novelty of our results lies in its explicit nature and the approach undertaken in substantiating these findings.…”

Section: Short Summary Of Our Resultsmentioning

confidence: 89%

“…Let G = A H where A = Z-span {e k : 1 ≤ k ≤ d} is a full rank lattice of 1 is a finite subgroup of GL (A) [5]. The group G is equipped with the following operation:…”

Section: Short Summary Of Our Resultsmentioning

confidence: 99%

Random walk on finite extensions of lattices

Oussa¹

2021

APAM

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“…In the spirit of the title of the Auslander and Tolimieri paper, the authors revisited the DFT and one of the most important algorithms for computing it, namely the Cooley-Tukey algorithm [17] (which often is used as a synonym of FFT; see [16,71] as well as the nice survey by Maslen and Rockmore [52]), by establishing a relation with the representation theory of the finite Heisenberg groups (see also [9]). The original methods for FFT (which, as was already mentioned, in fact goes back to Gauss) have been generalized by several authors (e.g., Good [29], Rader [60], Rose [62], and Winograd [74]) yielding a matrix and tensor product approach of a purely algebraic flavor.…”

Section: Book Reviewsmentioning

confidence: 99%

Book Review: Discrete harmonic analysis

Grigorchuk¹

2019

Bull. Amer. Math. Soc.

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What is discrete harmonic analysis? The book under review, by Tullio Ceccherini-Silberstein, Fabio Scarabotti, and Filippo Tolli, gives a comprehensive answer. It is a continuation of a series of wonderful books by the same authors, including Harmonic analysis on finite groups: Representation theory, Gelfand pairs and Markov chains [13] and Representation theory and harmonic analysis of wreath products of finite groups [14] (both published by Cambridge University Press), which witness the power of harmonic analysis on finite groups both in theory and in applications.The study of mathematical models involving infinite objects, via their approximation by finite counterparts and leading to a discretization of these models, is one of the features of modern mathematics. This also reflects a "competition" and interaction between pure and applied mathematics. The title of the famous article of Louis Auslander and Richard Tolimieri that appeared in 1979 in the Bulletin of the AMS [6] asks, "Is computing with the finite Fourier transform pure or applied mathematics?" The book of Ceccherini-Silberstein, Scarabotti, and Tolli gives an exposition and a clarification of many questions raised by Auslander and Tolimieri, and much more.In order to confirm the importance and popularity of discretization, let us recall a couple of examples. The statistical models on Z d , d ≥ 2 (including the Ising model), are studied by means of their approximations in finite "subdomains" [−n, n] d and by passing to the limit for n → ∞ (the van Hove limit). The spectral properties of the Laplacian on an infinite graph Γ can be retrieved from the spectral properties of a sequence (Γ n ) n∈N of finite graphs approximating Γ. Similarly, the L 2 -Betti numbers of an infinite residually finite group G can be computed via a sequence (G n ) n∈N of finite groups approximating G (the Lück approximation), and the Gibbs measures on such a group G are limits of Gibbs measures of the G n 's [34]. Finally, the classical Fourier transform on Z or on the unit circle S 1 can be approximated by the discrete Fourier transform (DFT for short) on the finite cyclic groups Z n . An additional advantage of this approach is the fact that the groups Z n are not only finite but self-dual (in the sense of Pontryagin), in contrast with Z and S 1 that are mutually dual.Jean-Baptiste Joseph Fourier [27], at the turn of the nineteenth century, showed that representing a function as a sum of trigonometric functions (the exponentials e 2iπnx ), namely its Fourier series, greatly simplifies the study of the heat transfer. The study of the decomposition process of a function into these trigonometric functions is called Fourier analysis, while the operation of building back the function from these pieces is known as Fourier synthesis. In parallel, the original concept of Fourier analysis has been extended to apply to more and more abstract and general situations: the general field is often known as abstract harmonic analysis.

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An Exercise(?) in Fourier Analysis on the Heisenberg Group

Cited by 13 publications

References 37 publications

Useful Bounds on the Extreme Eigenvalues and Vectors of Matrices for Harper’s Operators

Useful Bounds on the Extreme Eigenvalues and Vectors of Matrices for Harper’s Operators

Random walk on finite extensions of lattices

Book Review: Discrete harmonic analysis

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