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Dive into the research topics where Henry J. Landau is active.

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Featured researches published by Henry J. Landau.


IEEE Transactions on Information Theory | 1988

On the minimum distance problem for faster-than-Nyquist signaling

James E. Mazo; Henry J. Landau

The authors reconsider the problem of determining the minimum distance between output sequences of an ideal band-limiting channel that are generated by uncoded binary sequences transmitted at a rate exceeding the Nyquist rate. For signaling rates up to about 25% faster than the Nyquist rate, it is shown that the minimum distance does not drop below the value which it would have in the ideal case wherein there is not intersymbol interference. Mathematically, the problem is to decide if the best L/sup 2/ Fourier approximation to the constant 1 on the interval (- sigma pi , sigma pi ), 0 0, with coefficients restricted to be =1 or =0, occurs when all coefficients are zero. This is shown to be optimal for 0.802... >


Journal of Mathematical Analysis and Applications | 1980

Eigenvalue distribution of time and frequency limiting

Henry J. Landau; Harold Widom

The operator in (1) consists of restricting


Bulletin of the American Mathematical Society | 1987

Maximum entropy and the moment problem

Henry J. Landau

k(~) to the set cS, restricting the Fourier transform of the function so obtained to the set T, and viewing the result again on cS. We can therefore represent it compactly as A c = WV Q(T) w9, where P(O) and Q(A) represent orthogonal projections in the Hilbert space I%*(--co, 00) onto the subspaces of those functions which vanish outside of R, and those whose Fourier transform vanishes outside of A. The interest of the problem lies in the fact that the eigenvalues are useful in describing the geometry of these subspaces, and thereby provide information about how the energy of a function can be distributed over time and frequency. From a comparison of Ck A,(c) and 469 0022-241X/80/100329-15


Linear Algebra and its Applications | 1981

Bounds for eigenvalues of certain stochastic matrices

Henry J. Landau; Andrew M. Odlyzko

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IEEE Transactions on Information Theory | 1993

On the density of phase-space expansions

Henry J. Landau

Introduction. The trigonometric moment problem stands at the source of several major streams in analysis. From it flow developments in function theory, in spectral representation of operators, in probability, in approximation, and in the study of inverse problems. Here we connect it also with a group of questions centering on entropy and prediction. In turn, this will suggest a simple approach, by way of orthogonal decomposition, to the moment problem itself. In statistical estimation, one often wants to guess an unknown probability distribution, given certain observations based on it. There are generally infinitely many distributions consistent with the data, and the question of which of these to select is an important one. The notion of entropy has been proposed here as the basis of a principle of salience which has received considerable attention. We will show that, in the context of spectral analysis, this idea is linked to a certain question of prediction by the trigonometric moment problem, and that all three strongly illuminate one another. The phenomena we describe are known, but our object is to unify them conceptually and to reduce the analytic intricacy of the arguments. To this end, we give a completely elementary discussion, virtually free of calculation, which shows that all the facts, including those concerning the moment problem, can be understood as direct consequences of orthogonal decomposition in a finite-dimensional space. We then describe how, in its continuous version, this leads to a view of second-order Sturm-Liouville differential equations, and conclude with some questions concerning the connection between combinatorial ideas and orthogonality in this problem.


Journal of the American Mathematical Society | 1994

The inverse eigenvalue problem for real symmetric Toeplitz matrices

Henry J. Landau

Abstract We consider the class of stochastic matrices M generated in the following way from graphs: if G is an undirected connected graph on n vertices with adjacency matrix A, we form M from A by dividing the entries in each row of A by their row sum. Being stochastic, M has the eigenvalue λ=1 and possibly also an eigenvalue λ=-1. We prove that the remaining eigenvalues of M lie in the disk ¦λ¦⩽1–n-3, and show by examples that the order of magnitude of this estimate is best possible. In these examples, G has a bar-bell structure, in which n/3 of the vertices are arranged along a line, with n/3 vertices fully interconnected at each end. We also obtain better bounds when either the diameter of G or the maximal degree of a vertex is restricted.


Current and future directions in applied mathematics | 1997

On the well-posedness of the rational covariance extension problem

Christopher I. Byrnes; Henry J. Landau; Anders Lindquist

Phase-space decompositions for signals of finite energy have been used to formulate the intuitive but elusive idea that frequency content of a signal can vary with time as the signal evolves. The decompositions consist of subdividing the time and frequency axes into certain ranges and expanding signals stably in a two-parameter basis of fixed functions of time, in which the (k,m)th coefficient is viewed as describing the part of the signal that is concentrated in the kth time and mth frequency range. It is shown that in any such expansion, the time-frequency space must be subdivided, or, equivalently, the coefficients computed, at least at the Nyquist rate of two per unit of time and cycle of bandwidth. >


Journal of Functional Analysis | 1980

The classical moment problem: Hilbertian proofs

Henry J. Landau

We show that every set of n real numbers is the set of eigenvalues of an n x n real symmetric Toeplitz matrix; the matrix has a certain additional regularity. The argument-based on the topological degree-is nonconstructive. AT&T BELL LABORATORIES, MURRAY HILL, NEW JERSEY 07974 E-mail address: hjlfresearch. att .com This content downloaded from 157.55.39.29 on Tue, 12 Apr 2016 08:53:54 UTC All use subject to http://about.jstor.org/terms


Constructive Approximation | 1987

Polynomials orthogonal on the semicircle, II

Walter Gautschi; Henry J. Landau; Gradimir V. Milovanović

In this paper, we give a new proof of the solution of the rational covariance extension problem, an interpolation problem with historical roots in potential theory, and with recent application in speech synthesis, spectral estimation, stochastic systems theory, and systems identification. The heart of this problem is to parameterize, in useful systems theoretical terms, all rational, (strictly) positive real functions having a specified window of Laurent coefficients and a bounded degree. In the early 1980’s, Georgiou used degree theory to show, for any fixed “Laurent window”, that to each Schur polynomial there exists, in an intuitive systems-theoretic manner, a solution of the rational covariance extension problem. He also conjectured that this solution would be unique, so that the space of Schur polynomials would parameterize the solution set in a very useful form. In a recent paper, this problem was solved as a corollary to a theorem concerning the global geometry of rational, positive real functions. This corollary also asserts that the solutions are analytic functions of the Schur polynomials.


Archive | 1985

An Overview of Time and Frequency Limiting

Henry J. Landau

Abstract In view of its connection with a host of important questions, the classical moment problem deserves a central place in analysis. It is usually treated by methods striking in their virtuosity, but difficult to motivate. Here we describe an elementary approach which establishes the structural results and exposes the Hilbert space origins of the arguments.

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Zeph Landau

University of California

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Aaron Abrams

Washington and Lee University

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Eric Zaslow

Northwestern University

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Eric Babson

University of California

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