Associate Professor Jeffrey Hogan

Under minimal assumptions on a function ¿ we obtain wavelettype frames of the form ¿j,k(x) = r1/2nj¿(rjx - sk) j ¿ Z k ¿ Zn for some r > 1 and s > 0. This collection is shown to be a frame for a scale of Triebel-Lizorkin spaces (which includes Lebesgue, Sobolev and Hardy spaces) and the reproducing formula converges in norm as well as pointwise a.e. The construction follows from a characterization of those operators which are bounded on a space of smooth molecules. This characterization also allows us to decompose a broad range of singular integral operators in terms of smooth molecules.

One of the most basic applications of Fourier analysis is power spectrum estimation. Some historical comments on this age-old problem can be found in review articles by Robinson [279] and Benedetto [23], and in Percival and Walden¿s book [264]. We refer to Appendix A for basic definitions and properties, and references for stochastic processes. In the problem of estimating a wide-sense stationary random process from data, one has only finitely many samples, often of a single realization of the process, with which to work.

In this last chapter we explore briefly some connections among sampling and time and band limiting. The chapter begins by pointing out a general connection between the samples of eigenfunctions of time and band limiting and the eigenvectors of a certain matrix whose entries are, in essence, the samples of time-localized images of functions that interpolate the samples in the given Paley¿Wiener space. Next, a discrete method is considered for generating eigenfunctions of time¿frequency localizations to unions of sets from their separate localizations. We then reconsider the connection between eigenfunctions and their samples in the concrete context of localization to intervals of the real line, outlining work ofWalter and Shen [347] and of Khare and George [177].Walter and Shen provided L2-estimates for approximate prolate spheroidal wave functions (PSWFs) constructed from interpolation of their sample values within the time-localization interval. We provide a partial sharpening of their estimates by using a slightly enlarged set of samples.

When a = 2OT, the operator POQT corresponding to single time and frequency intervals has an eigenvalue ¿¿a¿¿1/2, as Theorem 4.1.2 belowwill show. The norm ¿0(a = 1) of the operator PQ1/2 satisfies ¿0(a = 1) = ¿sinc 1[1/2,1/2]¿ > 0.88. The trace of PQ1/2 is equal to a = 1, on the one hand and to S¿n on the other, so ¿1(a = 1) = 1¿0(a = 1) < 1/2. Suppose that T = 1 and S is a finite, pairwise disjoint union of a frequency intervals I1, ¿, Ia each of unit length. Then PSQ should have on the order of a eigenvalues of magnitude at least 1/2. Consider now the limiting case in which the frequency intervals become separated at infinity. Any function ¿j that is concentrated in frequency on Ij will be almost orthogonal over [T,T], in the separation limit, to any function ¿k that is frequency-concentrated on Ik when j ¿ k.

We provide here an overview of sampling theory that emphasizes real-variable aspects and functional analytic methods rather than analytic function-theoretic ones. While this approach does not justify the most powerful mathematical results, it does provide the basis for practical sampling techniques for band-limited and multiband signals.

This chapter is concerned with the role of the prolates in numerical analysis¿ particularly their approximation properties and application in the numerical solution of differential equations. The utility of the prolates in these contexts is due principally to the fact that they form a Markov system (see Defn. 2.1.6) of functions on [-11], a property that stems from their status as eigenfunctions of the differential operator P of (1.6), and allows the full force of the Sturm¿Liouville theory to be applied. The Markov property immediately gives the orthogonality of the prolates on [-11] (previously observed in Sect. 1.2 as the double orthogonality property) and also a remarkable collection of results regarding the zeros of the prolates as well as quadrature properties that are central to applications in numerical analysis.

Duration limiting, or time limiting, refers to restricting a signal by setting its values equal to zero outside of a finite time interval or, more generally, outside of a compact set. Bandwidth limiting, or band limiting, refers to restricting a signal by setting its amplitudes equal to zero outside of a finite frequency interval or again, more generally, outside of a compact set. This book addresses primarily the theory of time and band limiting whose core was developed by Landau, Pollak, and Slepian in a series of papers [195, 196, 303, 309] appearing in the Bell System Technical Journal in the early to middle part of the 1960s, and a broader body of work that grew slowly but steadily out of that core up until around 1980, with a resurgence since 2000, due in large part to the importance of time and band limiting in wireless communications. The 1960s Bell Labs theory of time and band limiting is but one aspect of the Bell Labs information theory. The foundations of this encompassing theory were laid, in large part, in Nyquist¿s fundamental papers ¿Certain Topics in Telegraph Transmission Theory¿ [247], which appeared in the Transactions of the American Institute of Electrical Engineers in 1928, and ¿Certain Factors Affecting Telegraph Speed,¿ published in April 1924 in the Bell System Technical Journal, along with Hartley¿s paper ¿Transmission of Information,¿ which also appeared in the Bell System Technical Journal in 1928 [137]. These papers quantified general ideas that were in the air, though certain specific versions were attributed to Kelvin and Wiener among others. Of course, Claude Shannon¿s seminal work, ¿A Mathematical Theory of Communication,¿ which appeared in the Bell System Technical Journal in July and October 1948 [293], is often cited as providing the basis for much of modern communications theory. His sampling theory plays a central role in Chap. 5 of this monograph. The works of Nyquist and Hartley however remain, in some ways, more germane to the study at hand.

This chapter develops aspects of previous work ([10] and [11]) that pertain distinctly to band-limited functions. In particular, we discuss some representation formulas for band-limited functions in terms of periodic nonuniform samples. In the case of multiband signals, periodic nonuniform sampling is often valid at a lower sampling rate than is uniform sampling, as will be discussed. Finally, we will consider some related questions about optimally time- and multiband-limited signals.

This chapter reviews several ideas that grew out of observations of Djokovic and Vaidyanathan to the effect that a generalized sampling method for bandlimited functions, due to Papoulis, could be carried over in many cases to the spline spaces and other shift-invariant spaces. Papoulis¿ method is based on the sampling output of linear, time-invariant systems. Unser and Zerubia formalized Papoulis¿ approach in the context of shift-invariant spaces. However, it is not easy to provide useful conditions under which the Unser-Zerubia criterion provides convergent and stable sampling expansions. Here we review several methods for validating the Unser-Zerubia approach for periodic nonuniform sampling, which is a very special case of generalized sampling. The Zak transform plays an important role.

We present an optimization approach to wavelet architecture that hinges on the Zak transform to formulate the construction as a minimization problem. The transform warrants parametrization of the quadrature mirror filter in terms of the possible integer sample values of the scaling function and the associated wavelet. The parameters are predicated to satisfy constraints derived from the conditions of regularity, compact support and orthonormality. This approach allows for the construction of nearly cardinal scaling functions when an objective function that measures deviation from cardinality is minimized. A similar objective function based on a measure of symmetry is also proposed to facilitate the construction of nearly symmetric scaling functions on the line.

We introduce a reformulation technique that converts a many-set feasibility problem into an equivalent two-set problem. This technique involves reformulating the original feasibility problem by replacing a pair of its constraint sets with their intersection, before applying Pierra¿s classical product space reformulation. The step of combining the two constraint sets reduces the dimension of the product spaces. We refer to this technique as the constraint reduction reformulation and use it to obtain constraint-reduced variants of well-known projection algorithms such as the Douglas¿Rachford algorithm and the method of alternating projections, among others. We prove global convergence of constraint-reduced algorithms in the presence of convexity and local convergence in a nonconvex setting. In order to analyze convergence of the constraint-reduced Douglas¿Rachford method, we generalize a classical result which guarantees that the composition of two projectors onto subspaces is a projector onto their intersection. Finally, we apply the constraint-reduced versions of Douglas¿Rachford and alternating projections to solve the wavelet feasibility problems and then compare their performance with their usual product variants.

The operator that first truncates to a neighborhood of the origin in the spatial domain then truncates to a neighborhood of the origin in the spectral domain is investigated in the case of Boolean cubes. This operator is self adjoint on the space of functions spanned by the Laplacian eigenvectors corresponding to small eigenvalues. The eigenspaces of this iterated projection operator are studied through reduced matrices based on certain invariant subspaces. They are shown to depend fundamentally on the neighborhood structure of the cube defined in terms of Hamming distance.

Discrete tori are ZmN thought of as vertices of graphs CmN whose adjacencies encode the Cartesian product structure. Space-limiting refers to truncation to a symmetric path neighborhood of the zero element and spectrum-limiting in this case refers to corresponding truncation in the isomorphic Fourier domain. Composition spatio-spectral limiting (SSL) operators are analogues of classical time and band limiting operators. Certain adjacency-invariant spaces of vectors defined on ZmN are shown to have bases consisting of Fourier transforms of eigenvectors of SSL operators. We show that when m= 3 or m= 4 , all eigenvectors of SSL arise in this way. We study the structure of corresponding invariant spaces when m= 5 and give an example to indicate that the relationship between eigenvectors of SSL and the corresponding adjacency-invariant spaces should extend to m= 5.

© 2014 Elsevier Inc. Abstract We provide conditions on a shift parameter and number of basic prolate spheroidal wave functions with a fixed bandwidth and time concentrated to a fixed duration such that the shifts of the basic prolates form a frame or a Riesz basis for the Paley-Wiener space consisting of all square integrable functions with the given bandlimit.

This letter provides a technique for numerical evaluation of certain eigenfunctions of the integral kernel operator corresponding to time truncation of a square-integrable function to a finite interval, followed by frequency limiting to frequencies in an annular band. © 2013 Springer Science+Business Media New York.

Gelfand¿Shilov spaces are spaces of entire functions defined in terms of a rate of growth in one direction and a concomitant rate of decay in an orthogonal direction. Gelfand and Shilov proved that the Fourier transform is an isomorphism among certain of these spaces. In this article we consider mapping properties of fractional Fourier transforms on Gelfand¿Shilov spaces. Just as the Fourier transform corresponds to rotation through a right angle in the phase plane, fractional Fourier transforms correspond to rotations through intermediate angles. Therefore, the aim of fractional Fourier estimates is to set up a correspondence between growth properties in the complex plane versus decay properties in phase space. © 2006, Taylor & Francis Group, LLC.

Under minimal assumptions on a function ¿ we obtain wavelettype frames of the form ¿j,k(x) = r1/2nj¿(rjx - sk) j ¿ Z k ¿ Zn for some r > 1 and s > 0. This collection is shown to be a frame for a scale of Triebel-Lizorkin spaces (which includes Lebesgue, Sobolev and Hardy spaces) and the reproducing formula converges in norm as well as pointwise a.e. The construction follows from a characterization of those operators which are bounded on a space of smooth molecules. This characterization also allows us to decompose a broad range of singular integral operators in terms of smooth molecules.

We give conditions for a vector-valued function u ¿ W1,n (O, Rn), satisfying det Du (x) 0 on a bounded domain O, which imply that det Du (x) is globally higher integrable on O. We also give conditions for u ¿ W1,n (O, Rn) such that det Du belongs to the Hardy space h1z (O) and exhibit some examples which show that our conditions are in some sense optimal. Applications to the weak convergence of Jacobians follow. Div-curl type extensions of these results to forms are also considered. © 2000 Editions scientifiques et médicales Elsevier SAS.

We refer to eigenfunctions of the kernel corresponding to truncation in a time interval followed by truncation in a frequency band as bandpass prelates (BPPs). We prove frame bounds for certain families of shifts of bandpass prolates, and we numerically construct dual frames for finite dimensional analogues. In the continuous case, the corresponding families produce wavelet frames for the space of square-integrable functions.