Joseph D. Lakey

Divergence-Free Multiwavelets on the Half Plane

We use the biorthogonal multiwavelets related by differentiation constructed in previous work to construct compactly supported biorthogonal multiwavelet bases for the space of vector fields on the upper half plane R2 + such that the reconstruction wavelets are divergence-free and have vanishing normal components on the boundary of R2 +. Such wavelets are suitable to study the Navier–Stokes equations on a half plane when imposing a Navier boundary condition.

Sampling and Time-Frequency Localization of Band-Limited and Multiband Signals

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.

Bandpass pseudo prolate shift frames and Riesz bases

We investigate frames and Riesz bases for the space of square-integrable functions on the line whose Fourier transforms are supported on the union of two disjoint intervals (bandpass signals). By suitably modulating a frame (resp. Riesz basis) for the Paley-Wiener space PWΩ which is generated by the shifts of prolate spheroidal wave functions, we generate frames (reps. Riesz bases) for the bandpass space, and show that the frame (resp. Riesz) bounds are the same as those of the baseband frame (resp. Riesz basis).

Prolate shift frames and their duals

We investigate frames for the Paley-Wiener space PW_Ω of square-integrable functions bandlimited to [-Ω/2, Ω/2] generated by translates φ_n (t - αℓ) of prolate spheroidal wave-functions ψ_n (where α > 0 and ℓ is an integer). We estimate frame bounds and give a Fourier construction of the dual frames. An ℓ² estimate on the decay of uniform samples of prolate functions is given to show that the computation of the duals can be done efficiently.

Riesz bounds for prolate shifts

We investigate shifts of prolate spheroidal wave functions and more particularly the shift-invariant spaces generated by the shifts of the first N of these functions. The Markov property satisfied by the prolates is used to show that at the Nyquist rate, such collections form a Riesz basis for the associated Paley-Wiener space, although explicit Riesz bounds cannot be derived. The fact that the prolate functions are eigenfunctions of the finite Fourier transform and a quadrature estimate for integrals of complex exponentials is used to provide Riesz bounds when the sampling rate is much lower than Nyquist. In this case, the Riesz basis will typically not span all of the Paley-Wiener space.

Frame Properties of Shifts of Prolate and Bandpass Prolate Functions

Frame properties of shifts of certain prolate spheroidal wave functions and corresponding bandpass prolates are reviewed and analogues between continuous and finite dimensional theories are pointed out. An application to the study of phase locking in electroencephalography channel signals is also reviewed.

On the Numerical Computation of Certain Eigenfunctions of Time and Multiband Limiting

A method is given for local numerical approximation of functions ψ that are multiband limited to a finite union of frequency bands and approximately time-limited to an interval [−T, T] in the sense that ψ is an eigenvalue of an operator that time limits then band limits to the corresponding sets, with an eigenvalue close to one. The local construction involves writing such an eigenfunction as an approximate sum of eigenfunctions of time- and band-limiting where the band-limiting interval has unit length. This is expressed as a discrete eigenvalue problem whose solution boils down to approximating local inner products of such functions and is reduced to the efficient numerical problem of approximating a corresponding local inner product of their sequences of integer samples. A numerical method for estimating the latter is also discussed.

Time and Band Limiting of Multiband Signals

When ( {mathcal a} ) = 2ΩT, the operator PΩQ T corresponding to single time and frequency intervals has an eigenvalue λ∟ ( {}_{mathcal a} ) ∟ ≈1/2, as Theorem 4.1.2 below will show. The norm λ0(( {mathcal a} ) = 1) of the operator PQ1/2 satisfies λ0(( {mathcal a} ) = 1) ≥ ∥sinc [−1/2, 1/2]∥ > 0.88. The trace of PQ1/2 is equal to ( {mathcal a} ) = 1, on the one hand and to Σλ n on the other, so λ1(( {mathcal a} ) = 1) ≤ 1−λ0(( {mathcal a} ) = 1) < 1/2. Suppose that T = 1 and Σ is a finite, pairwise disjoint union of ( {mathcal a} ) frequency intervals I1, … , I(_{} {mathcal a} ) each of unit length. Then PΣQ should have on the order of ( {mathcal a} ) eigenvalues of magnitude at least 1/2. Consider now the limiting case in which the frequency intervals become separated at infinity. Any function ψ j that is concentrated in frequency on I j will be almost orthogonal over [−T,T], in the separation limit, to any function ψ k that is frequency-concentrated on I k when j ≠ k.

Sampling of Band-limited and Multiband Signals

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.

Numerical Aspects of Time and Band Limiting

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 ( {mathcal 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.