Jeffrey A. Hogan

Global higher integrability of Jacobians on bounded domains

We give conditions for a vector-valued function View the MathML source, satisfying View the MathML source on a bounded domain Ω, which imply that View the MathML source is globally higher integrable on Ω. We also give conditions for View the MathML source such that View the MathML source belongs to the Hardy space View the MathML source 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.

Harmonic Analysis of Dirac Operators on Lipschitz Domains

We survey some results concerning Clifford analysis and the L2 theory of boundary value problems on domains with Lipschitz boundaries. Some novelty is introduced when using Rellich inequalities to invert boundary operators.

An Initial Analysis of River Discharge and Rainfall in Coastal New South Wales, Australia Using Wavelet Transforms

In many coastal catchments of south eastern New South Wales, Australia, changes in river morphology are a response to human impact superimposed on spatial and temporal patterns of variability in precipitation and discharge. Understanding, and preferably quantifying, spatial and temporal patterns of hydrologic variability are essential to understanding natural changes, and to separate these from artificial changes in river systems. Prediction and management of water resources are also dependent upon this understanding. We assess the variability in precipitation and discharge using the wavelet transform which projects the time series of data into a three dimensional surface of frequency, amplitude and time. The analysis reveals that changes across time often reflect changes in individual seasons and may be linked to changes in particular seasonal atmospheric circulation systems. Strong perturbations in the analysis of one catchment are consistent with documented, geomorphically-effective, flooding sequences. The characteristics of the series in the transformed data reveal interesting differences at certain times and scales which may be a reflection of changes in larger scale atmospheric processes.

Embeddings and Uncertainty Principles for Generalized Modulation Spaces

Modulation norms provide measures of joint time-frequency localization of a function f by replacing the L2-norm of the short-time Fourier transform of f by a mixed LP-norm. To have a firm understanding of how these norms measure smoothness versus decay of f it is important to establish embedding theorems, for example, from weighted Lp-spaces into the modulation spaces. Several ways of doing this are discussed herein, along with precise interpretations in terms of localization of information and the uncertainty principle. The techniques are intimately related to validating a variety of sampling methods.

SAMPLING AND OVERSAMPLING IN SHIFT-INVARIANT AND MULTIRESOLUTION SPACES I: VALIDATION OF SAMPLING SCHEMES

We ask what conditions can be placed on generators φ of principal shift invariant spaces to ensure the validity of analogues of the classical sampling theorem for bandlimited signals. Critical rate sampling schemes lead to expansion formulas in terms of samples, while oversampling schemes can lead to expansions in which function values depend only on nearby samples. The basic techniques for validating such schemes are built on the Zak transform and the Poisson summation formula. Validation conditions are phrased in terms of orthogonality, smoothness, and self-similarity, as well as bandlimitedness or compact support of the generator. Effective sampling rates which depend on the length of support of the generator or its Fourier transform are derived.

Periodic Nonuniform Sampling in Shift-Invariant Spaces

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.

Hardy’s theorem and rotations

We prove an extension of Hardy’s classical characterization of real Gaussians of the form

Wavelet frames generated by bandpass prolate functions

e^{-\pi\alpha x^2}, \alpha > 0

Sampling for shift-invariant and wavelet subspaces

, to the case of complex Gaussians in which α is a complex number with positive real part. Such functions represent rotations in the complex plane of real Gaussians. A condition on the rate of decay of analytic extensions of a function

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

f