Fourier Analysis: Space, Frequency, and Direction

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Jean Baptiste Joseph Fourier can rightly claim the title of forefather of frequency decompositions. Having postulated in 1807 the theory that essentially arbitrary functions can be expressed as sums of basic sinusoidal waves of various frequencies, Fourier resisted fervent theoretical objections to paint a landscape full of applications to concrete problems. Since then, the development of analysis has furnished new perspectives that have resulted in extraordinary accomplishments in several fields of mathematics and the sciences. Recent techniques include sensitive decompositions localized in both space and phase, which take into consideration directional aspects of the problems. Such decompositions have led to solutions of long-standing conjectures such as the almost everywhere convergence of Fourier series, and the boundedness of the bilinear Hilbert transform. The principal investigator proposes to embark on an extensive study of problems in Fourier Analysis and applications that have a common feature: they require a delicate balance between space, frequency, and direction. The intellectual merit of this proposal lies in the historical value of the problems studied, many of which have naturally arisen over time and have a reputation for their difficulty. This proposal consists of three parts: positive theory, quest for counterexamples, and concrete applications. Proposed work on the positive theory includes extension of the range of boundedness of the bilinear Hilbert transform and other rough singular integrals and a study of m-linear orthogonality and Littlewood-Paley theory. Geometric aspects of frequency decompositions play a crucial role in this study. Counterexamples are sought for the bilinear disc multiplier in the nonlocal square-integrable case and the Carleson-Hunt operator on certain spaces of functions. Concrete applications focus on directional sensitivity in computerized tomography.Fourier Analysis provides decompositions of functions in parts that have common spatial, frequency, and directional characteristics. Just as symphonic music can be analyzed as a finite union of simple notes, certain complicated operations can be represented by their actions on a spectrum of frequencies. Irregularities of signals and images are better located once they are decomposed into small pieces that can be studied individually. The alteration of frequency via multiplication by a typically nonsmooth function, such as intermittent television transmission, calls for a systematic study of preservation of information contained in a signal. Preservation of integrability under frequency alterations serves as a good model to study preservation of information and is the main focus of the theoretical part of this proposal. The broader impact of this proposal is exactly this point, i.e. to provide a solid theoretical foundation or groundwork for modeling protection against the loss of information contained in a signal or image. Applied problems addressed in this project focus on improved results in computerized tomography based on frequency decompositions sensitive to direction. Such applications may lead to sharper computerized tomography images for moving subjects during standard MRI tests.