Multiparameter harmonic analysis on the Heisenberg group

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Harmonic analysis is a highly active field of mathematics, which has had many recent applications to the analysis and manipulation of signals such as speech, images, electrocardiograms, as well as more general digital data sets. It provides tools for unraveling signals and extracting features at different scales, as well as methods of compressing information which are useful for storage and computation. Harmonic analysis is concerned, generally speaking, with breaking information up into simpler pieces. This can be nicely illustrated with the model of music. When a tuning fork for middle C is struck, it produces nearly a pure tone; that is, a sine wave variation in the air pressure. On the other hand, when a musical instrument such as a violin plays middle C, it produces a sum of pure tones: superimposed at various intensities on the same ground or fundamental frequency, are sine waves with frequencies which are multiples of this. These are called higher harmonics. It is the presence of these higher harmonics which is responsible for the character, or timbre, of an instrument. Although exactly the same note is being played in the two cases, the two instruments sound slightly different. The human ear, in recognizing this subtle difference in the character of the sound caused by the presence of these higher harmonics at various amplitudes, is doing harmonic analysis. My work largely concerns multiplier theory, which is a very important tool in harmonic analysis. It can be described very simply in the musical model. If we consider some sound, or signal, which has been decomposed in this way into constituent frequencies at various amplitudes, and suppose we are interested in damping certain of those frequencies, and perhaps magnifying others: we do this, for example, when adjusting the equalizer on a stereo set. Multiplier theory is the study of what effect this would have on the signal as a whole. Multiplier theory has important applications in many areas of mathematics, such as partial differential equations, analytic number theory, differential geometry, and even plays a role in the Navier Stokes problem, the solution of which is now worth US\$1m.