Mathematical Sciences: Geometric Harmonic Analysis and Spectral Theory

The basis for this work is the strong connections which exist between geometric and harmonic analysis. The latter subject concerns the decomposition of mathematical functions into simpler pieces and the reconstruction (synthesis) of functions from these pieces. Often the pieces arise naturally as eigenfunctions of differential operators such as the Laplace operator defined on a surface or manifold. However, eigenfunctions of operators occur in many contexts where the ideas of harmonic analysis are just beginning to penetrate. The object of this research is to foster further development in this direction. A simply-connected Lie nilpotent group of step 3 or higher has a sub-elliptic Laplacian representable as the sum of the squares of the generators. Work will be done in determining the eigenfunctions of this operator. For step 2 groups such as the Heisenberg group, the eigenfunctions are described in terms of Hermite functions. The present work should lead to new classes of special functions. Other work will study sets of fractional Hausdorff dimension. These sets which have long played a role in harmonic analysis are now coming into prominence in many areas of mathematics. One goal of this project will be to examine more closely recently discovered inequalities between the mean-square integral of Fourier transforms and the same integral of the originating function taken with respect to alpha-dimensional measure. In particular, work will concentrate on determining what fractal properties of the support of such a measure can be obtained by considering the asymptotic behavior of Fourier transforms of functions multiplied against the measure. A new line of investigation will take up the now classical theory of almost periodic functions developed early in the century by Besicovitch. One of the drawbacks to the theory was the fact that when a norm was defined for the space of such functions, it produced a large null space. This work will look for new ways of producing metrics on the space, developing a distributional point of view and deriving a useful theory of generalized harmonic analysis on the space, with special emphasis placed on the expansion of the Fourier transform.

The basis for this work is the strong connections which exist between geometric and harmonic analysis. The latter subject concerns the decomposition of mathematical functions into simpler pieces and the reconstruction (synthesis) of functions from these pieces. Often the pieces arise naturally as eigenfunctions of differential operators such as the Laplace operator defined on a surface or manifold. However, eigenfunctions of operators occur in many contexts where the ideas of harmonic analysis are just beginning to penetrate. The object of this research is to foster further development in this direction. A simply-connected Lie nilpotent group of step 3 or higher has a sub-elliptic Laplacian representable as the sum of the squares of the generators. Work will be done in determining the eigenfunctions of this operator. For step 2 groups such as the Heisenberg group, the eigenfunctions are described in terms of Hermite functions. The present work should lead to new classes of special functions. Other work will study sets of fractional Hausdorff dimension. These sets which have long played a role in harmonic analysis are now coming into prominence in many areas of mathematics. One goal of this project will be to examine more closely recently discovered inequalities between the mean-square integral of Fourier transforms and the same integral of the originating function taken with respect to alpha-dimensional measure. In particular, work will concentrate on determining what fractal properties of the support of such a measure can be obtained by considering the asymptotic behavior of Fourier transforms of functions multiplied against the measure. A new line of investigation will take up the now classical theory of almost periodic functions developed early in the century by Besicovitch. One of the drawbacks to the theory was the fact that when a norm was defined for the space of such functions, it produced a large null space. This work will look for new ways of producing metrics on the space, developing a distributional point of view and deriving a useful theory of generalized harmonic analysis on the space, with special emphasis placed on the expansion of the Fourier transform.