Research Interests

My hope is to identify and solve mathematical problems that arise in applications like signal processing and the solution of partial differential equations. Often those problems involve linear algebra. As an example, the wavelet transform has the valuable property that it is executed by a banded matrix that has a banded inverse. Both the transform and its inverse are then "local" and the time domain description of the signal remains visible (in comparison with the Fourier transform). A natural question is to identify all banded matrices with this exceptional property that the inverse is also banded (which makes the transforms fast). In this and many other applications, the underlying problem is "the choice of a good basis." My contribution is to identify the properties that make a basis good, to construct good ones, and to analyze the discrepancy between a continuous problem (in function space) and a corresponding discrete problem that uses the chosen basis.

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Primary Section

Section 32: Applied Mathematical Sciences