I do research on Dyadic Harmonic Analysis. This area, whose roots lie in the physics of vibration, uses integration to decompose (integrable) functions into piece-wise constant components by generating numbers (called Walsh-Fourier coefficients) and infinite series (Walsh-Fourier series). These numbers and series can be used to approximate and to characterize the original function. I am particularly interested in problems of uniqueness (under which conditions are the coefficients of an otherwise arbitrary Walsh series the Walsh-Fourier coefficients of a some integrable function?) and growth (how fast do the partial sums or the Cesaro means of a Walsh-Fourier series grow?). Specific results can be obtained by looking at the papers in my bibliography.
Dyadic Harmonic analysis has many applications. Using Walsh-Fourier series to approximate a given function makes it possible to transmit data efficiently (e.g., multiplexing), to filter data (e.g., remove noise from weak video signals), and for data compression (e.g., transmit hundreds of signals through a single fiber optic cable). Using Walsh-Fourier coefficients to characterize functions makes it possible to recognize patterns (e.g., read handwritten zip-codes). Walsh functions have also been used to design genetic algorithms, methods to optimize non-differentiable problems for which the standard approach via calculus will not work.
I am also interested in Math History. I enjoy discovering how concepts were discovered and what purposes they serve in the 21st century. I use this material to make myself a better teacher, and to enliven my course in Math History.
