Computing the Fourier transform and its inverse is important in many applications of mathematics, such as frequency analysis, signal modulation, and filtering. Two methods will be derived for numerically computing the inverse Fourier transforms, and they will be compared to the standard inverse discrete Fourier transform (IDFT) method. The first computes the inverse Fourier transform through direct use of the Laguerre expansion of a function. The second employs the Riesz projections, also known as Hilbert projections, to numerically compute the inverse Fourier transform. For some smooth functions with slow decay in the frequency domain, the Laguerre and Hilbert methods will work better than the standard IDFT. Applications of the Hilbert transform method are related to the numerical solutions of nonlinear inverse scattering problems and may have implications for the associated reconstruction algorithms.

Author Bio

As a double major in Mathematics and Electrical Engineering, I found myself particularly interested in mathematics with applications in the sciences and engineering. One such example is the Fourier transform which has many applications not only in engineering but also in disciplines such as economics and geophysics. My research on Inverse Fourier Transform methods was my Senior Research at Seattle University and I found much motivation for this project. Because of my research experiences, including two summer REUs in addition to this project, I have decided to pursue a Ph.D. degree in mathematics at U.C. Berkeley to gain more knowledge about the applied mathematics that I enjoy so much.