Date of Graduation

Spring 2015

Degree

Master of Science in Mathematics

Department

Mathematics

Committee Chair

William Bray

Keywords

Fourier analysis, harmonic analysis, Fourier transform, Fourier series, Bessel functions, Schwartz space, convolution

Subject Categories

Mathematics

Abstract

The field of Fourier analysis encompasses a vast spectrum of mathematics and has far reaching applications in all STEM fields. Here we introduce and study the Fourier transform and Fourier series on Euclidean space. After defining the Fourier transform, establishing its basic properties, and presenting some classical results we looked into what impact the smoothness of a function has on the growth and integrability of its Fourier transform. This endeavor also involved a brief study of Bessel functions and interpolation of operators. Having established several results indicating that the behavior of a function's Fourier transform is largely dictated by the smoothness of the function, the thesis concludes with a look into Fourier series and Bochner-Riesz eans.

Copyright

© Joseph William Roberts

Open Access

Included in

Mathematics Commons

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