MATH 502 Harmonic Analysis

Starting with the foundational work of J. Fourier on solving differential equations using trigonometric series, we will cover the Fourier series, the Fourier transform, and their algebraic and spectral theoretic properties and generalizations, which highlight their essential unity. We then will concentrate on the Euclidean spaces and introduce oscillatory integrals, singular integrals and maximal operators, which altogether form the three main branches of study in modern Euclidean harmonic analysis, and which are deeply interrelated. Oscillatory integrals generalize the Fourier transform, singular integrals generalize the Hilbert transform and are related to the Cauchy integrals of complex analysis, and maximal operators generalize the Hardy-Littlewood maximal function and are extremely relevant to proving existence of limits. We then deepen our investigation of oscillatory integrals by investigating multipliers. We finally turn our attention to analyzing functions by dividing them into building blocks other than characters. This subject is known as time-frequency analysis and is of profound importance in engineering as it allows better localized and less structured building blocks.