MATH 7378-1

Mathematical analysis is a base on which much of modern science and engineering is built, and perhaps the most widely used tools in the applied sciences are Fourier transforms and Fourier series. This part of harmonic analysis is closely related to the solution of differential equations and reconstruction of functions from frequency information, but it lacked the possibility of localization in both space and frequency. Thus in order to study the spectral behavior of an analog signal from its Fourier transform, we need to know the full function/signal over all time and that would even include future information! Therefore, the Fourier transform is not the ideal tool to cope with the explosive growth of digital information, digital transmission of data, and the corresponding need of effective computational tools for compression of information to minimize storage and speed up transmission.

This has given rise to the theory of wavelets. The goal of most modern wavelet research is to create a set of basis functions and transforms that will give an informative, efficient and useful description of a function or signal. If the signal is represented as a function of time, wavelets provide efficient localization in both time and frequency or scale. Fundamentally important is the construction of orthonormal bases - or more generally frames - generated by dyadic dilation and integer translation from one function. The choice of this ''generating'' function depends on the concrete problem at hand and is often only defined indirectly or by an algorithmic prescription whose implementation requires substantial computational power.

This lecture will be an introduction in Fourier analysis and modern theory of wavelets. The main topics will be:

• Periodic functions and Fourier series
• Convergence of Fourier series.
• The L²-theory
• The Fourier integral in Euclidean space
• Rapidly decreasing functions
• Distributions and the Fourier transform of tempered distributions
• The Fourier transform of compactly supported functions and the Paley-Wiener Theorem
• Some applicatipons:
  • Application to differential equations, in particular the wave equation
  • Uncertainity
  • Shannon sampling theorem
• Wavelet theory
• The Haar wavelet
• Multiresolution analysis
• Compactly supported wavelets
• Higher dimensiona
• The final topic will depend on the audience. We might discuss the connection to representation theory