Eva introduced the basics of multiresolution analyses, following 2.1 and 2.2 of Hernandez and Weiss, and showed how to construct a wavelet given a multiresolution analysis. Some facts about Fourier transforms for periodic functions and sequences indexed by the integers were needed. Recommended background reading on this topic includes Krantz or Dym and McKean.
A side note: the normalizing function that we need to divide a "pre-scaling" function (one whose translates give us a Reisz basis) by in order to get a scaling function (one whose translates give us an orthonormal basis) is an autocorrelation function. According to his abstract, Jon will talk about autocorrelation functions in his talk on Wednesday, September 29. There should be some connection (correlation?) between the two.