Codes and Expansions (CodEx) Seminar

Palle Jorgensen (University of Iowa)
Fourier expansions for classes of fractals

Starting with classical Fourier analysis, the talk focuses on classes of fractals, and their wider role in harmonic analysis. We begin with a construction by the author and S. Pedersen of explicit orthogonal Fourier expansions, and fractals in the large, for certain affine fractals, as well as early work by Strichartz. We will cover several new directions, each one dealing with new aspects of the wider subject, including (among others) joint work with Dorin Dutkay, and with Eric Weber and John Herr (via infinite-dimensional Kaczmarz algorithms.) In our work with Weber and Herr, the orthogonality constraint for the Fourier expansions is relaxed. This is accomplished with a new infinite-dimensional Kaczmarz algorithm. We stress how Fourier expansions for self similar fractals contrast to their classical counterparts. By now the general theme of Fourier series, and harmonic analysis, on Fractals has taken off in a number of diverse directions; e.g., (i) wavelets on fractals, or frames; (ii) non-commutative analysis on graph limits, (iii) discrete approximations; to mention only three. A general question is: What kind of fractals admit what kind of Fourier series?