"Cesaro Summability and its Applications
to Classical Fourier Series"
Abstract: This is Senior Project presentation. We present some ideas and results of Cesaro summability for series and sequences with the intention of showing how useful they are in order to answer two of main questions for classical Fourier series, more precisely: The Convergence Question: Given a real periodic function $f$ and integrable on $[-\pi,\pi]$, does the Fourier series of $f$ converge to $f$? The Uniqueness Question: If a trigonometric series $S$ converges to some function $f$ integrable on $[-\pi,\pi]$, is $S$ the Fourier series of $f$? Using the Dirichlet kernel and the Fejer kernel approach we show how the idea of Cesaro summability is used in order to obtain important theorems on these questions. One of the main result included in this presentation is Hardy's Tauberian Theorem which is used to provide an answer to the Convergence Question, known as the Dirichlet-Jordan Theorem.
Date: Wednesday, December 23, 2015
Place: Mathematics Seminar, SA-141
Tea and cookies will be served before the seminar.