**"****Cesaro Summability and its Applications
to Classical Fourier Series**

By

**ALİ DENİZ**

**(BİLKENT UNIVERSITY)**

**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**

**Time: ****14:00**

**Place: ****Mathematics Seminar,
SA-141**

Tea and cookies will be served before the
seminar.