### Abstract:

A function, or a signal as electrical engineers like to call it, can be decomposed as a sum, possibly infinite, of sines and cosines, called its Fourier series expan- sion. The coefficients in this decomposition represent the various frequencies that are present in the signal. However the Fourier expansion fails to give information on what part of the function the frequencies come from. Thus if a signal is a function of time, a Fourier analysis of this signal does not tell us when various of its frequencies occurred. This poses a serious drawback in many applications. A lot of effort has gone into a search for alternative mecha- nisms, mechanisms whereby a signal can be decomposed into constituents that bear information both on its frequencies and where these frequencies occur in the signal. Luckily, the last 20 years has seen considerable progress due to the discovery of various functions, called wavelets, that roughly speaking replace the cosine and sine functions in the Fourier expansion scheme. Our objective here is to provide the reader with the basics of wavelet construction for the analysis of periodic functions on Z. In spite of the drawback that Fourier anal- ysis has, we do not want to give the impression that it is an outdated method. It has served mathematics and the applied sciences extremely well for over 4 centuries and it remains indispensable in various modern applications. In fact the construction of wavelets depends heavily on Fourier analysis and the prac- ticality and usefulness of wavelets derives from features that are built into the Discrete Fourier Transform, a concept we will look into first. I first came across wavelets in the Student-Faculty Seminar. This is a seminar on wavelets that was run by the department of Mathematical Sciences at Ball State during the 2003/2004 academic year. For this seminar, I was required to have had Linear Algebra experience. Recommendations included Complex Analysis and minor information from Real Analysis. Overall, seminars like these are very under- graduate friendly, with the textbook going through many linear algebra, real analysis, and complex ideas for the theorems needed. If ever the textbook is not helpful, the professors are more than willing to help out with any questions you have.