Fourier Analysis and Wavelets

In this chapter, we provide an introduction to the theory of Fourier analysis and wavelets. Fourier analysis is a large branch of mathematics, and it is useful in a wide spectrum of applications, such as in solving differential equations arising in sciences and engineering, and in signal processing. The first three sections of the chapter will be devoted to the Fourier series, the Fourier transform, and the discrete Fourier transform, respectively. The Fourier transform converts a function of a time or space variable into a function of a frequency variable. When the original function is periodic, it is sufficient to consider integer multiples of the base frequency, and we are led to the notion of the Fourier series. For a general non-periodic function, we need coefficients of all possible frequencies, and the result is the Fourier transform.