# CHAPTER 15FOURIER SERIES

In 1807, Fourier announced in a seminal paper that a large class of functions can be written as linear combinations of sines and cosines. Today, infinite series representation of functions in terms of sinusoidal functions is called the Fourier series, which has become an indispensable tool in signal analysis. Spectroscopy is the branch of science that deals with the analysis of a given signal in terms of its components. Image processing and data compression are among other important areas of application for the Fourier series.

## 15.1 ORTHOGONAL SYSTEMS OF FUNCTIONS

After the introduction of Fourier series, it became clear that they are only a part of a much more general branch of mathematics called the theory of **orthogonal functions**. Legendre polynomials, Hermite polynomials, and Bessel functions are among the other commonly used orthogonal function sets. Certain features of this theory are incredibly similar to geometric vectors, where in dimensions a given vector can be written as a linear combination of linearly independent basis vectors. In the theory of orthogonal functions, we can express almost any *arbitrary* function as the linear combination of a set of basis functions. Many of the tools used in the study of ordinary vectors have counterparts ...

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