Fourier Series. Analysis and construction

Fourier series is a mathematical technique used to represent periodic functions as an infinite sum of sine and cosine functions. They were developed by the French mathematician Joseph Fourier in the 19th century as a tool for studying heat conduction in solids.

The basic idea behind Fourier series is that any periodic function can be decomposed into a series of sinusoids of different frequencies and amplitudes. This means that if we know the frequencies and amplitudes of the sinusoids that make up a periodic function, we can represent that function as a sum of those sinusoids. The representation of a function in terms of Fourier series allows us to analyze its behavior at different frequencies and is used in areas such as engineering, physics and telecommunications.

The decomposition of a function in terms of Fourier series is performed by calculating integrals. There are different techniques to perform this decomposition, among them are the Fourier Fourier series, the complex Fourier series and the trigonometric Fourier series.

Fourier series have a wide variety of applications in areas such as signal processing, partial differential equation solving and mechanical vibration analysis. They are also an important tool in communication theory, since they allow the representation of signals in terms of frequencies.