a method of expressing a periodic function in terms of sinusoidal basis functions

The Fourier series is a method of expressing a periodic function in terms of sinusoidal basis functions.

an enlargement of a periodic function f(x)

A Fourier series is an enlargement of a periodic function f(x) with respect to an infinite sum of sines and cosines.

which is a method of expressing an arbitrary periodic function as a sum of cosine terms

This applet demonstrates Fourier series, which is a method of expressing an arbitrary periodic function as a sum of cosine terms.

a way to represent a function as the sum of sine waves

If you remember your schooling on Fourier series, you recall that Fourier series is a way to represent a function as the sum of sine waves.

a series whose terms are composed of trigonometric functions

A Fourier series is a series whose terms are composed of trigonometric functions.

a mathematical tool

For better understanding of better understanding , read Fourier series' Fourier series is a mathematical tool.

an expansion of a periodic function

A Fourier series is an expansion of a periodic function.

The representation of a periodic function

The representation of a periodic function as a sum of sinusoids is known as a Fourier series.

Such a decomposition of periodic signals

Such a decomposition of periodic signals is called a Fourier series.

periodic functions

In signal processing you encounter the problem, that Fourier series represent periodic functions and that Fourier series satisfy convolution theorems (i.e. convolution of Fourier series corresponds to multiplication of represented periodic function and vice versa), but periodic functions cannot be convolved with the usual definition, since the involved integrals diverge.

an infinite series of sines and cosines

This function can be easily written (e.g. defined piece-wise), but the Fourier series is an infinite series of sines and cosines, and we all know infinite series pose notorious conceptual challenges for students at all levels.

this method of decomposing periodic functions into sines

Depending on how the original signal is expressed, this method of decomposing periodic functions into sines is known as Fourier series or Fourier transform.

a sum of sine and cosine functions

The Fourier series is a sum of sine and cosine functions.

a sum of this form

A Fourier series is a sum of this form: where each of the squares () is a different number, and one is adding infinitely many terms.

This superposition or linear combination

This superposition or linear combination is called the Fourier series.

an infinite sum of trigonometric terms

Likewise, a Fourier series is an infinite sum of trigonometric terms.