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 expansion of a periodic function

f(x) would not be periodic and it would not work because a Fourier series is an expansion of a periodic function in terms of 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.

weighted sums of sines and cosines that can represent any periodic signal

Fourier series are weighted sums of sines and cosines that can represent any periodic signal.

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.

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.

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.

a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short

The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short.

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

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.