"A Fourier series is an expansion of a periodic function into a sum of trigonometric functions."
A mathematical technique used to analyze periodic functions by breaking them down into a series of simpler functions.
"The Fourier series is an example of a trigonometric series, but not all trigonometric series are Fourier series."
"By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood."
"For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation."
"Fourier series cannot be used to approximate arbitrary functions because most functions have infinitely many terms in their Fourier series, and the series do not always converge."
"Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function."
"The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions."
"The study of the convergence of Fourier series focuses on the behaviors of the partial sums."
"Fourier series are closely related to the Fourier transform, which can be used to find the frequency information for functions that are not periodic."
"Periodic functions can be identified with functions on a circle, for this reason, Fourier series are the subject of Fourier analysis on a circle."
"The Fourier transform is also part of Fourier analysis, but is defined for functions on R^n."
"Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered."
"All of which are consistent with one another, but each of which emphasizes different aspects of the topic."
"Some of the more powerful and elegant approaches are based on mathematical ideas and tools that were not available in Fourier's time."
"Fourier analysis has birthed an area of mathematics called Fourier analysis."
"Fourier originally defined the Fourier series for real-valued functions of real arguments."
"Many other Fourier-related transforms have since been defined, extending his initial idea to many applications."
"...birthing an area of mathematics called Fourier analysis." Note: N/A Note: N/A Note: N/A Note: N/A Note: N/A