Fourier series expansion

The Fourier series expansion of a periodic function or a function over a limited interval is the function's series expansion consisting of the function's equivalent Fourier series. The terms of the series are called the Fourier coefficients, and for an arbitrary function, they can be calculated using the Fourier transform. The Fourier coefficients of a particular function may be described as "within Fourier space" or "within the Fourier domain" (though these phrases often are used with more particular meanings, e.g., the term space is generally used for a variant of Fourier series expansion of two or three dimensional functions).

As a Fourier series, the Fourier series expansion may be written in terms of trigonometric functions or alternately in equivalent exponential functions (as per Euler's formula) and may use complex numbers to include representation of phase information. Summing the series expansion including the phase information reconstructs the original (periodic) function, and summing just a number of the first terms produces an approximation of the original function.