The **Fourier transform**, named after Joseph Fourier, is a mathematical transform with many applications in physics and engineering. Very commonly it transforms a mathematical function of time, into a new function, sometimes denoted by or whose argument is frequency with units of cycles or radians per second. The new function is then known as the **Fourier transform** and/or the frequency spectrum of the function The Fourier transform is also a reversible operation. Thus, given the function one can determine the original function, (See Fourier inversion theorem.) and are also respectively known as *time domain* and *frequency domain* representations of the same "event". Most often perhaps, is a real-valued function, and is complex valued, where a complex number describes both the amplitude and phase of a corresponding frequency component. In general, is also complex, such as the analytic representation of a real-valued function. The term "Fourier transform" refers to both the transform operation and to the complex-valued function it produces.

In the case of a periodic function (for example, a continuous but not necessarily sinusoidal musical sound), the Fourier transform can be simplified to the calculation of a discrete set of complex amplitudes, called Fourier series coefficients. Also, when a time-domain function is sampled to facilitate storage or computer-processing, it is still possible to recreate a version of the original Fourier transform according to the Poisson summation formula, also known as discrete-time Fourier transform. These topics are addressed in separate articles. For an overview of those and other related operations, refer to Fourier analysis or List of Fourier-related transforms.

Read more about Fourier Transform: Definition, Introduction, Properties of The Fourier Transform, Fourier Transform On Euclidean Space, Fourier Transform On Other Function Spaces, Alternatives, Other Notations, Other Conventions, Tables of Important Fourier Transforms

### Famous quotes containing the word transform:

“But I must needs take my petulance, contrasting it with my accustomed morning hopefulness, as a sign of the ageing of appetite, of a decay in the very capacity of enjoyment. We need some imaginative stimulus, some not impossible ideal which may shape vague hope, and *transform* it into effective desire, to carry us year after year, without disgust, through the routine- work which is so large a part of life.”

—Walter Pater (1839–1894)