Fourier transform

This page is a sub-page of our page on Infinitesimal Calculus of One Real Variable.

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Related sources of information:

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Fourier Transform Intuition (Better Explained on YouTube):

An interactive introduction to the Fourier Transform

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Introduction to the Fourier Transform (Part 1) (Brian Douglas on YouTube):

Introduction to the Fourier Transform (Part 2) (Brian Douglas on YouTube):

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But what is the Fourier Transform? A visual introduction (by 3Blue1Brown):

TheFourierTransform.com Presents A Simple Explanation for the Fourier Transform:

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Layman’s Explaination of the Fourier Transform (by studioTTTguTTT):

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What is a Fast Fourier Transform (FFT)? The Cooley-Tukey Algorithm (LeiosOS on YouTube):

/////// Quoting Wikipedia on “Fourier transforms”

The Fourier transform (FT) decomposes (analyzes) a function of time (a signal) into its constituent frequencies. This is similar to the way a musical chord can be expressed in terms of the volumes and frequencies (or pitches) of its constituent notes. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time.

The Fourier transform of a function of time is itself a complex-valued function of frequency, whose magnitude (modulus) represents the amount of that frequency present in the original function, and whose argument is the phase offset of the basic sinusoid in that frequency.

The Fourier transform is not limited to functions of time, but the domain of the original function is commonly referred to as the time domain.

There is also an inverse Fourier transform that mathematically synthesizes the original function (of time) from its frequency domain representation.

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Fourier transform:

$\hat f(\xi) = \int_{-\infty}^{+\infty} f(x)\,e^{-2 \pi i \xi x} \,dx$.

Inverse Fourier transform:

$f(x) = \int_{-\infty}^{+\infty} \hat f(\xi)\,e^{2 \pi i \xi x} \,d\xi$.

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Linear operations performed in one domain (time or frequency) have corresponding operations in the other domain, which are sometimes easier to perform. The operation of differentiation in the time domain corresponds to multiplication by the frequency, so some differential equations are easier to analyze in the frequency domain.

Also, convolution in the time domain corresponds to ordinary multiplication in the frequency domain (see Convolution theorem). After performing the desired operations, transformation of the result can be made back to the time domain.

Harmonic analysis is the systematic study of the relationship between the frequency and time domains, including the kinds of functions or operations that are “simpler” in one or the other. Harmonic analysis has deep connections to many areas of modern mathematics.

Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle.

The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution (e.g., diffusion).

Joseph Fourier introduced the transform in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation.

The Fourier transform can be formally defined as an improper Riemann integral, making it an integral transform, although this definition is not suitable for many applications requiring a more sophisticated integration theory. For example, many relatively simple applications use the Dirac delta function, which can be treated formally as if it were a function, but the justification requires a mathematically more sophisticated viewpoint.

The Fourier transform can also be generalized to functions of several variables on Euclidean space, sending a function of 3-dimensional ‘position space’ to a function of 3-dimensional momentum (or a function of space and time to a function of 4-momentum).

This idea makes the spatial Fourier transform very natural in the study of waves, as well as in quantum mechanics, where it is important to be able to represent wave solutions as functions of either position or momentum and sometimes both.

In general, functions to which Fourier methods are applicable are complex-valued, and possibly vector-valued.

Still further generalization is possible to functions on groups, which, besides the original Fourier transform on ℝ or ℝn (viewed as groups under addition), notably includes the discrete-time Fourier transform (DTFT, group = ℤ), the discrete Fourier transform (DFT, group = ℤ mod N) and the Fourier series or circular Fourier transform (group = S1, the unit circle ≈ closed finite interval with endpoints identified). The latter is routinely employed to handle periodic functions. The fast Fourier transform (FFT) is an algorithm for computing the DFT.

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Fast Fourier transform:

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Heisenberg uncertainty relation

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