Harmonic analysis on Euclidean spaces Definition

Fourier analysis

Builds on Harmonic analysis Fourier series Fourier transform Complex exponential

Definition

Fourier analysis is the part of harmonic analysis that studies the decomposition of functions into complex exponentials and the reconstruction of functions from these components. It covers the following transforms:

the Fourier series of a $2\pi$ -periodic function, with components $e^{inx}$ for $n \in \Z$ ;
the Fourier transform of a function on $\R^n$ , with components $e^{2\pi i x \cdot \xi}$ for $\xi \in \R^n$ ;
the discrete Fourier transform of a function on $\Z / N\Z$ , with components $e^{2\pi i jk/N}$ for $k \in \Z / N\Z$ .

Remarks

Example: the $2\pi$ -periodic function $f(x) = \sin^3 x$ satisfies $f(x) = \frac{3}{4} \sin x - \frac{1}{4} \sin 3x$ , by the identity $\sin 3x = 3 \sin x - 4 \sin^3 x$ . So its Fourier decomposition has only the frequencies $\pm 1$ and $\pm 3$ . Non-example: writing a function as a power series $\sum a_n x^n$ is an expansion into monomials, not into exponentials, so it is not Fourier analysis. The word analysis refers to the decomposition step, and the word synthesis to the reconstruction step.

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Tag	`D7PH59MNTV`
URI	`m2://D7PH59MNTV`
Kind	definition
Writing	W0 stub
MSC	42A16, 42A38, 42B10
Defines	Fourier analysis
Uses	Harmonic analysis, Fourier series, Fourier transform, Complex exponential

Fourier analysis

Definition

Remarks

Reading path

Further reading

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