Use Fourier identities

a repository of mathematical know-how

Use Fourier identities

Title: *

Area of mathematics: *

A comma-separated list of areas of mathematics to which this article applies. Use ">" to tag in a subcategory. Example: Analysis > Harmonic analysis, Combinatorics

Keywords:

A comma-separated list of keywords associated with this article. Example: free group

Used in:

A comma-separated list of examples of where this technique is used. Example: Cauchy-Schwarz inequality

Parent articles:		Order

Body:

[QUICK DESCRIPTION]

The Fourier transform is a powerful tool that allows one to express very general types of functions on a symmetric domain (e.g. a locally compact abelian group) in terms of special functions, such as characters.  Sometimes, when studying an expression involving some function $f(x)$, it is often useful to expand $f$ into Fourier coefficients (e.g. as Fourier series $f(x) = \sum_n \hat f(n) e^{2\pi i nx}$ if the domain is the unit circle $\R/\Z$, or as Fourier integrals $f(x) = \int_{\R^d} \hat f(\xi) e^{2\pi i \xi \cdot x}$ if the domain is a Euclidean space $\R^d$).

Generally speaking, Fourier decomposition is advantageous under one of the following conditions:

* '''Fourier phases already appear in the expression'''.  If the expression already contains terms such as $e^{itx}$, then it is natural to try to manipulate it using, say, the Fourier inversion formula.
* '''The expression one is studying would be easier to manipulate if $f$ was a character'''.  For instance, suppose one was trying to understand an expectation ${\Bbb E} f(X_1+X_2)$, where $X_1, X_2$ were two random variables.  For arbitrary $f$, there is not much one can obviously do, but if $f$ were an exponential function such as $f(x) = e^{i t x}$, then it looks like there is a way to separate $X_1$ and $X_2$ from each other.  So Fourier analysis could be a useful tool here.
* '''The expression one is studying enjoys some sort of translation invariance'''.  For instance, expressions involving convolution, constant-coefficient differential operators, addition of random variables, addition of sets in a group, or finding patterns such as arithmetic progressions, would fall into this category.  Fourier analysis can be interpreted as the representation theory of the translation action, so it is reasonable to suspect that it will be a useful tool in translation-invariant settings.
* '''The function $f$ has a particularly simple, or well understood Fourier transform'''.  For instance, if $f$ lies in a Sobolev space $H^s(\R^n)$, then its Fourier transform lies in a weighted $L^2$ space, which is a more elementary object than $H^s$, so it can be profitable to work in the Fourier domain as much as possible when dealing with problems involving $L^2$ Sobolev spaces.
* '''One expects the "worst case" to be when $f$ behaves like a character'''.  In such cases, taking a Fourier-analytic perspective is likely to isolate the dominant features of $f$, while making it easier to control all the other features.

[PREREQUISITES]

Fourier analysis

[EXAMPLE]

(Counting additive quadruples $a_1+a_2=a_3+a_4$ in a set)

[EXAMPLE]

(Bilinear Hilbert transform)

[EXAMPLE]

(Kato local smoothing estimate for the Schrodinger equation)

[EXAMPLE]

(Central limit theorem)

[EXAMPLE]

(Product estimates for $H^s(\R^n)$ - the point is that it converts an oscillatory problem into a non-oscillatory one)

[GENERAL DISCUSSION]

See also "[[If your problem can be expressed in terms of convolutions and inner products then take the Fourier transform]]".

Not every translation-invariant expression is simplified by the Fourier transform.  There are some expressions of "quadratic" expressions and higher which seem to require some sort of "higher-order" Fourier analysis (e.g. quadratic Fourier analysis) to control properly.  The theory of higher-order Fourier analysis is not nearly as well developed as linear Fourier analysis, but is the subject of ongoing research.

This is a stub

A stub is an article that is not sufficiently complete to be interesting.

Notifications

File attachments

Changes made to the attachments are not permanent until you save this post. The first "listed" file will be included in RSS feeds.

Attach new file:

Images are larger than 640x480 will be resized. The maximum upload size is 1 MB. Only files with the following extensions may be uploaded: jpg jpeg gif png svg.

Revision information

Log message:

An explanation of the additions or updates being made to help other authors understand your motivations.

Use Fourier identities

Recent articles

Active forum topics

Recent comments