The _Fourier transform_ is a decomposition of a function, $f(x),$%%which functions is a question here. just leave this unlinked for now. this is nontrivial and takes a lot of analysis.%% into an infinite sum of [sinusoids,](Fourier%20analysis%20(Index).md#Sinusoids) denoted by $\mathcal{F}[f(x)]_k$ such that $\mathcal{F}[f(x)]_k=\int dx \, f(x) e^{-ixk}.$ Here in the subscript we specify that we're transforming to a function of whatever the subscript is (usually $k$ unless we need to specify more variables). ^7b1c33
The inverse of a [Fourier transform](Fourier%20transform.md) is defined in terms of a Fourier transformed function $\widetilde f(k)=\mathcal{F}[f(x)]_k$, as $\mathcal{F}^{-1}[\widetilde f(k)]_x =\frac{1}{2\pi}\int dk f(x)\, e^{ixk},$ where this is shown to be the inverse [here.](Fourier%20transform.md#Reciprocity%20of%20$%20mathcal{F}%20{-1}$%20with%20$%20mathcal{F}$)^d634e6
A [Fourier transform](Fourier%20transform.md) may be thought of as a [change of basis between function spaces](Fourier%20transform.md#Fourier%20transforms%20as%20basis%20transformations) that is realized via an [Integral transform.](Integral%20transform.md) %%Why is it here that there are no limits to the integral? This is likely intentional on the part of the authors of Altland and von Delft because there may be situations where you can still define a fourier transform without the infinite integrals.%%
Since [Fourier transforms](Fourier%20transform.md) contain a [complex exponential term,](Fourier%20analysis%20(Index).md#Euler's%20formula) it may help to also decompose a Fourier transformed function $\widetilde f(k)=\mathcal{F}[f(x)]_k$ into its real and complex parts such that $\widetilde f(k) = \mathrm{Re}(\widetilde f(k))+\mathrm{Im}(\widetilde f(k)) = |\widetilde f(k)|e^{i\theta(k)},$ where we define $\theta(k)$ to be the _phase angle of the Fourier transform_.
# Derivation of the Fourier transform from the Fourier series
The _Fourier transform_ is a [Fourier series](Fourier%20series.md) expansion of a function in the continuum limit.
Given a Fourier Series,
[$f(x)=\frac{1}{L}\sum_k e^{ikx}\widetilde{f}_k(x)$](Fourier%20series.md#^7fe728)
as $L\rightarrow \infty$ where the spacing between the Fourier modes becomes $\delta k = \frac{2\pi}{k} \rightarrow 0,$ the Fourier series becomes a [Riemann sum](Riemann%20integral.md#Riemann%20Sum), and thus
$\frac{1}{L}\sum_k (...) = \frac{1}{2\pi}\delta k \sum_n (...) \rightarrow \frac{1}{2\pi}\int_{-\infty}^{\infty}dk (...)$
In the continuum. [Fourier coefficients](Fourier%20series#Fourier%20Coefficients) become continuous functions $\widetilde{f}_k(x)$ such that
$\widetilde{f}(k) = \int_{-\infty}^{\infty}dx f(x) e^{-ixk}$
Where the _Fourier transform_ of $f(x)$ emerges as the following [integral transform](Integral%20transform.md): [$\mathcal{F}[f(x)]_k=\int dx \, f(x) e^{-ixk}$](Fourier%20transform#^7b1c33) with its inverse being [$\mathcal{F}^{-1}[\widetilde f(k)]_x =\frac{1}{2\pi}\int dk f(x)\, e^{ixk}.$](Fourier%20transform#^d634e6)
%%Rethink the use of the phrase continuum limit here. This is a phrase from physics and I haven't seen it in Analysis! This section needs to be grounded in an understanding of analysis.%%
# Fourier transform completeness relation
The form of the integrand in the Fourier transform definition comes about as a result of the [completeness relation for the Fourier series](Fourier%20series#Completeness%20Relation). In the limit where we go from a discrete Fourier series to a continuum, its completeness relation becomes
$\delta(x) = \frac{1}{2\pi}\int_{-\infty}^\infty dk \, e^{ixk} = \mathcal{F}^{-1}[1]_x$ ^1b547b
and
$2\pi \delta(k) = \int_{-\infty}^\infty dx\, e^{-ixk} = \mathcal{F}[1]_k$
where $\delta(x)$ is the [[Dirac delta function]]. These expressions can be equivalently thought of as both the [[Fourier transform]] and inverse Fourier transform of $f(x)=1.$
## Reciprocity of $\mathcal{F}^{-1}$ with $\mathcal{F}$
The function $f(x)$ is preserved in taking $\mathcal{F}^{-1}[\mathcal{F}[f(x)]]$ and we can show this by direct calculation, where also the [completeness relation](Fourier%20transform.md#Fourier%20transform%20completeness%20relation) appears in the integrand:
$\frac{1}{2\pi}\int dk e^{iky}\bigg(\int dx\,e^{-ikx}f(x)\bigg) = \int dx \bigg(\frac{1}{2\pi}\int dk\, e^{ik(y-x)}\bigg)f(x) = \int dx \delta(y-x)f(x)=f(y)$
# Multi-dimensional Fourier transforms
([... see more](Multi-dimensional%20Fourier%20transform.md))
# Time-frequency Fourier transforms
([... see more](Time-frequency%20Fourier%20transform.md))
# Existence of Fourier transforms
Stating generic criteria for the [[Existence of Fourier transforms]] is a large topic on its own, however _generally_ functions that can be integrated in the presence of a _convergence factor_.
([... see more](Existence%20of%20Fourier%20transforms.md))
%%Fix the non objective wording here.%%
# Properties of Fourier transforms
%%This section needs to be reorganized in terms such that these properties are enumerated in a list as is the more typical format here and furthermore more properties need to be incorporated from page 47 of E.O. Brigham's fft book.%%
## Property 1
$\mathcal{F}[f(0)]_k=\int dx f(x)\;\;\;\; f(0) = \frac{dk}{2\pi}\int\mathcal{F}[f(k)]_x$
## Property 2
The Fourier transform of an even function is also even, and the Fourier transform of an odd function is likewise also odd.
## Property 3
In general, under complex conjugation,
$(\mathcal{F}[f(k)]_x)^* =\mathcal{F}[(f(-k))^*]_x$ ^daa42d
%%What is "in general" mean here?%%
## Property 4
For real valued functions
becomes
$(\mathcal{F}[f(k)]_x)^*=\pm\mathcal{F}[f(k)]_x$
## Property 5
The Fourier transform of an exponential function is a [[Dirac delta function]] and the Fourier transform of a Dirac delta function is an exponential function: $\mathcal{F}[e^{iqx}]_k = 2\pi\delta(k-q)\;\;\;\; \mathcal{F}[\delta(y-x)]_y = e^{-iky}$ where for $q=0$ and $y=0$ we obtain the [completeness relation](Fourier%20transform.md#Fourier%20transform%20completeness%20relation)s.
%%In the analysis section exponential functions should be listed as a prereq topic at some point%%
## Property 6
Fourier transforms reduce derivatives into multiplicative factors. That is, $\mathcal{F}[\partial_x f(x)]_k = ik\mathcal{F}[f(x)]_k$
and $\partial_k\mathcal{F}[f(x)]_k=-i\mathcal{F}[xf(x)]_k$
## Property 7
## Discussion of Fourier transform properties
[Property 1](Fourier%20transform.md#Property%201) follows immediately from plugging in 0 into the exponential in the Fourier transform definition. [Property 2](Fourier%20transform.md#Property%202) can be shown in one of two ways as shown below. [Property 3](Fourier%20transform.md#Property%203) and [Property 4](Fourier%20transform.md#Property%204) follow immediately from applying the definition of the Fourier transform applied to some function $f(-x)$. [Property 5](Fourier%20transform.md#Property%205) follows from the definition of the $\delta$ function.
## Unitarity of Fourier Transforms
The Fourier transform is a [unitary operator.](Unitary%20operators.md) This unitarity is given by [Parseval's theorem](Parseval's%20theorem.md).
([... see more](Parseval's%20theorem.md))
### Fourier transforms as basis transformations
The [Fourier transform](Fourier%20transform.md) is a [change of basis](Change%20of%20basis%20between%20function%20spaces) between [$L^2(I)$ spaces](L2(I)%20space.md)s.
Its role as a change of basis operator is hinted at by the fact that the Fourier transform is a [unitary operator](Unitary%20operators.md) according to [Property 8](Fourier%20transform.md#Unitarity%20of%20Fourier%20Transforms),
%%Find where you got this claim. Also are all basis changes unitary? And there needs to be a section for change of basis between function spaces.%%
# Discrete Fourier transforms
([... see more](Discrete%20Fourier%20transforms.md))
---
# Proofs and examples
# Examples of Fourier transforms
%%Here you should link to a md file or a series of md files that are just Fourier transforms of different functions plotted in matplotlib. The intuition for Fourier transforms is best understood in terms of taking them for many functions and showing what that looks like.%%
## Proof of [Property 2](Fourier%20transform.md#Property%202)
## Proof of [Property 6](Fourier%20transform.md#Property%206)
#MathematicalFoundations/Analysis/FourierAnalysis/Integrals
#MathematicalFoundations/Analysis/Functions/Functionals/Integrals/IntegralTransforms