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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Fourier transform} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{harmonic_analysis}{}\paragraph*{{Harmonic analysis}}\label{harmonic_analysis} [[!include harmonic analysis - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak \noindent\hyperlink{basic_idea}{Basic idea}\dotfill \pageref*{basic_idea} \linebreak \noindent\hyperlink{generalizations}{Generalizations}\dotfill \pageref*{generalizations} \linebreak \noindent\hyperlink{general_definition}{General definition}\dotfill \pageref*{general_definition} \linebreak \noindent\hyperlink{over_the_circle_and_the_integers}{Over the circle and the integers}\dotfill \pageref*{over_the_circle_and_the_integers} \linebreak \noindent\hyperlink{over_compact_abelian_groups_and_discrete_groups}{Over compact abelian groups and discrete groups}\dotfill \pageref*{over_compact_abelian_groups_and_discrete_groups} \linebreak \noindent\hyperlink{over_cyclic_groups_the_discretized_circle}{Over cyclic groups (the discretized circle)}\dotfill \pageref*{over_cyclic_groups_the_discretized_circle} \linebreak \noindent\hyperlink{FourierTransformOnCartesianSpaces}{Over Cartesian spaces}\dotfill \pageref*{FourierTransformOnCartesianSpaces} \linebreak \noindent\hyperlink{on_functions_with_rapidly_decreasing_partial_derivatives}{On functions with rapidly decreasing partial derivatives}\dotfill \pageref*{on_functions_with_rapidly_decreasing_partial_derivatives} \linebreak \noindent\hyperlink{FourierTransformOnTemperedDistributionsOverCartesianSpace}{On tempered distributions}\dotfill \pageref*{FourierTransformOnTemperedDistributionsOverCartesianSpace} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{Idea}{}\subsection*{{Idea}}\label{Idea} \hypertarget{basic_idea}{}\subsubsection*{{Basic idea}}\label{basic_idea} Generally, a \emph{Fourier transform} is an [[isomorphism]] between the algebra of [[complex numbers|complex]]-valued [[functions]] on a suitable [[topological group]] and a [[convolution product]]-algebra structure on the [[Pontrjagin dual]] group. The study of Fourier transforms is also called \emph{Fourier analysis}. Typically, such as in the case over [[Cartesian space]] (def. \ref{FourierTransformSmoothFunctionsWithRapidlyDecayingDerivativesOnCartesianSpace} below) this means to decompose any suitable [[function]] as a [[superposition]] of \emph{[[complex number|complex]] [[plane waves]]}, which may be thought of as the ``harmonics'' of the given function. Therefore one speaks of \emph{[[harmonic analysis]]}. \hypertarget{generalizations}{}\subsubsection*{{Generalizations}}\label{generalizations} The concept of Fourier transforms of functions generalizes in a variety of ways. Core part of the subject of Fourier analysis is the generalization to \emph{[[Fourier transform of distributions]]} (def. \ref{FourierTransformOnTemperedDistributions} below). The asymptotic growth of the [[Fourier transform of distributions]] reflects the singularity structure of the distributions, in dependence of the [[direction of a vector|direction]] of the [[wave vector]] (the ``[[wave front set]]''). The study of this behaviour is called \emph{[[microlocal analysis]]}. If the role of complex [[plane waves]] in the Fourier transform are replaced by [[wavelets]], one speaks of the \emph{[[wavelet transform]]}. For [[noncommutative topology|noncommutative topological]] groups, instead of continuous characters one should consider [[irreducible representation|irreducible]] [[unitary representations]], which makes the subject much more difficult. There are also generalizations in [[noncommutative geometry]], see \emph{[[quantum group Fourier transform]]}. \hypertarget{general_definition}{}\subsection*{{General definition}}\label{general_definition} Let $G$ be a [[locally compact space|locally compact]] [[Hausdorff topological space|Hausdorff]] [[abelian group|abelian]] [[topological group]] with [[Haar measure|invariant (= Haar) measure]] $\mu$. Then for each $f\in L_1(G,\mu)$, define its \textbf{Fourier transform} $\hat{f}$ as a function on its [[Pontrjagin dual]] group $\hat{G}$ given by \begin{displaymath} \hat{f}(\chi) = \int_G f(x) \widebar{\chi(x)} d\mu(x),\,\,\,\chi\in\hat{G}. \end{displaymath} The Fourier transform of $f\in L_1(G,\mu)$ is always continuous and bounded on $\hat{G}$; the transform of the [[convolution]] of two functions is the product of the transforms of each of the functions separately. \hypertarget{over_the_circle_and_the_integers}{}\subsection*{{Over the circle and the integers}}\label{over_the_circle_and_the_integers} In the classical case of \textbf{Fourier series}, where $G=\mathbb{Z}$ (the additive group of [[integers]]) and $\hat{G}=S^1$ (the [[circle group]]), the Fourier transform restricts to a unitary operator between the [[Hilbert spaces]] $L_2(S^1,d t)$ and $l_2(\mathbb{Z})$ and the Fourier coefficients are the numbers \begin{displaymath} c_n := \hat{f}(\chi_n) = \int_0^1 f(t) e^{-2\pi i n t} d t, \end{displaymath} for $n\in\mathbb{Z}$, where the functions $\chi_n(t)= e^{2\pi i n t}$ form an orthonormal basis of $L_2(S^1,d t)$. The Fourier transform $\hat{\chi_n}$ is then viewed as the $\mathbb{Z}$-series $\delta_n$ which in the $n$-th place has $1$ and elsewhere $0$. The Fourier transform replaces the operator of differentiation $d/d t$ by the operator of multiplication by the series $\{2\pi i n\}_{n\in\mathbb{Z}}$. \hypertarget{over_compact_abelian_groups_and_discrete_groups}{}\subsection*{{Over compact abelian groups and discrete groups}}\label{over_compact_abelian_groups_and_discrete_groups} In general, if $G$ is a [[compact space|compact]] abelian group (whose [[Pontrjagin dual]] is [[discrete group|discrete]]), one can normalize the invariant measure by $\mu(G)=1$ and $\hat{\mu}(X)=card(X)$ for $X\subset\hat{G}$. Then the Fourier transform restricts to a unitary operator from $L_2(X,\mu)$ to $L_2(\hat{G},\hat{\mu})$. \hypertarget{over_cyclic_groups_the_discretized_circle}{}\subsection*{{Over cyclic groups (the discretized circle)}}\label{over_cyclic_groups_the_discretized_circle} \begin{itemize}% \item [[discrete Fourier transform]] \end{itemize} \hypertarget{FourierTransformOnCartesianSpaces}{}\subsection*{{Over Cartesian spaces}}\label{FourierTransformOnCartesianSpaces} Throughout, let $n \in \mathbb{N}$ and write $\mathbb{R}^n$ for the [[Cartesian space]] of [[dimension]] $n$ and write $(-) \cdot (-)$ for the canonical [[inner product]] on $\mathbb{R}^n$: \begin{displaymath} k \cdot x \;\coloneqq\; \underoverset{a = 1}{n}{\sum} k_n x^n \,. \end{displaymath} In the following by a [[smooth function]] $f \in C^\infty(\mathbb{R}^n)$ on $\mathbb{R}^n$ we mean a smooth function with values in the [[complex numbers]]. For $f \in C^\infty(\mathbb{R}^n)$, we write $f^\ast \in C^\infty(\mathbb{R}^n)$ for its pointwise [[complex conjugate]]: \begin{displaymath} f^\ast(x) \coloneqq (f(x))^\ast \,. \end{displaymath} \hypertarget{on_functions_with_rapidly_decreasing_partial_derivatives}{}\paragraph*{{On functions with rapidly decreasing partial derivatives}}\label{on_functions_with_rapidly_decreasing_partial_derivatives} \begin{defn} \label{SchwartzSpace}\hypertarget{SchwartzSpace}{} \textbf{([[Schwartz space]] of [[functions with rapidly decreasing partial derivatives]])} A [[complex number|complex]]-valued [[smooth function]] $f \in C^\infty(\mathbb{R}^n)$ is said to have \emph{[[rapidly decreasing function|rapidly decreasing]] [[partial derivatives]]} if for all $\alpha,\beta \in \mathbb{N}^{n}$ we have \begin{displaymath} \underset{x \in \mathbb{R}^n}{sup} {\vert x^\beta \partial^\alpha f(x) \vert} \;\lt\; \infty \,. \end{displaymath} Write \begin{displaymath} \mathcal{S}(\mathbb{R}^n) \hookrightarrow C^\infty(\mathbb{R}^n) \end{displaymath} for the sub-[[vector space]] on the functions with rapidly decreasing partial derivatives regarded as a [[topological vector space]] for the [[Frechet space]] struzcture induced by the [[seminorms]] \begin{displaymath} p_{\alpha, \beta}(f) \coloneqq \underset{x \in \mathbb{R}^n}{sup} {\vert x^\beta \partial^\alpha f(x) \vert} \,. \end{displaymath} This is also called the \emph{[[Schwartz space]]}. \end{defn} (e.g. \hyperlink{Hoermander90}{H\"o{}rmander 90, def. 7.1.2}) \begin{example} \label{CompactlySupportedSmoothFunctionsAreFunctionsWithRapidlyDecreasingDerivatives}\hypertarget{CompactlySupportedSmoothFunctionsAreFunctionsWithRapidlyDecreasingDerivatives}{} \textbf{([[compactly supported function|compactly supported]] [[smooth function]] are [[functions with rapidly decreasing partial derivatives]])} Every [[compactly supported function|compactly supported]] [[smooth function]] ([[bump function]]) $b \in C^\infty_{cp}(\mathbb{R}^n)$ rapidly decreasing partial derivatives (def. \ref{SchwartzSpace}): \begin{displaymath} C^\infty(\mathbb{R}^n) \hookrightarrow \mathcal{S}(\mathbb{R}^n) \,. \end{displaymath} \end{example} \begin{prop} \label{ConvolutionProductOnSchwartzSpace}\hypertarget{ConvolutionProductOnSchwartzSpace}{} \textbf{(pointwise product and [[convolution product]] on [[Schwartz space]])} The [[Schwartz space]] $\mathcal{S}(\mathbb{R}^n)$ (def. \ref{SchwartzSpace}) is closed under the following operations on smooth functions $f,g \in \mathcal{S}(\mathbb{R}^n) \hookrightarrow C^\infty(\mathbb{R}^n)$ \begin{enumerate}% \item pointwise product: \begin{displaymath} (f \cdot g)(x) \coloneqq f(x) \cdot g(x) \end{displaymath} \item [[convolution product]]: \begin{displaymath} (f \star g)(x) \coloneqq \underset{y \in \mathbb{R}^n}{\int} f(y)\cdot g(x-y) \, dvol(y) \,. \end{displaymath} \end{enumerate} \end{prop} \begin{proof} By the [[product law]] of [[differentiation]]. \end{proof} \begin{prop} \label{RapidlyDecreasingFunctionsAreIntegrable}\hypertarget{RapidlyDecreasingFunctionsAreIntegrable}{} \textbf{([[rapidly decreasing functions]] are [[integrable functions|integrable]])} Every [[rapidly decreasing function]] $f \colon \mathbb{R}^n \to \mathbb{R}$ (def. \ref{SchwartzSpace}) is an [[integrable function]] in that its [[integral]] exists: \begin{displaymath} \underset{x \in \mathbb{R}^n}{\int} f(x) \, d^n x \;\lt\; \infty \end{displaymath} In fact for each $\alpha \in \mathbb{N}^n$ the product of $f$ with the $\alpha$-power of the [[coordinate functions]] exists: \begin{displaymath} \underset{x \in \mathbb{R}^n}{\int} x^\alpha f(x)\, d^n x \;\lt\; \infty \,. \end{displaymath} \end{prop} \begin{defn} \label{FourierTransformSmoothFunctionsWithRapidlyDecayingDerivativesOnCartesianSpace}\hypertarget{FourierTransformSmoothFunctionsWithRapidlyDecayingDerivativesOnCartesianSpace}{} \textbf{([[Fourier transform]] of [[functions with rapidly decreasing partial derivatives]])} The \emph{[[Fourier transform]]} is the [[continuous linear functional]] \begin{displaymath} \widehat{(-)} \;\colon\; \mathcal{S}(\mathbb{R}^n) \longrightarrow \mathcal{S}(\mathbb{R}^n) \end{displaymath} on the [[Schwartz space]] of [[functions with rapidly decreasing partial derivatives]] (def. \ref{SchwartzSpace}), which is given by [[integration]] against the [[exponential function|exponential]] [[plane wave]] functions \begin{displaymath} x \mapsto e^{- i k \cdot x} \end{displaymath} times the standard [[volume form]] $d^n x$: \begin{equation} \hat f(k) \;\colon\; \int_{x \in \mathbb{R}^n} e^{- i \, k \cdot x} f(x) \, d^n x \,. \label{IntegralExpressionForFourierTransform}\end{equation} Here the argument $k \in \mathbb{R}^n$ of the Fourier transform is also called the \emph{[[wave vector]]}. \end{defn} (e.g. \hyperlink{Hoermander90}{H\"o{}rmander, lemma 7.1.3}) \begin{defn} \label{FourierInversion}\hypertarget{FourierInversion}{} \textbf{([[Fourier inversion theorem]])} The [[Fourier transform]] $\widehat{(-)}$ (def. \ref{FourierTransformSmoothFunctionsWithRapidlyDecayingDerivativesOnCartesianSpace}) on the [[Schwartz space]] $\mathcal{S}(\mathbb{R}^n)$ (def. \ref{SchwartzSpace}) is an [[isomorphism]], with [[inverse function]] the \emph{[[inverse Fourier transform]]} \begin{displaymath} \widecheck {(-)} \;\colon\; \mathcal{S}(\mathbb{R}^n) \longrightarrow \mathcal{S}(\mathcal{R}^n) \end{displaymath} given by \begin{displaymath} \widecheck g (x) \;\coloneqq\; \underset{k \in \mathbb{R}^n}{\int} g(k) e^{i k \cdot x} \, \frac{d^n k}{(2\pi)^n} \,. \end{displaymath} Hence in the language of [[harmonic analysis]] the function $\widecheck g \colon \mathbb{R}^n \to \mathbb{C}$ is the [[superposition]] of [[plane waves]] in which the plane wave with [[wave vector]] $k\in \mathbb{R}^n$ appears with [[amplitude]] $g(k)$. \end{defn} (e.g. \hyperlink{Hoermander90}{H\"o{}rmander, theorem 7.1.5}) \begin{prop} \label{BasicPropertiesOfFourierTransformOverCartesianSpaces}\hypertarget{BasicPropertiesOfFourierTransformOverCartesianSpaces}{} \textbf{(basic properties of the [[Fourier transform]])} The [[Fourier transform]] $\widehat{(-)}$ (def. \ref{FourierTransformSmoothFunctionsWithRapidlyDecayingDerivativesOnCartesianSpace}) on the [[Schwartz space]] $\mathcal{S}(\mathbb{R}^n)$ (def. \ref{SchwartzSpace}) satisfies the following properties, for all $f,g \in \mathcal{S}(\mathbb{R}^n)$: \begin{enumerate}% \item (interchanging [[coordinate]] [[multiplication]] with [[partial derivatives]]) \begin{equation} \widehat{ x^a f } = + i \partial_a \widehat f \phantom{AAAAA} \widehat{ - i\partial_a f} = k_a \widehat f \label{FourierTransformInterchangesCoordinateProductWithDerivative}\end{equation} \item (interchanging pointwise multiplication with [[convolution product]], remark \ref{ConvolutionProductOnSchwartzSpace}): \begin{equation} \widehat {(f \star g)} = \widehat{f} \cdot \widehat{g} \phantom{AAAA} \widehat{ f \cdot g } = (2\pi)^{-n} \widehat{f} \star \widehat{g} \label{FourierTransformInterchangesPointwiseProductWithConvolution}\end{equation} \item ([[unitary operator|unitarity]], [[Parseval's theorem]]) \begin{displaymath} \underset{x \in \mathbb{R}^n}{\int} f(x) g^\ast(x)\, d^n x \;=\; \underset{k \in \mathbb{R}^n}{\int} \widehat{f}(k) \widehat{g}^\ast(k) \, d^n k \end{displaymath} \item \begin{equation} \underset{k \in \mathbb{R}^n}{\int} \widehat{f}(k) \cdot g(k) \, d^n k \;=\; \underset{x \in \mathbb{R}^n}{\int} f(x) \cdot \widehat{g}(x) \, d^n x \label{FourierTransformInIntegralOfProductMayBeShiftedToOtherFactor}\end{equation} \end{enumerate} \end{prop} (e.g \hyperlink{Hoermander90}{H\"o{}rmander 90, lemma 7.1.3, theorem 7.1.6}) \hypertarget{FourierTransformOnTemperedDistributionsOverCartesianSpace}{}\paragraph*{{On tempered distributions}}\label{FourierTransformOnTemperedDistributionsOverCartesianSpace} The [[Schwartz space]] of [[functions with rapidly decreasing partial derivatives]] (def. \ref{SchwartzSpace}) serves the purpose to support the [[Fourier transform]] (def. \ref{FourierTransformSmoothFunctionsWithRapidlyDecayingDerivativesOnCartesianSpace}) together with its inverse (prop. \ref{FourierInversion}), but for many applications one needs to apply the Fourier transform to more general functions, and in fact to \emph{[[generalized functions]]} in the sense of [[distributions]] (via \href{non-singular+distribution#NonSingularDistributionsAreDenseInAllDistributions}{this prop.}). But with the [[Schwartz space]] in hand, this generalization is readily obtained by [[formal duality]]: \begin{defn} \label{TemperedDistribution}\hypertarget{TemperedDistribution}{} \textbf{([[tempered distribution]])} A \emph{[[tempered distribution]]} is a [[continuous linear functional]] \begin{displaymath} u \;\colon\; \mathcal{S}(\mathbb{R}^n) \longrightarrow \mathbb{C} \end{displaymath} on the [[Schwartz space]] (def. \ref{SchwartzSpace}) of [[functions with rapidly decaying partial derivatives]]. The [[vector space]] of all tempered distributions is canonically a [[topological vector space]] as the [[dual space]] to the [[Schwartz space]], denoted \begin{displaymath} \mathcal{S}'(\mathbb{R}^n) \;\coloneqq\; \left( \mathcal{S}(\mathbb{R}^n) \right)^\ast \,. \end{displaymath} \end{defn} e.g. (\hyperlink{Hoermander90}{H\"o{}rmander 90, def. 7.1.7}) \begin{example} \label{SomeNonSingularTemperedDistributions}\hypertarget{SomeNonSingularTemperedDistributions}{} \textbf{(some [[non-singular distribution|non-singular]] [[tempered distributions]])} Every [[function with rapidly decreasing partial derivatives]] $f \in \mathcal{S}(\mathbb{R}^n)$ (def. \ref{SchwartzSpace}) induces a [[tempered distribution]] $u_f \in \mathcal{S}'(\mathbb{R}^n)$ (def. \ref{TemperedDistribution}) by [[integration|integrating]] against it: \begin{displaymath} u_f \;\colon\; g \mapsto \underset{x \in \mathbb{R}^n}{\int} g(x) f(x)\, d^n x \,. \end{displaymath} This construction is a linear inclusion \begin{displaymath} \mathcal{S}(\mathbb{R}^n) \overset{\text{dense}}{\hookrightarrow} \mathcal{S}'(\mathbb{R}^n) \end{displaymath} of the [[Schwartz space]] into its [[dual space]] of [[tempered distributions]]. This is a [[dense subspace]] inclusion. In fact already the restriction of this inclusion to the [[compactly supported function|compactly supported]] [[smooth functions]] (example \ref{CompactlySupportedSmoothFunctionsAreFunctionsWithRapidlyDecreasingDerivatives}) is a [[dense subspace]] inclusion: \begin{displaymath} C^\infty_{cp}(\mathbb{R}^n) \overset{dense}{\hookrightarrow} \mathcal{S}'(\mathbb{R}^n) \,. \end{displaymath} This means that every [[tempered distribution]] is a [[limit of a sequence|limit]] of a [[sequence]] of ordinary [[functions with rapidly decreasing partial derivatives]], and in fact even the [[limit of a sequence|limit]] of a [[sequence]] of [[compactly supported function|compactly supported]] [[smooth functions]] ([[bump functions]]). It is in this sense that [[tempered distributions]] are ``generalized functions''. \end{example} (e.g. \hyperlink{Hoermander90}{H\"o{}rmander 90, lemma 7.1.8}) \begin{example} \label{CompactlySupportedDistibutionsAreTemperedDistributions}\hypertarget{CompactlySupportedDistibutionsAreTemperedDistributions}{} \textbf{([[compactly supported distributions]] are [[tempered distributions]])} Every [[compactly supported distribution]] is a [[tempered distribution]] (def. \ref{TemperedDistribution}), hence there is a [[linear map|linear]] [[injection|inclusion]] \begin{displaymath} \mathcal{E}'(\mathbb{R}^n) \hookrightarrow \mathcal{S}'(\mathbb{R}^n) \,. \end{displaymath} \end{example} \begin{example} \label{DiracDeltaDistribution}\hypertarget{DiracDeltaDistribution}{} \textbf{([[delta distribution]])} Write \begin{displaymath} \delta_0(-) \;\in\; \mathcal{E}'(\mathbb{R}^n) \end{displaymath} for the [[distribution]] given by point evaluation of functions at the origin of $\mathbb{R}^n$: \begin{displaymath} \delta_0(-) \;\colon\; f \mapsto f(0) \,. \end{displaymath} This is clearly a [[compactly supported distribution]]; hence a [[tempered distribution]] by example \ref{CompactlySupportedDistibutionsAreTemperedDistributions}. We write just ``$\delta(-)$'' (without the subscript) for the corresponding [[generalized function]] (example \ref{SomeNonSingularTemperedDistributions}), so that \begin{displaymath} \underset{x \in \mathbb{R}^n}{\int} \delta(x) f(x) \, d^n x \;\coloneqq\; f(0) \,. \end{displaymath} \end{example} \begin{example} \label{SquareIntegrableFunctionsInduceTemperedDistributions}\hypertarget{SquareIntegrableFunctionsInduceTemperedDistributions}{} \textbf{([[square integrable functions]] induce [[tempered distributions]])} Let $f \in L^p(\mathbb{R}^n)$ be a function in the $p$th [[Lebesgue space]], e.g. for $p = 2$ this means that $f$ is a [[square integrable function]]. Then the operation of [[integration]] against the [[measure]] $f dvol$ \begin{displaymath} g \mapsto \underset{x \in \mathbb{R}^n}{\int} g(x) f(x) \, d^n x \end{displaymath} is a [[tempered distribution]] (def. \ref{TemperedDistribution}). \end{example} (e.g. \hyperlink{Hoermander90}{H\"o{}rmander 90, below lemma 7.1.8}) Property \eqref{FourierTransformInIntegralOfProductMayBeShiftedToOtherFactor} of the ordinary [[Fourier transform]] on [[functions with rapidly decreasing partial derivatives]] motivates and justifies the fullowing generalization: \begin{defn} \label{FourierTransformOnTemperedDistributions}\hypertarget{FourierTransformOnTemperedDistributions}{} \textbf{([[Fourier transform of distributions]] on [[tempered distributions]])} The \emph{[[Fourier transform of distributions]]} of a [[tempered distribution]] $u \in \mathcal{S}'(\mathbb{R}^n)$ (def. \ref{TemperedDistribution}) is the [[tempered distribution]] $\widehat u$ defined on a smooth function $f \in \mathcal{S}(\mathbb{R}^n)$ in the [[Schwartz space]] (def. \ref{SchwartzSpace}) by \begin{displaymath} \widehat{u}(f) \;\coloneqq\; u\left( \widehat f\right) \,, \end{displaymath} where on the right $\widehat f \in \mathcal{S}(\mathbb{R}^n)$ is the [[Fourier transform]] of functions from def. \ref{FourierTransformSmoothFunctionsWithRapidlyDecayingDerivativesOnCartesianSpace}. \end{defn} (e.g. \hyperlink{Hoermander90}{H\"o{}rmander 90, def. 1.7.9}) \begin{example} \label{FourierTransformOfDistributionsIndeedGeneralizedOrdinaryFourierTransform}\hypertarget{FourierTransformOfDistributionsIndeedGeneralizedOrdinaryFourierTransform}{} \textbf{([[Fourier transform of distributions]] indeed generalizes [[Fourier transform]] of [[functions with rapidly decreasing partial derivatives]])} Let $u_f \in \mathcal{S}'(\mathbb{R}^n)$ be a [[non-singular distribution|non-singular]] [[tempered distribution]] induced, via example \ref{SomeNonSingularTemperedDistributions}, from a [[function with rapidly decreasing partial derivatives]] $f \in \mathcal{S}(\mathbb{R}^n)$. Then its [[Fourier transform of distributions]] (def. \ref{FourierTransformOnTemperedDistributions}) is the [[non-singular distribution]] induced from the [[Fourier transform]] of $f$: \begin{displaymath} \widehat{u_f} \;=\; u_{\hat f} \,. \end{displaymath} \end{example} \begin{proof} Let $g \in \mathcal{S}(\mathbb{R}^n)$. Then \begin{displaymath} \begin{aligned} \widehat{u_f}(g) & \coloneqq u_f\left( \widehat{g}\right) \\ & = \underset{x \in \mathbb{R}^n}{\int} f(x) \hat g(x)\, d^n x \\ & = \underset{x \in \mathbb{R}^n}{\int} \hat f(x) g(x) \, d^n x \\ & = u_{\hat f}(g) \end{aligned} \end{displaymath} Here all equalities hold by definition, except for the third: this is property \eqref{FourierTransformInIntegralOfProductMayBeShiftedToOtherFactor} from prop. \ref{BasicPropertiesOfFourierTransformOverCartesianSpaces}. \end{proof} \begin{example} \label{CompactlySupportedDistributionFourierTransform}\hypertarget{CompactlySupportedDistributionFourierTransform}{} \textbf{([[Fourier transform of distributions|Fourier transform]] of [[compactly supported distributions]])} Under the identification of [[smooth functions]] of bounded growth with [[non-singular distributions|non-singular]] [[tempered distributions]] (example \ref{SomeNonSingularTemperedDistributions}), the [[Fourier transform of distributions]] (def. \ref{FourierTransformOnTemperedDistributions}) of a [[tempered distribution]] that happens to be [[compactly supported distribution|compactly supported]] (example \ref{CompactlySupportedDistibutionsAreTemperedDistributions}) \begin{displaymath} u \in \mathcal{E}'(\mathbb{R}^n) \hookrightarrow \mathcal{S}'(\mathbb{R}^n) \end{displaymath} is simply \begin{displaymath} \widehat{u}(k) = u\left( e^{- i k \cdot (-)}\right) \,. \end{displaymath} \end{example} (\hyperlink{Hoermander90}{H\"o{}rmander 90, theorem 7.1.14}) \begin{defn} \label{FourierTransformOfDeltaDistribution}\hypertarget{FourierTransformOfDeltaDistribution}{} \textbf{([[Fourier transform of distributions|Fourier transform]] of the [[delta-distribution]])} The [[Fourier transform of distributions|Fourier transform]] (def. \ref{FourierTransformOnTemperedDistributions}) of the [[delta distribution]] (def. \ref{DiracDeltaDistribution}), via example \ref{CompactlySupportedDistributionFourierTransform}, is the [[constant function]] on 1: \begin{displaymath} \begin{aligned} \widehat {\delta}(k) & = \underset{x \in \mathbb{R}^n}{\int} \delta(x) e^{- i k x} \, d x \\ & = 1 \end{aligned} \end{displaymath} This implies by the [[Fourier inversion theorem]] (prop. \ref{FourierInversionTheoremForDistributions}) that the [[delta distribution]] itself has equivalently the following expression as a [[generalized function]] \begin{displaymath} \begin{aligned} \delta(x) & = \widecheck{\widehat {\delta_0}}(x) \\ & = \underset{k \in \mathbb{R}^n}{\int} e^{i k \cdot x} \, \frac{d^n k}{ (2\pi)^n } \end{aligned} \end{displaymath} in the sense that for every [[function with rapidly decreasing partial derivatives]] $f \in \mathcal{S}(\mathbb{R}^n)$ (def. \ref{SchwartzSpace}) we have \begin{displaymath} \begin{aligned} f(x) & = \underset{y \in \mathbb{R}^n}{\int} f(y) \delta(y-x) \, d^n y \\ & = \underset{y \in \mathbb{R}^n}{\int} \underset{k \in \mathbb{R}^n}{\int} f(y) e^{i k \cdot (y-x)} \, \frac{d^n k}{(2\pi)^n} \, d^n y \\ & = \underset{k \in \mathbb{R}^n}{\int} e^{- i k \cdot x} \underset{= \widehat{f}(-k) }{ \underbrace{ \underset{y \in \mathbb{R}^n}{\int} f(y) e^{i k \cdot y} \, d^n y } } \,\, \frac{d^n k}{(2\pi)^n} \\ & = + \underset{k \in \mathbb{R}^n}{\int} e^{i k \cdot x} \underset{= \widehat{f}(k) }{ \underbrace{ \underset{y \in \mathbb{R}^n}{\int} f(y) e^{- i k \cdot y} \, d^n y } } \,\, \frac{d^n k}{(2\pi)^n} \\ & = \widecheck{\widehat{f}}(x) \end{aligned} \end{displaymath} which is the statement of the [[Fourier inversion theorem]] for smooth functions (prop. \ref{FourierInversion}). (Here in the last step we used [[change of integration variables]] $k \mapsto -k$ which introduces one sign $(-1)^{n}$ for the new volume form, but another sign $(-1)^n$ from the re-[[orientation]] of the integration domain. ) Equivalently, the above computation shows that the [[delta distribution]] is the [[neutral element]] for the [[convolution product of distributions]]. \end{defn} \begin{prop} \label{PaleyWienerSchwartzTheorem}\hypertarget{PaleyWienerSchwartzTheorem}{} \textbf{([[Paley-Wiener-Schwartz theorem]])} Let $u \in \mathcal{E}'(\mathbb{R}^n) \hookrightarrow \mathcal{S}'(\mathbb{R}^n)$ be a [[compactly supported distribution]] regarded as a [[tempered distribution]] by example \ref{CompactlySupportedDistibutionsAreTemperedDistributions}. Then its [[Fourier transform of distributions]] (def. \ref{FourierTransformOnTemperedDistributions}) is a [[non-singular distribution]] induced from a [[smooth function]] that grows at most exponentially. \end{prop} \begin{prop} \label{FourierInversionTheoremForDistributions}\hypertarget{FourierInversionTheoremForDistributions}{} \textbf{([[Fourier inversion theorem]] for [[Fourier transform of distributions]])} The operation of forming the [[Fourier transform of distributions]] $\widehat{u}$ (def. \ref{FourierTransformOnTemperedDistributions}) [[tempered distributions]] $u \in \mathcal{S}'(\mathbb{R}^n)$ (def. \ref{TemperedDistribution}) is an [[isomorphism]], with [[inverse]] given by \begin{displaymath} \widecheck{ u } \;\colon\; g \mapsto u\left( \widecheck{g}\right) \,, \end{displaymath} where on the right $\widecheck{g}$ is the ordinary [[inverse Fourier transform]] of $g$ according to prop. \ref{FourierInversion}. \end{prop} \begin{proof} By def. \ref{FourierTransformOnTemperedDistributions} this follows immediately from the [[Fourier inversion theorem]] for smooth functions (prop. \ref{FourierInversion}). \end{proof} We have the following distributional generalization of the basic property \eqref{FourierTransformInterchangesPointwiseProductWithConvolution} from prop. \ref{BasicPropertiesOfFourierTransformOverCartesianSpaces}: \begin{prop} \label{FourierTransformOfDistributionsInterchangesConvolutionOfDistributionsWithPointwiseProduct}\hypertarget{FourierTransformOfDistributionsInterchangesConvolutionOfDistributionsWithPointwiseProduct}{} \textbf{([[Fourier transform of distributions]] interchanges [[convolution of distributions]] with pointwise product)} Let \begin{displaymath} u_1 \in \mathcal{S}'(\mathbb{R}^n) \end{displaymath} be a [[tempered distribution]] (def. \ref{TemperedDistribution}) and \begin{displaymath} u_2 \in \mathcal{E}'(\mathbb{R}^n) \hookrightarrow \mathcal{S}'(\mathbb{R}^n) \end{displaymath} be a [[compactly supported distribution]], regarded as a [[tempered distribution]] via example \ref{CompactlySupportedDistibutionsAreTemperedDistributions}. Observe here that the [[Paley-Wiener-Schwartz theorem]] (prop. \ref{PaleyWienerSchwartzTheorem}) implies that the [[Fourier transform of distributions]] of $u_1$ is a [[non-singular distribution]] $\widehat{u_1} \in C^\infty(\mathbb{R}^n)$ so that the product $\widehat{u_1} \cdot \widehat{u_2}$ is always defined. Then the [[Fourier transform of distributions]] of the [[convolution product of distributions]] is the product of the [[Fourier transform of distributions]]: \begin{displaymath} \widehat{u_1 \star u_2} \;=\; \widehat{u_1} \cdot \widehat{u_2} \,. \end{displaymath} \end{prop} (e.g. \hyperlink{Hoermander90}{H\"o{}rmander 90, theorem 7.1.15}) \begin{remark} \label{ProductOfDistributionsViaFourierTransformOfConvolution}\hypertarget{ProductOfDistributionsViaFourierTransformOfConvolution}{} \textbf{([[product of distributions]] via [[Fourier transform of distributions]])} Prop. \ref{FourierTransformOfDistributionsInterchangesConvolutionOfDistributionsWithPointwiseProduct} together with the [[Fourier inversion theorem]] (prop. \ref{FourierInversionTheoremForDistributions}) suggests to \emph{define} the [[product of distributions]] $u_1 \cdot u_2$ for [[compactly supported distributions]] $u_1, u_2 \in \mathcal{E}'(\mathbb{R}^n) \hookrightarrow \mathcal{S}'(\mathbb{R}^n)$ by the formula \begin{displaymath} \widehat{ u_1 \cdot u_2 } \;\coloneqq\; (2\pi)^n \widehat{u_1} \star \widehat{u_2} \end{displaymath} which would complete the generalization of of property \eqref{FourierTransformInterchangesPointwiseProductWithConvolution} from prop. \ref{BasicPropertiesOfFourierTransformOverCartesianSpaces}. For this to make sense, the [[convolution product]] of the [[smooth functions]] on the right needs to exist, which is not guaranteed (prop. \ref{ConvolutionProductOnSchwartzSpace} does not apply here!). The condition that this exists is the [[Lars Hörmander|Hörmander]]-condition on the \emph{[[wave front set]]} of $u_1$ and $u_2$. See at \emph{[[product of distributions]]} for more. \end{remark} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[Fourier integral operator]] \item [[Fourier transform of distributions]] \item [[pseudodifferential operator]] \item [[Poisson summation formula]] \item [[Laplace transform]], [[Fourier-Laplace transforms]] \item [[Mellin transform]] \item [[wavefront set]] \item [[wavelet transform]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} Lecture notes include \begin{itemize}% \item [[John Peacock]], \emph{Fourier analysis} 2013 (\href{http://www.roe.ac.uk/japwww/teaching/fourier/fourier_lectures_part1.pdf}{part 1 pdf}, \href{http://www.roe.ac.uk/japwww/teaching/fourier/fourier_lectures_part2.pdf}{part 2 pdf}, \href{http://www.roe.ac.uk/japwww/teaching/fourier/fourier_lectures_part3.pdf}{part 3 pdf}, \href{http://www.roe.ac.uk/japwww/teaching/fourier/fourier_lectures_part4.pdf}{part 4 pdf}, \href{http://www.roe.ac.uk/japwww/teaching/fourier/fourier_lectures_part5.pdf}{part 5 pdf}) \item Gerald B. Folland, \emph{A course in abstract harmonic analysis}, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995. x+276 pp. \href{http://books.google.com/books?hl=en&lr=&id=0VwYZI1DypUC}{gBooks} \end{itemize} Discussion in the broader context of [[functional analysis]] and [[distribution]] theory: \begin{itemize}% \item [[Lars Hörmander]], chapter 7 of \emph{The analysis of linear partial differential operators}, vol. I, Springer 1983, 1990 \item [[Sergiu Klainerman]], chapter 5 of of \emph{Lecture notes in analysis}, 2011 (\href{https://web.math.princeton.edu/~seri/homepage/courses/Analysis2011.pdf}{pdf}) \end{itemize} category: analysis [[!redirects Fourier transforms]] [[!redirects Fourier series]] [[!redirects Fourier integral]] [[!redirects Fourier integrals]] [[!redirects Fourier analysis]] [[!redirects Fourier mode]] [[!redirects Fourier modes]] [[!redirects Fourier expansion]] [[!redirects Fourier expansions]] \end{document}