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\begin{document}

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\section*{integral transform}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{linear_algebra}{}\paragraph*{{Linear algebra}}\label{linear_algebra}

[[!include homotopy - contents]]

\hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis}

[[!include functional analysis - contents]]

\hypertarget{higher_category_theory}{}\paragraph*{{Higher category theory}}\label{higher_category_theory}

[[!include higher category theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{History}{History}\dotfill \pageref*{History} \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}

An \emph{integral transform} on [[functions]] is a [[linear map]] between functions on [[spaces]] $X$, $Y$ encoded by a function $K$ (or [[generalized function]], e.g. [[distribution of two variables]]) on the [[product]] space $X \times Y$ and given by a formula of the type

\begin{displaymath}
(function\;f\;on\;X) \mapsto
  \left(y \mapsto
  \int_{X} f(x) K(x,y) \right)
  \,,
\end{displaymath}
where on the right we have some kind of [[integration]] over $X$.

Here $K$ is called the integral \textbf{kernel} of the transformation.

Typically the definition of an integral transform on functions involves some delicate technical issues concerning the precise nature of the function space, the [[measure]] with respect to which the integral is defined, etc.

On the other hand, one may understand the general form of an integral transform as the [[decategorification]] of a very natural general abstract construction in [[higher category theory]]: that of [[integral transforms on sheaves]] given by spans of [[base change geometric morphism]]s.

Special cases of such [[categorification|categorified]] integral transforms are discussed at

\begin{itemize}%
\item [[Schwartz kernel]] ([[distribution of two variables]])


\item [[Fourier-Mukai transform]], [[topological T-duality]]


\item [[groupoidification]]


\item [[geometric infinity-function theory]].



\end{itemize}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item The simplest example is [[matrix multiplication]], which corresponds to the case where $X$ and $Y$ are [[discrete space]]s.


\item Standard examples involving genuine [[functional analysis]] are for instance the [[Fourier transform]] and related [[Laplace transform]], [[Mellin transform]], [[Hankel transform]], [[Hilbert transform]].


\item Also the [[path integral]] in [[quantum mechanics]] and [[quantum field theory]] is supposed to be a class of examples of a (secondary) integral kernel.


\item [[Fourier-Mukai transform]]


\item [[Penrose transform]]


\item [[Hecke transform]]


\item [[Harish Chandra transform]]



\end{itemize}
\hypertarget{History}{}\subsection*{{History}}\label{History}

In [[noncommutative algebraic geometry]], one of the most important results is [[Dmitri Orlov]]`s representability theorem (1997) which states that every fully faithful [[triangulated functor]] between the [[derived categories]] of [[coherent sheaves]] on two [[smooth scheme|smooth]] [[projective varieties]] is representable by some ``[[integral kernel]]'', i.e., a coherent complex on the [[product]]. One would like to remove the ``fully faithful'' assumption, but this has proved extremely difficult so far. In the context of [[dg-categories]] the analogous result, discovered by [[Bertrand Toen]] in 2004, does have the clean form one would like it to. Along with other problems with [[triangulated categories]], this has been one of the motivations for people like [[Maxim Kontsevich]] and [[Goncalo Tabuada]] to start doing [[noncommutative geometry]] with [[pretriangulated dg-categories]] instead of triangulated categories.

On the other hand [[pretriangulated dg-categories]] are known to provide a model for linear [[stable (infinity,1)-categories]]. Using a different model, like [[quasi-categories]], would be more convenient for extending Toen's theorem from (smooth proper) schemes to (smooth proper) [[derived stacks]]. This was done in \hyperlink{BZFN08}{Ben-Zvi \& Francis \& Nadler 08}. In the followup (\hyperlink{BZNP13}{Ben-Zvi \& Nadler \& Preygel 13}) the authors have also extended these results to the non-smooth case.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[polynomial functor]]


\item [[Schwartz kernel]]



\end{itemize}
\hypertarget{References}{}\subsection*{{References}}\label{References}

Discussion of integral kernels in the sense of [[functional analysis]] (as n the [[Schwartz kernel theorem]]) is in

\begin{itemize}%
\item [[François Trèves]], \emph{Topological Vector Spaces, Distributions and Kernels} (Academic Press, New York, 1967)

\end{itemize}
Discussion of integral transforms in [[derived algebraic geometry]] (see also at \emph{[[geometric infinity-function theory]]}) is in

\begin{itemize}%
\item [[David Ben-Zvi]], [[John Francis]], [[David Nadler]], \emph{Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry} (\href{http://arxiv.org/abs/0805.0157}{arXiv:0805.0157})


\item [[David Ben-Zvi]], [[David Nadler]], Anatoly Preygel, \emph{Integral transforms for coherent sheaves} (\href{http://arxiv.org/abs/1312.7164}{arXiv:1312.7164})



\end{itemize}
Comments on the formalization of integral transforms and [[quantization]] in [[dependent linear type theory]] are at

\begin{itemize}%
\item \emph{[[schreiber:Type-semantics for quantization]]}

\end{itemize}
category: analysis, geometry [[!redirects integral transforms]] [[!redirects integral kernel]] [[!redirects integral kernels]]



\end{document}