I think this will be unexpected, but from the point of view of stereotype theory, Fourier transform is ubiquitous because it is an example of a general categorical construction -- envelope. This is not a vague idea, but a formal construction that describes in the language of category theory different mathematical operations of "taking nearest exterior of a given class" (in contrast to the dual construction of taking the "interior enrichment", which is called refinement). The examples of envelopes are

• the completion of a locally convex space,

• the Stone–Čech compactification of a topological space,

• the Arens-Michael envelope of a topological algebra, etc.

Fourier transform is also an example because in different "big geometries" it turns out to be a special case of the key envelopes used in the construction of these geometries. In particular, at the "branch of this tree" that can be called "topology" we obtain the following result:

for each locally compact abelian group $G$ the Fourier transform ${\mathcal F}:{\mathcal C}^\star(G)\to {\mathcal C}(\widehat{G})$ is a continuous envelope of the stereotype group algebra ${\mathcal C}^\star(G)$ of measures with compact support on $G$.

The same results are true in other big geometries: in differential geometry (with the smooth envelope as the key construction) and in complex geometry (with the Arens-Michael envelope, see details here and here).

I think this will be unexpected, but from the point of view of stereotype theory, Fourier transform is ubiquitous because it is an example of a general categorical construction -- envelope. This is a formal construction that describes in the language of category theory different mathematical operations of "taking nearest exterior of a given class" (in contrast to the dual construction of taking the "interior enrichment", which is called refinement). The examples of envelopes are

• the completion of a locally convex space,

• the Stone–Čech compactification of a topological space,

• the Arens-Michael envelope of a topological algebra, etc.

Fourier transform is also an example because in different "big geometries" it turns out to be a special case of the key envelopes used in the construction of these geometries. In particular, at the "branch of this tree" that can be called "topology" we obtain the following result:

for each locally compact abelian group $G$ the Fourier transform ${\mathcal F}:{\mathcal C}^\star(G)\to {\mathcal C}(\widehat{G})$ is a continuous envelope of the stereotype group algebra ${\mathcal C}^\star(G)$ of measures with compact support on $G$.

The same results are true in other big geometries: in differential geometry (with the smooth envelope as the key construction) and in complex geometry (with the Arens-Michael envelope, see details here and here).

I think this will be unexpected, but from the point of view of stereotype theory, Fourier transform is ubiquitous because it is an example of a general categorical construction -- envelope. This is not a vague idea, but a formal construction that describes in the language of category theory different mathematical operations of "taking nearest exterior of a given class" (in contrast to the dual construction of taking the "interior enrichment", which is called refinement). The examples of envelopes are

• the completion of a locally convex space,

• the Stone–Čech compactification of a topological space,

• the Arens-Michael envelope of a topological algebra, etc.

Fourier transform is also an example because in different "big geometries" it turns out to be a special case of the key envelopes used in the construction of these geometries. As an example, at the "branch of this tree" that can be called "topology" we obtain the following result:

for each locally compact abelian group $G$ the Fourier transform ${\mathcal F}:{\mathcal C}^\star(G)\to {\mathcal C}(\widehat{G})$ is a continuous envelope of the stereotype group algebra ${\mathcal C}^\star(G)$ of measures with compact support on $G$.

The same results are true in other big geometries: in differential geometry (with the smooth envelope as the key construction) and in complex geometry (with the Arens-Michael envelope, see details here and here).

I think this will be unexpected, but from the point of view of stereotype theory, Fourier transform is ubiquitous because it is an example of a general categorical construction -- envelope. This is not a vague idea, but a formal construction that describes in the language of category theory different mathematical operations of "taking nearest exterior of a given class" (in contrast to the dual construction of taking the "interior enrichment", which is called refinement). The examples of envelopes are

• the completion of a locally convex space,

• the Stone–Čech compactification of a topological space,

• the Arens-Michael envelope of a topological algebra, etc.

Fourier transform is also an example because in different "big geometries" it turns out to be a special case of the key envelopes used in the construction of these geometries. In particular, at the "branch of this tree" that can be called "topology" we obtain the following result:

for each locally compact abelian group $G$ the Fourier transform ${\mathcal F}:{\mathcal C}^\star(G)\to {\mathcal C}(\widehat{G})$ is a continuous envelope of the stereotype group algebra ${\mathcal C}^\star(G)$ of measures with compact support on $G$.

The same results are true in other big geometries: in differential geometry (with the smooth envelope as the key construction) and in complex geometry (with the Arens-Michael envelope, see details here and here).