TITLE: Laplace, Legendre, Fourier, Hankel, Mellin, Hilbert, Borel, Z...: unified treatment of transforms?
QUESTION [70 upvotes]: I understand "transform methods" as recipes, but beyond this they are a big mystery to me.  
There are two aspects of them I find bewildering.
One is the sheer number of them.  Is there a unified framework that includes all these transforms as special cases?
The second one is heuristic: what would lead anyone to discover such a transform in the course of solving a problem?
(My hope is to find a unified treatment of the subject that simultaneously addresses both of these questions.)

REPLY [3 votes]: There is actually a unified treatment of transform theory. This is what chapter seven of 
Keener's book is about.
There is a relationship between Linear differential equations, spectrum,
Green functions, contour integration,  Dirac Deltas, and Transforms.
The story goes like this. Given a linear operator (differential) $L$,
form the operator $L - \lambda = 0$, with given boundary conditions,
find its Green function $G(x, \xi, \lambda)$.  Then
\begin{equation}
\delta(x - \xi) = -\frac{1}{2 \pi \mathrm{i}} \int_{C_{\infty}}
G(x, \xi , \lambda) \, d \lambda,
\end{equation}
where the contour $C_{\infty}$ is a circle having all the spectrum of $L$ inside. 
This particular Dirac delta representation is the product of the direct and inverse transforms.
Here are a few examples:

Sine transform pair:
Functions continuously differentiable in $[0,1]$.
\begin{equation}
   Lu = - u'' \quad , \quad u(0)=u(1)=0.
  \end{equation}
result:

\begin{eqnarray*}
  U_k &=& 2 \int_0^1 d \xi \sin( k \pi \xi) u( \xi ) \\
  u(x) &=& \sum_{k=1}^{\infty} U_k \sin k \pi x.
\end{eqnarray*}

Cosine transform pair:
\begin{equation}
   Lu = - u'' \quad , \quad u'(0)=u'(1)=0.
  \end{equation}
result:

\begin{eqnarray*}
  U_k &=& 2 \int_0^1 d \xi \cos( k \pi \xi) u( \xi ) \\
  u(x) &=& \sum_{k=1}^{\infty} U_k \cos k \pi x.
\end{eqnarray*}

Sine transform integral
Functions in $L^2[0, \infty)$.
\begin{equation}
   Lu = - u'' \quad , \quad u(0)=0 \quad , \quad \lim_{x \to \infty} u(x) = 0.
  \end{equation}
result:

\begin{eqnarray*}
  U(\mu) &=& \frac{2}{\pi} \int_0^{\infty}  dx u(x) \sin \mu x \\
  u(\xi) &=& \ \int_0^{\infty}  d \mu U(\mu) \sin \mu x \\
\end{eqnarray*}

cosine transform integral
\begin{equation}
   Lu = - u'' \quad , \quad u'(0)=0 \quad , \quad  \lim_{x \to \infty} u(x) = 0.
  \end{equation}
result:

\begin{eqnarray*}
  U(\mu) &=& \frac{2}{\pi} \int_0^{\infty}  dx u(x) \cos \mu x \\
  u(\xi) &=& \ \int_0^{\infty}  d \mu U(\mu) \cos \mu x \\
\end{eqnarray*}

Fourier transform
Functions in $L^2(-\infty, \infty)$.
\begin{equation}
   Lu = - u'' \quad ,  \quad , \quad  \lim_{x \to \pm \infty} u(x) = 0.
  \end{equation}
result:

\begin{eqnarray*}
  U(\mu) &=& \int_{-\infty}^{\infty}  dx \; u(x) \; \mathrm{e}^{\mathrm{i} \mu \xi} \\
  u(\xi) &=& \frac{1}{2 \pi} \int_{-\infty}^{\infty} U(\mu) \mathrm{e}^{-\mathrm{i} \mu \xi}
  d \mu.
\end{eqnarray*}
The list follows: Mellin, Hankel, etc. Keener shows the linear operators with boundary conditions for these.
I do not know which operator and boundary conditions generate the Laplace
transform. I opened this question in StackExchange Mathematics for this particular problem.
I am writing some notes about this matter 
here.
UPDATE:
I solved the problem of connecting the Laplace transform to an ODE with boundary conditions. Please see here
Thanks.