What does the notation "$\;(f·g)(x)\;$" mean? Suppose I have two functions $f(x)$ and $g(x)$. What does it mean to have $(f·g)(x)$?
This notation can be found in a paper on Partial Differential Equations and Image Processing:

What we call a multiscale analysis is a family of transforms $(T_t)_{t>0}$ of an image $u^o(x)$, such that $u(t,x) = (T_t \cdot u^o)(x)$ can be seen as the image $u^o$ at a coarser scale $t$.

 A: In the most general setting, considering an arbitrary set $A$, a magma $(M, \cdot)$ -- which is a structure obtained by equipping the arbitrary set $M$ with an arbitrary binary operation $\cdot \colon M \times M \to M$ -- and two maps $f, g \colon A \to M$ one can define the product of the two maps as the map given by:
$$\begin{align}
fg \colon A &\to M \\
\left(fg \right)(x)&＝f(x)g(x),
\end{align}$$
where the product mentioned on the right-hand side of the second line refers of course to the binary operation $\cdot$ on $M$.
If one considers the diagonal direct product:
$$\begin{align}
f \underline{\times} g \colon A &\to M \times M \\
\left(f \underline{\times} g\right)(x)&＝\left(f(x), g(x)\right),
\end{align}$$
then we have the relation $fg＝\cdot \circ \left(f \underline{\times} g\right)$ expressing the product with respect to the diagonal direct product and the original underlying multiplication.
P.S. Upon closer inspection it seems that the notation used in the article you are referring to has nothing to do with the algebraic construction described above and is most likely an example of vaguely and sloppily used syntax, a custom which to the disappointment of some has become quite widespread nowadays...
A: When working with functions there are several standard notations that you should have always present:
The $\cdot$ is used to describe the usual product (sometimes called pointwise product):$$f(x)\cdot g(x)=f(x)g(x).$$
You can also compose both functions using $\circ$ in the following way:$$f\circ g = f(x)\circ g(x)=f(g(x)).$$
Finally, the notation used in the paper you mentioned, has to do with the convolution. The symbol $*$ is used to express the convolution of $f$ with $g$ (it is read this way), and it is defined in the following way:$$f*g(x)=(f*g)(x)=\int_{-\infty}^{\infty}f(\xi)g(x-\xi)d\xi.$$ This operation is closely related to the notion of Fourier transform and very popular in the real analysis, signal processing and differential equations fields.
