I read this somewhere:
$U$ and $V$ are defined on the set $\mathbb{Z}$ of integers. The convolution of $u$ and $v$, noted as $(u*v) (k)\,\theta$ or $ (u \otimes v) (k)\theta$, is a new sequence whose general term is given by $(u*v) (k) = (u \otimes v) (k) = \sum_{i=-\infty}^{\infty} u(i)v(k-i)\theta$
Can I take $\{-1, -2, -3, \ldots\}$ for $U$ and $\{1, 2, 3, \ldots\}$ for $V$?
How to interpret this: $(u*v) (k)\,\theta$
Why do we insert this part $ (k) \,\theta$ in the formula?
Why does $i$ go from $-\infty$ to $\infty$?
Could someone give me an example of this formula?
The $k$ in the formula indicates that you are talking about the $k$th term in the sequence $u\ast v$.
I do not know for sure what the $\theta$ adds anything, unless that is part of the author's notation for making a modification of these sequences.
If summed from $-\infty$ to $\infty$, that sum is often going to be divergent: is it possible the $\theta$ is some restriction that zeros out all but finitely many terms of the series? Please consult your source.
As for "give me an example":
Take $u(x)$ and $v(x)$ to be integer polynomials, and then interpret them as sequences in the obvious way: i.e. you put the $i$th term to be the coefficient of $x^i$. Then you'll find that $u\ast v$ is the sequence representing the coefficients of $u(x)v(x)$!
Example: $u(x)=x-2$, $v(x)=3x^2+x$. The sequences are: $$ \dots u_{-2}=u_{-1}=0;\ \ u_0=-2;\ \ u_1=1; \ \ u_2=u_3=\dots=0 $$ $$ \dots v_{-2}=v_{-1}=v_0=0;\ \ v_1=1;\ \ v_2=3;\ \ v_3=v_4=\dots=0 $$
The convoluted sequence is:
$$ \dots (v\ast u)_{-2}=(v\ast u)_{-1}=(v\ast u)_0=0;\ \ (v\ast u)_1=-2;\ \ (v\ast u)_2=-5;\ \ (v\ast u)_3=3;\ \ (v\ast u)_4=(v\ast u)_5=\dots=0 $$
And that turns out to be exactly the sequence for $u(x)v(x)=3x^3-5x^2-2x$.
You can use this to show that the sequences of finite support contained in the nonnegative integers, equipped with the $\ast$ operation are isomorphic to $\Bbb{Z}[x]$. If you consider all sequences of finite support, then that is isomorphic to the "Laurent polynomials" $\Bbb Z[x,x^{-1}]$.