Structure of the functional space $\int_ {- \infty} ^ \infty f (x) dx = 1 $ Please, help me with studying of useful practical features of the following functional space:
$$\int_{-\infty}^\infty f(x) \, dx = 1$$
For example:
1) What basis types are most convenient for representation an element from the space?
2) How to find an element of best approximation for the given data set $\{(x_i, y_i) | i = \overline{1,n} \}$?

The integral condition says very little about $f$; it selects an affine subspace of codimension $1$, which is not any more manageable than the space you began with. For any  $f\in L^1(\mathbb R)$, the function $$f-c\chi_{[0,1]},\quad \text{where } c = \int_{-\infty}^\infty f -1 $$ satisfies your condition. So, if you have a convenient representation here, you have it for all $L^1(\mathbb R)$. 
The Haar system is a convenient basis for  the linear space $\{f\in L^1(\mathbb R):\int_{-\infty}^\infty f=0\}$. Therefore, every function in your space can be written as 
$$f = \chi_{[0,1]}+ \sum_{n,k}c_{n,k} \psi_{n,k}$$ 
with $\psi_{n,k}$ being Haar functions. You can find $c_{n,k}$ using the fact that the Haar system is orthogonal in $L^2$ (even though your function need not be in $L^2$): $c_{n,k} = \int_{-\infty}^\infty f\psi_{n,k} $.
