Lauricella's theorem

{{short description|Orthogonal functions theorem}}

In the theory of orthogonal functions, Lauricella's theorem provides a condition for checking the closure of a set of orthogonal functions, namely:

Theorem – A necessary and sufficient condition that a normal orthogonal set $\backslash \{u\_k\backslash \}$ be closed is that the formal series for each function of a known closed normal orthogonal set $\backslash \{v\_k\backslash \}$ in terms of $\backslash \{u\_k\backslash \}$ converge in the mean to that function.

The theorem was proved by Giuseppe Lauricella in 1912.