Lauricella's theorem explained

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

\{u _{k\}} be closed is that the formal series for each function of a known closed normal orthogonal set \{v _{k\}} in terms of \{u _{k\}} converge in the mean to that function.

The theorem was proved by Giuseppe Lauricella in 1912.

References

• G. Lauricella: Sulla chiusura dei sistemi di funzioni ortogonali, Rendiconti dei Lincei, Series 5, Vol. 21 (1912), pp. 675 - 85.