## Picone's Theorem

Let be integrable in , let be of bounded variation in , let denote the least upper bound of in , and let denote the total variation of in . Given the function

then the terms of its Legendre Series

where is a Legendre Polynomial, satisfy the inequalities

for (Sansone 1991).

References

Picone, M. Appunti di Analise Superiore. Naples, Italy,, p. 260, 1940.

Sansone, G. Orthogonal Functions, rev. English ed. New York: Dover, pp. 203-205, 1991.