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

(1)

then the terms of its Fourier-Legendre series

(2)

(3)

where  is a Legendre polynomial, satisfy the inequalities

(4)

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.