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Picone's Theorem

Let $f(x)$ be integrable in $[-1,1]$, let $(1-x^2)f(x)$ be of bounded variation in $[-1,1]$, let $M'$ denote the least upper bound of $\vert f(x)(1-x^2)\vert$ in $[-1,1]$, and let $V'$ denote the total variation of $f(x)(1-x^2)$ in $[-1,1]$. Given the function

F(x)=F(-1)+\int_1^x f(x)\,dx,

then the terms of its Legendre Series

F(x)\sim \sum_{n=0}^\infty a_nP_n(x)

a_n={\textstyle{1\over 2}}(2n+1)\int_{-1}^1 F(x)P_n(x)\,dx,

where $P_n(x)$ is a Legendre Polynomial, satisfy the inequalities

\vert a_nP_n(x)\vert<\cases{
2(M'+V')n^{-1} & for $\vert x\vert\leq 1$\cr}

for $n\geq 1$ (Sansone 1991).

See also Jackson's Theorem, Legendre Series


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.

© 1996-9 Eric W. Weisstein