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

$$\vert a_nP_n(x)\vert<\cases{ 8\sqrt{2\over\pi}{M'+V'\over(1-... ...\leq\delta<1$\cr 2(M'+V')n^{-1} & for $\vert x\vert\leq 1$\cr}$$

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

See also Jackson's Theorem, Legendre Series

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.