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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 |f(x)(1-x^2)| 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^xf(x)dx,
(1)

then the terms of its Fourier-Legendre series

F(x)∼sum_(n=0)^inftya_nP_n(x)
(2)
a_n=1/2(2n+1)int_(-1)^1F(x)P_n(x)dx,
(3)

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

|a_nP_n(x)|<{8sqrt(2/pi)(M^'+V^')/((1-delta^2)^(1/4))n^(-3/2) for |x|<=delta<1; 2(M^'+V^')n^(-1) for |x|<=1
(4)

for n>=1 (Sansone 1991).

See also

Fourier-Legendre Series, Jackson's Theorem

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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.

Referenced on Wolfram|Alpha

Picone's Theorem

Cite this as:

Weisstein, Eric W. "Picone's Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PiconesTheorem.html

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