Picone,s Theorem
المؤلف:
Picone, M
المصدر:
Appunti di Analise Superiore. Naples, Italy
الجزء والصفحة:
...
17-11-2021
1698
Picone's Theorem
Let 
be integrable in ![[-1,1]](https://mathworld.wolfram.com/images/equations/PiconesTheorem/Inline2.gif)
, let 
be of bounded variation in ![[-1,1]](https://mathworld.wolfram.com/images/equations/PiconesTheorem/Inline4.gif)
, let 
denote the least upper bound of 
in ![[-1,1]](https://mathworld.wolfram.com/images/equations/PiconesTheorem/Inline7.gif)
, and let 
denote the total variation of 
in ![[-1,1]](https://mathworld.wolfram.com/images/equations/PiconesTheorem/Inline10.gif)
. 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.
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