Taylor Series are Limits of Legendre Expansions - Gvsu - Grand ...

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Bounding the Series Terms The key is to establish the existence of a fixed positive quantity r strictly less than 1 such that for all h sufficiently small, lim n→∞ ( ∣ 2n + 1 2h ∫ t=h t=−h ) f(t)P n (t/h)dt P n (z/h) ∣ 1 n ≤ r < 1. Recall from earlier that lim n→∞ ∣ 2n + 1 2h ∫ t=h t=−h f(t)P n (t/h)dt ∣ 1 n ≤ 1 R h + √ (Rh ) 2 − 1 Taylor Series areLimits of Legendre Expansions – p.12/15
Bounding P n (z/h) To estimate |P n (z/h)|, we use a result of Mauro Picone: M. Picone, Maggiorazione di un polinomio di Legendre e delle derivate in un’ellisse a quello confocale, Bollettino dell’Unione Matematica Italiana (3), 8, 1953, 237-242. Taylor Series areLimits of Legendre Expansions – p.13/15
Page 1 and 2: Taylor Series are Limits of Legendr
Page 3 and 4: A Common Question in Calculus Can w
Page 5 and 6: Taylor’s Theorem If f is analytic
Page 7 and 8: Neumann’s Theorem (1862) If f is
Page 9 and 10: Legendre Expansions on [−h, h] If
Page 11 and 12: The Limit of the Legendre Expansion
Page 13 and 14: The Limit of the Legendre Expansion
Page 15 and 16: The Proof Assume first that we may
Page 17 and 18: The Proof • By Taylor’s Theorem
Page 19 and 20: The Proof • By Taylor’s Theorem
Page 21 and 22: The Proof Combining all these ideas
Page 23: Bounding the Series Terms The key i
Page 27 and 28: Bounding the Series Terms Applying
Page 29: Acknowledgements • Nathanial Burc

Taylor Series are Limits of Legendre Expansions - Gvsu - Grand ...

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