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

Taylor Series are
Limits of Legendre Expansions
Paul Fishback
Grand Valley State University
Allendale, MI 49401
fishbacp@gvsu.edu
Taylor Series areLimits of Legendre Expansions – p.1/15

A Common Question in Calculus
Can we determine the interval of convergence of a Taylor
series corresponding to a function f simply by looking at f
itself?
Taylor Series areLimits of Legendre Expansions – p.2/15

A Common Question in Calculus
Can we determine the interval of convergence of a Taylor
series corresponding to a function f simply by looking at f
itself?
0.8
0.7
0.6
f(x) = 1
x 2 + 4
y
0.5
0.4
0.3
0.2
0.1
–2 –1 0
1 2
x
Taylor Series areLimits of Legendre Expansions – p.2/15

Taylor’s Theorem
If f is analytic in the open disk of radius R > 0 , then we
may write for z in this disk,
f(z) =
∞∑
n=0
a n z n ,
Taylor Series areLimits of Legendre Expansions – p.3/15

Taylor’s Theorem
If f is analytic in the open disk of radius R > 0 , then we
may write for z in this disk,
f(z) =
∞∑
n=0
a n z n ,
2
where
y
–2 –1 0 1 2
x
–1
–2
1
a n = f(n) (0)
n!
and
1
R = lim
n→∞ |a n| 1 n .
Taylor Series areLimits of Legendre Expansions – p.3/15

Legendre Polynomials
P 0 (x) = 1,
P 1 (x) = x
P 2 (x) = 1 2 (3x2 − 1), P 3 (x) = 1 2 (5x3 − 3x)
In general, P n (x) = 2n − 1 xP n−1 (x) − n − 1
n
n P n−2(x).
Legendre polynomials satisfy the important orthogonality
relation,
∫ x=1
x=−1
P k (x)P j (x)dx =
{
2
2k+1
if k = j;
0 otherwise.
Taylor Series areLimits of Legendre Expansions – p.4/15

Neumann’s Theorem (1862)
If f is analytic in an ellipse containing the closed unit disk
and having foci at (±1, 0), then for z contained in this ellipse,
we may write
f(z) =
∞∑
n=0
a n P n (z), where a n = 2n + 1
2
∫ t=1
t=−1
f(t)P n (t)dt.
Taylor Series areLimits of Legendre Expansions – p.5/15

Neumann’s Theorem (1862)
If f is analytic in an ellipse containing the closed unit disk
and having foci at (±1, 0), then for z contained in this ellipse,
we may write
(x, y) =
“
1
2
f(z) =
“
ρ + 1 ρ
∞∑
n=0
a n P n (z), where a n = 2n + 1
2
”
cos(θ), 1 2
“ ” ”
ρ − 1 sin(θ)
ρ
∫ t=1
t=−1
f(t)P n (t)dt.
If ρ denotes the semiaxes
y
2
1
of this ellipse, then
–2 –1 0
1 2
x
–1
1
ρ = lim
n→∞ |a n| 1 n ≤
1
R + √ R 2 − 1 .
–2
Taylor Series areLimits of Legendre Expansions – p.5/15

Legendre Expansions on [−h, h]
If f is analytic in a disk of radius R > 0 about the origin,
then for h sufficiently small and positive, one may write
(
∞∑
∫ )
2n + 1 t=h
f(z) =
f(t)P n (t/h)dt P n (z/h).
2h
n=0
t=−h
y
(–h,0) (h,0)
x
Moreover,
2n + 1
lim
n→∞ ∣ 2h
≤
∫ t=h
R
h
+
f(t)P n (t/h)dt
∣
1
√ (Rh )
.
2
− 1
t=−h
1
n
Taylor Series areLimits of Legendre Expansions – p.6/15

The Limit of the Legendre Expansion
What happens to the first few terms of
(
∞∑
∫ )
2n + 1 t=h
f(z) =
f(t)P n (t/h)dt
2h
n=0
t=−h
P n (z/h) as h → 0?
Taylor Series areLimits of Legendre Expansions – p.7/15

The Limit of the Legendre Expansion
What happens to the first few terms of
(
∞∑
∫ )
2n + 1 t=h
f(z) =
f(t)P n (t/h)dt
2h
n=0
t=−h
P n (z/h) as h → 0?
• n = 0:
1
2h
∫ t=h
t=−h
f(t)dt · 1 → f(0) as h → 0;
Taylor Series areLimits of Legendre Expansions – p.7/15

The Limit of the Legendre Expansion
What happens to the first few terms of
(
∞∑
∫ )
2n + 1 t=h
f(z) =
f(t)P n (t/h)dt
2h
n=0
t=−h
P n (z/h) as h → 0?
• n = 0:
• n = 1:
(
1
2h
(
3
2h
∫
3 t=h
2h 3 t=−h
∫ t=h
t=−h
∫ t=h
t=−h
tf(t)dt
f(t)dt · 1 → f(0) as h → 0;
f(t)P 1 (t/h)dt
)
· z ...
)
P 1 (z/h) simplifies to
Taylor Series areLimits of Legendre Expansions – p.7/15

The Limit of the Legendre Expansion
What happens to the first few terms of
(
∞∑
∫ )
2n + 1 t=h
f(z) =
f(t)P n (t/h)dt
2h
n=0
t=−h
P n (z/h) as h → 0?
• n = 0:
• n = 1:
(
1
2h
(
3
2h
∫
3 t=h
2h 3 t=−h
∫ t=h
t=−h
∫ t=h
t=−h
tf(t)dt
f(t)dt · 1 → f(0) as h → 0;
f(t)P 1 (t/h)dt
)
· z ...
)
P 1 (z/h) simplifies to
... which → LGD(f)(0)z = f ′ (0)z as h → 0.
Taylor Series areLimits of Legendre Expansions – p.7/15

A Limit Theorem
Theorem: For f analytic in the open disk of radius R > 0 at
the origin and for z interior to this disk,
(
n=∞
∑
∫ )
2n + 1 t=h
n=∞
∑ f (n) (0)
lim
f(t)P n (t/h)dt P n (z/h) = z n .
h→0 + 2h
n!
n=0
t=−h
n=0
In other words, the limiting value of the Legendre series expansion
at z is merely the Taylor series expansion.
Taylor Series areLimits of Legendre Expansions – p.8/15

The Proof
Assume first that we may pass the limit as h → 0 inside the
series. We obtain the desired result by noting that
(
2n + 1
lim
h→0 + 2h
∫ t=h
t=−h
f(t)P n (t/h)dt
( 2n + 1
lim
h→0 + 2
∫ t=1
t=−1
and combining the following ideas.
)
P n (z/h) =
)
f(th)P n (t)dt P n (z/h)
Taylor Series areLimits of Legendre Expansions – p.9/15

The Proof
Taylor Series areLimits of Legendre Expansions – p.10/15

The Proof
• By Taylor’s Theorem,
f(th) = f(0)+f ′ (0)th+...+ f(n) (0)
n!
(th) n +O ( h n+1) as h → 0.
Taylor Series areLimits of Legendre Expansions – p.10/15

The Proof
• By Taylor’s Theorem,
f(th) = f(0)+f ′ (0)th+...+ f(n) (0)
n!
(th) n +O ( h n+1) as h → 0.
• For each 0 ≤ m ≤ n − 1, t m is orthogonal to P n .
Taylor Series areLimits of Legendre Expansions – p.10/15

The Proof
• By Taylor’s Theorem,
f(th) = f(0)+f ′ (0)th+...+ f(n) (0)
n!
(th) n +O ( h n+1) as h → 0.
• For each 0 ≤ m ≤ n − 1, t m is orthogonal to P n .
• 2n+1
2
∫ t=1
t=−1
t n P n (t)dt = 2n (n!) 2
(2n)!
Taylor Series areLimits of Legendre Expansions – p.10/15

The Proof
• By Taylor’s Theorem,
f(th) = f(0)+f ′ (0)th+...+ f(n) (0)
n!
(th) n +O ( h n+1) as h → 0.
• For each 0 ≤ m ≤ n − 1, t m is orthogonal to P n .
• 2n+1
2
∫ t=1
t=−1
t n P n (t)dt = 2n (n!) 2
(2n)!
• P n (z/h) = 1 ( )
(2n)!
h n 2 n (n!) 2 · zn + O(h)
as h → 0.
Taylor Series areLimits of Legendre Expansions – p.10/15

The Proof
Combining all these ideas in just the right way yields
( ∫ )
2n + 1 t=h
lim
f(t)P n (t/h)dt P n (z/h) = f(n) (0)
h→0 + 2h
n!
for each n ≥ 0.
t=−h
Taylor Series areLimits of Legendre Expansions – p.11/15

The Proof
Combining all these ideas in just the right way yields
( ∫ )
2n + 1 t=h
lim
f(t)P n (t/h)dt P n (z/h) = f(n) (0)
h→0 + 2h
n!
for each n ≥ 0.
t=−h
So now the question becomes, how do we justify passing
the limit as h → 0 inside the series in the first place?
Taylor Series areLimits of Legendre Expansions – p.11/15

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.
Taylor Series areLimits of Legendre Expansions – p.12/15

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

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.
Picone’s result allows us to assert that
|P n (z/h)| ≤
(2n − 1)!!
n!
( |z|
h + 1 ) n
.
Taylor Series areLimits of Legendre Expansions – p.13/15

Bounding the Series Terms
Applying Picone’s result, our previous estimate on the
Legendre coefficients, and Stirling’s approximation, we
arrive at
lim
n→∞
(
∣
2n + 1
2h
∫ t=h
t=−h
)
f(t)P n (t/h)dt P n (z/h)
∣
1
n
≤
2|z| + 2h
R + √ R 2 − h 2.
Taylor Series areLimits of Legendre Expansions – p.14/15

Bounding the Series Terms
Applying Picone’s result, our previous estimate on the
Legendre coefficients, and Stirling’s approximation, we
arrive at
lim
n→∞
(
∣
2n + 1
2h
∫ t=h
t=−h
)
f(t)P n (t/h)dt P n (z/h)
∣
1
n
≤
2|z| + 2h
R + √ R 2 − h 2.
Since |z| < R, this latter quantity is strictly less than 1 for all
h sufficiently small, and the proof is complete.
Taylor Series areLimits of Legendre Expansions – p.14/15

Acknowledgements
• Nathanial Burch, GVSU;
• Gisella Licari, GVSU;
• Mario Martelli, Claremont McKenna College;
• Paul Nevai, Ohio State University;
• GVSU Research and Development Center.
Taylor Series areLimits of Legendre Expansions – p.15/15