Suppose you wish to fit a function of the form f ( t ) = c + p sin ( t ) + q cos ( t ) to a given continuous function g ( t ) on the closed interval from 0 to 2 π . One approach is to choose n equally spaced points a between 0 and 2 π [ a i = i ⋅ ( 2 π / n ) , for i = 1 , ... , n , say ] . We can fit a function f n ( t ) = c n + p n sin ( t ) + q n cos ( t ) to the data points ( a i , g ( a i ) ) , for i = 1 , ... , n . Now examine what happens to the coefficients c n , p n , q n of f n ( t ) as n approaches infinity. To find f n ( t ) , we make an attempt to solve the equations f n ( a i ) = g ( a i ) , for i = 1 , ... , n , or | c n + p n sin ( a 1 ) + q n cos ( a 1 ) = g ( a 1 ) c n + p n sin ( a 2 ) + q n cos ( a 2 ) = g ( a 2 ) ⋮ ⋮ c n + p n sin ( a n ) + q n cos ( a n ) = g ( a , ) | , or A n [ c n p n q n ] = b → n , where A n = [ 1 sin ( a 1 ) cos ( a 1 ) 1 sin ( a 2 ) cos ( a 2 ) ⋮ ⋮ ⋮ 1 sin ( a 2 ) cos ( a n ) ] , b → n = [ g ( a 1 ) g ( a 2 ) ⋮ g ( a n ) ] a. Find the entries of the matrix A n T A n and the components of the vector A n T b → n . b. Find lim n → ∞ ( 2 π n A n T A n ) and lim n → ∞ ( 2 π n A n T b → ) . Hint: Interpret the entries of the matrix ( 2 π / n ) A n T A n and thecomponents of the vector ( 2 π / n ) A T b → as Riemann sums. Then the limits are the corresponding Riemann integrals. Evaluate as many integrals as you can. Note that lim n → ∞ ( 2 π n A n T A n ) is a diagonal matrix c. Find lim n → ∞ [ c n p n q n ] = lim n → ∞ ( A n T A n ) − 1 A n T b → n = lim n → ∞ [ ( 2 π n A n T A n ) − 1 ( 2 π n A n T b → n ) ] = [ lim n → ∞ ( 2 π n A n T A n ) ] − 1 lim n → ∞ ( 2 π n A n T b → n ) The resulting vector [ c p q ] gives you the coefficient of the desired function f ( t ) = lim n → ∞ f n ( t ) . Write f ( t ) . The function f ( t ) is called the first Fourier approximation of g ( t ) . The Fourier approximation satisfies a “continuous” least-squares condition, an idea we will make more precise in the next section.
Suppose you wish to fit a function of the form f ( t ) = c + p sin ( t ) + q cos ( t ) to a given continuous function g ( t ) on the closed interval from 0 to 2 π . One approach is to choose n equally spaced points a between 0 and 2 π [ a i = i ⋅ ( 2 π / n ) , for i = 1 , ... , n , say ] . We can fit a function f n ( t ) = c n + p n sin ( t ) + q n cos ( t ) to the data points ( a i , g ( a i ) ) , for i = 1 , ... , n . Now examine what happens to the coefficients c n , p n , q n of f n ( t ) as n approaches infinity. To find f n ( t ) , we make an attempt to solve the equations f n ( a i ) = g ( a i ) , for i = 1 , ... , n , or | c n + p n sin ( a 1 ) + q n cos ( a 1 ) = g ( a 1 ) c n + p n sin ( a 2 ) + q n cos ( a 2 ) = g ( a 2 ) ⋮ ⋮ c n + p n sin ( a n ) + q n cos ( a n ) = g ( a , ) | , or A n [ c n p n q n ] = b → n , where A n = [ 1 sin ( a 1 ) cos ( a 1 ) 1 sin ( a 2 ) cos ( a 2 ) ⋮ ⋮ ⋮ 1 sin ( a 2 ) cos ( a n ) ] , b → n = [ g ( a 1 ) g ( a 2 ) ⋮ g ( a n ) ] a. Find the entries of the matrix A n T A n and the components of the vector A n T b → n . b. Find lim n → ∞ ( 2 π n A n T A n ) and lim n → ∞ ( 2 π n A n T b → ) . Hint: Interpret the entries of the matrix ( 2 π / n ) A n T A n and thecomponents of the vector ( 2 π / n ) A T b → as Riemann sums. Then the limits are the corresponding Riemann integrals. Evaluate as many integrals as you can. Note that lim n → ∞ ( 2 π n A n T A n ) is a diagonal matrix c. Find lim n → ∞ [ c n p n q n ] = lim n → ∞ ( A n T A n ) − 1 A n T b → n = lim n → ∞ [ ( 2 π n A n T A n ) − 1 ( 2 π n A n T b → n ) ] = [ lim n → ∞ ( 2 π n A n T A n ) ] − 1 lim n → ∞ ( 2 π n A n T b → n ) The resulting vector [ c p q ] gives you the coefficient of the desired function f ( t ) = lim n → ∞ f n ( t ) . Write f ( t ) . The function f ( t ) is called the first Fourier approximation of g ( t ) . The Fourier approximation satisfies a “continuous” least-squares condition, an idea we will make more precise in the next section.
Solution Summary: The author enumerates the enteries of matrix A_nTStackrelto and its components.
Suppose you wish to fit a function of the form
f
(
t
)
=
c
+
p
sin
(
t
)
+
q
cos
(
t
)
to a given continuous function
g
(
t
)
on the closed interval from 0 to
2
π
. One approach is to choose n equally spaced points a between 0 and
2
π
[
a
i
=
i
⋅
(
2
π
/
n
)
,
for
i
=
1
,
...
,
n
,
say
]
. We can fit a function
f
n
(
t
)
=
c
n
+
p
n
sin
(
t
)
+
q
n
cos
(
t
)
to the data points
(
a
i
,
g
(
a
i
)
)
, for
i
=
1
,
...
,
n
. Now examine what happens to the coefficients
c
n
,
p
n
,
q
n
of
f
n
(
t
)
as n approaches infinity.
To find
f
n
(
t
)
, we make an attempt to solve the equations
f
n
(
a
i
)
=
g
(
a
i
)
,
for
i
=
1
,
...
,
n
,
or
|
c
n
+
p
n
sin
(
a
1
)
+
q
n
cos
(
a
1
)
=
g
(
a
1
)
c
n
+
p
n
sin
(
a
2
)
+
q
n
cos
(
a
2
)
=
g
(
a
2
)
⋮
⋮
c
n
+
p
n
sin
(
a
n
)
+
q
n
cos
(
a
n
)
=
g
(
a
,
)
|
, or
A
n
[
c
n
p
n
q
n
]
=
b
→
n
, where
A
n
=
[
1
sin
(
a
1
)
cos
(
a
1
)
1
sin
(
a
2
)
cos
(
a
2
)
⋮
⋮
⋮
1
sin
(
a
2
)
cos
(
a
n
)
]
,
b
→
n
=
[
g
(
a
1
)
g
(
a
2
)
⋮
g
(
a
n
)
]
a. Find the entries of the matrix
A
n
T
A
n
and the components of the vector
A
n
T
b
→
n
. b. Find
lim
n
→
∞
(
2
π
n
A
n
T
A
n
)
and
lim
n
→
∞
(
2
π
n
A
n
T
b
→
)
. Hint: Interpret the entries of the matrix
(
2
π
/
n
)
A
n
T
A
n
and thecomponents of the vector
(
2
π
/
n
)
A
T
b
→
as Riemann sums. Then the limits are the corresponding Riemann integrals. Evaluate as many integrals as you can. Note that
lim
n
→
∞
(
2
π
n
A
n
T
A
n
)
is a diagonal matrix c. Find
lim
n
→
∞
[
c
n
p
n
q
n
]
=
lim
n
→
∞
(
A
n
T
A
n
)
−
1
A
n
T
b
→
n
=
lim
n
→
∞
[
(
2
π
n
A
n
T
A
n
)
−
1
(
2
π
n
A
n
T
b
→
n
)
]
=
[
lim
n
→
∞
(
2
π
n
A
n
T
A
n
)
]
−
1
lim
n
→
∞
(
2
π
n
A
n
T
b
→
n
)
The resulting vector
[
c
p
q
]
gives you the coefficient of the desired function
f
(
t
)
=
lim
n
→
∞
f
n
(
t
)
. Write
f
(
t
)
. The function
f
(
t
)
is called the first Fourier approximation of
g
(
t
)
. The Fourier approximation satisfies a “continuous” least-squares condition, an idea we will make more precise in the next section.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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