MAT128A: Numerical Analysis, Section 2

MAT128A: Numerical Analysis, Section 2

1. Show that when the n-point periodic trapezoidal rule is used to evaluate the integral
Z π
−π
exp(ikt) dt,
the result is



(−1)|k| 2π if k = m · n for some nonzero integer m
2π if k = 0
0 otherwise.
2. Suppose that f is a continuously differentiable 2π-periodic function. Show that if f is even —
meaning that f(−x) = f(x) for all 0 < x ≤ π — then f can be represented via a convergent series
of the form
f(t) = X∞
n=0
bn cos(nt).
3. Suppose that f is a continuously differentiable 2π-periodic function. Show that if f is odd —
meaning that f(−x) = −f(x) for all 0 < x ≤ π — then f can be represented via a convergent
series of the form
f(t) = X∞
n=1
cn sin(nt).
4. Suppose that
f(t) = cos(2t) + cos(4t) + cos(6t) + . . . + cos(20t).
What is the exact value of
Z π
−π
f(t) dt? (1)
How long is the periodic trapezoidal rule of minimum length which evaluates (1) exactly? That is,
what is the least positive integer n such that
Z π
−π
f(t) dt =
2π
n
nX−1
j=0
f
ˆ
−π +
2π
n
j
˙
? (2)
Here, we are assuming that exact arithmetic is used to perform the calculations so that we need
not worry about roundoff error.
5. Let
f(t) = X∞
n=1
an exp(int)
1
with
|an| ≤
1
n2
.
Show that the error incurred when the periodic trapezoidal rule of length N is used to evaluate the
integral
1
2π
Z π
−π
f(t) dt
is bounded above by
π
2
6
1
N2
.
Again, we are assuming that exact arithmetic is used to perform the calculations so that we needn’t
worry about roundoff error.
(Hint: look at the solutions from the previous homework assignment to find the value of P∞
n=1
1
n2 ).
6. Let
f(t) = X∞
n=0
an exp(int)
with
|an| ≤
1
2
n
.
Show that the error incurred when the periodic trapezoidal rule of length N is used to evaluate the
integral
1
2π
Z π
−π
f(t) dt
is bounded above by
1
2N − 1
.
Once again, we are assuming that exact arithmetic is used to perform the calculations so that we
needn’t worry about roundoff error.
7. Find the Fourier series for the function
f(t) = 2
exp(it) − 2
by using the identity
1
1 − z
=
X∞
j=0
z
j
,
which holds for all complex-valued z such that |z|< 1.
2