Math 118 :: Winter 2009 :: Homework 4 :: Due March 5
1. (Analysis of the potential for Chebyshev interpolation.) Show that the complex function
√
|z − z 2 − 1|
φ(z) = log
2
takes on the constant value − log 2 on [−1, 1].
2. (Exact interpolation)
(a) Describe the set of functions f (θ) of θ ∈ [0, 2π] which are interpolated without error,
by the method of bandlimited interpolation, from the knowledge of f (θj ) at equispaced
nodes θj = jh, h = 2π/N , j = 1, . . . , N .
(b) Describe the set of functions f (x) of x ∈ [−1, 1] which are interpolated without error, by
the method of Chebyshev interpolation, from the knowledge of f (xj ) at the Chebyshev
nodes xj = cos(jπ/N ), j = 0, . . . , N .
[Hint: there may be different ways of answering these questions. One way to properly justify
your answers is via a Poisson summation formula.]
Note that any function interpolated without error will also be differentiated or integrated
without error! (in exact arithmetic.)
3. (Going from an estimate in k to an estimate in x)
(a) Assume that two functions û, v̂ of k obey
|û(k) − v̂(k)| ≤ ,
for all k ∈ [−π/h, π/h].
Let uj , vj be the inverse semidiscrete Fourier transforms of û(k), v̂(k) respectively. Show
that
|uj − vj | ≤ C ,
for all j ∈ Z,
h
for some constant C that you should determine.
(b) Prove part (a) of Theorem 4 on p.34 of the textbook. [Hint: use Theorem 3 on p.33,
part (a) of this exercise, and recall that differentiation is done by multiplication by ik
in the Fourier domain.]
4. Formulate an estimate of accuracy of the “centered rectangles” rule for integration on the
interval [0, 1]. The quadrature formula for this rule is
h
N
−1
X
f
j=0
1
j+
2
h .
Assume that f has 3 bounded derivatives.
[Hint: expand f (x) in Taylor series around xj = (j + 12 )h, and compare the integral of f over
[jh, (j + 1)h] to the value hf (xj )].
(see backside)
5. Consider the periodic function
f (θ) =
1 if 0 ≤ θ < 1;
0 if 1 ≤ θ < 2π.
(a) How accurate is the N -point bandlimited quadrature rule for
the theorem seen in class.]
R 2π
0
f (θ)dθ? [Hint: invoke
(b) Since bandlimited quadrature is nothing but the trapezoidal rule, confirm the approximation rate obtained in (a) by a geometric argument of comparison of areas.
(c) How would your answer to (b) be modified if the discontinuity point is at θ = π instead,
and the value of the function at that isolated point is taken to be f (π) = 1/2?
6. Consider a function f in the space of bandlimited functions, which have a Fourier transform
compactly supported in the interval [−B, B]. Find a differentiable function g(x) which obeys
the decay law
1
|g(x)| ≤ C
,
1 + x2
for some constant C > 0, and such that f does not change when convolved with g:
Z
f (y)g(x − y) dy = f (x).
R
Justify your answer.
[Hint: it’s enough to specify the Fourier transform of g.]
7. Bonus question (100 points, only if you have time and some affinity with real analysis). In the
setting of question 6, show that it’s impossible to find a function g which decays exponentially,
and such that f = f ? g.