Description
Problem 1. Suppose that f : r´1, 1s Ñ R is a continuous function, and that tanu are its
Chebyshev coefficients. That is, the sequence tanu is defined by the formula
an “
2
π
ż 1
´1
fpxqTnpxq
dx
?
1 ´ x
2
.
Moreover, suppose that for each positive integer N, pN denotes the polynomial
pN pxq “ ÿ
N
n“0
1
anTnpxq.
Indicate whether each of the following statements is true or false. You do not need to justify
your answers.
1. If f is continously differentiable, then }pN ´ f}8 Ñ 0 as N Ñ 8.
2. If f is k-times continuously differentiable, then
|an| “ O
ˆ
1
nk
˙
.
3. If f is infinitely differentiable, then there exists an r ą 0 such that |an| “ expp´rnq.
4. If |an| ď 2
´n
for all nonnegative integers n, then
|fpxq ´ PN pxq| ď 2
´N
for all positive integers N.
5. For each positive integer N, pN is the unique polynomial of degree N which interpolates
f at the points
cos ˆ
j `
1
2
N ` 1
π
˙
, j “ 0, 1, . . . , N.
Problem 2. Compute
ż π
´π
fptq dt,
where f : r´π, πs Ñ C is the function defined by the Fourier series
fptq “ ÿ8
n“0
2
n`1
exppintq.
Problem 3. Let κf pxq denote the condition number of evaluation of the function f at the point
x. Find a function f which is infinitely differentiable on the interval p0, 1q (but which may have
singularities at x “ 0) such that
lim
xÑ0`
κf pxq “ 8.
Problem 4. Suppose that f : r0, 1s Ñ R is a twice differentiable function such that
|f
2
pxq| ď 1 for all 0 ď x ď 1.
Let p be the unique polynomial of degree 1 which interpolates f at the points 0 and 1. Show
that
|ppxq ´ fpxq| ď 1
8
for all x P r0, 1s.
Problem 5. Suppose that f : r´1, 1s Ñ R is an infinitely differentiable function, and that tanu
is the sequence of Chebyshev coefficients of f — that is, tanu is defined via the formula
an “
2
π
ż π
0
fpxqTnpxq
dx
?
1 ´ x
2
.
Suppose also that N is a positive integer, and that p is the polynomial
ppxq “ ÿ
N
n“0
1
anTnpxq.
Show that if q is any polynomial of degree N then
|aN`1| ď 4
π
sup
´1ďxď1
|fpxq ´ qpxq|
for all x P r´1, 1s.
Hint: you may use the fact that
ż 1
´1
|Tn`1pxq| dx
?
1 ´ x
2
“ 2.
for all n ě 0 without proving it.
