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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.