MAT 330: Fourier Series and Partial Diﬀerential Equations
Professor Sergiu Klainerman
Problem Set 5
Rik Sengupta
rsengupt@princeton.edu
March 30, 2010
1.
Stein and Shakarchi, p. 162, problem 4
Bump functions.
Examples of compactly supported functions in
S
(
R
) are very handy in
many applications in analysis. Some examples are:
(a) Suppose
a < b
, and
f
is the function such that
f
(
x
) = 0 if
x
≤
a
or
x
≥
b
and
f
(
x
) =
e

1
/
(
x

a
)
e

1
/
(
b

x
)
if
a < x < b.
Show that
f
is indeﬁnitely diﬀerentiable on
R
.
(b) Prove that there exists an indeﬁnitely diﬀerentiable function
F
on
R
such that
F
(
x
) = 0
if
x
≤
a
,
F
(
x
) = 1 is
x
≥
b
, and
F
is strictly increasing on [
a,b
].
(c) Let
δ >
0 be so small that
a
+
δ < b

δ
. Show that there exists and indeﬁnitely
diﬀerentiable function
g
such that
g
is 0 if
x
≤
a
or
x
≥
b
,
g
is 1 on [
a
+
δ,b

δ
], and
g
is strictly monotonic on [
a,a
+
δ
] and [
b

δ,b
].
Solution.
We will solve the parts one by one.
(a) First of all, note that
f
is clearly inﬁnitely diﬀerentiable at all points other than
a
and
b
,
from the fact that the product of two inﬁnitely diﬀerentiable functions is also inﬁnitely
diﬀerentiable (since
d
(
uv
) =
udv
+
vdu
). We will only consider the point
x
=
a
, because
for
x
=
b
the argument is completely symmetric and therefore identical (a fact that is
evident from the expression for
f
).
Clearly, the term
e

1
/
(
b

x
)
will not give us any problems, because for all small enough
neighborhoods we will not face diﬀerentiability issues. So consider the term
e

1
/
(
x

a
)
.
The following proof is presented in its generality, though we would have been justiﬁed in
considering
a
= 0 to simplify the proof, because of translation invariance. The derivative
of
e

1
/
(
x

a
)
is
e

1
/
(
x

a
)
(
x

a
)
2
, and also, approaching this point from the left will always give
us 0 from the deﬁnition, and so we need only consider the case of approaching from the
right. Therefore, the problem reduces to proving
lim
x
→
a
+
e

1
/
(
x

a
)
(
x

a
)
2
= 0
.
We will use a small but neat trick to prove this fact. Since
1
3!