Problems on Distributions
1. Compute the distributional derivatives:
Ýd/dxÞ k | x| n for k, n = 1, 2, ...
2. Compute the distributional derivatives:
Ýd/dxÞ k | sinÝxÞ|
3. Compute / x f, / y f
and
fÝx, yÞ =
4. Compute / x f
for k = 1, 2, ...
and
fÝx, yÞ =
/ xy f
1 if
?1 < x + y < 1
0
otherwise
/ xy f
if :
1 if
xy > 0,
0
otherwise
5. For n = 1, 2, ... let
H n ÝxÞ = X
x
?1
sinÝntÞ
dt
^t
Lim n¸K H n ÝxÞ =
ÝaÞ Show that
if :
1 if x > 0
0 if x < 0
Is this convergence uniform? pointwise? distributional?
ÝbÞ Prove that
d
dx
H n ÝxÞ ¸ NÝxÞ as n ¸ K, in the sense of distributions.
sinÝnxÞ
6. Show that dxd H n ÝxÞ =
is a tempered function and compute its Fourier
^x
transform. To what does this transform converge as n ¸ K, and does it follow
that this is the transform of NÝxÞ?
7.Show that fÝxÞ = | x|, is a tempered function and compute its Fourier transform.
Compute the distributional derivative of f and find its Fourier transform.
1