30 I. Theory of Distributions
8. Problems
1. Let f(x), x
R1,
be a piecewise absolutely continuous function,
i.e., there are finitely many points c1,...,cn such that f(x) is abso-
lutely continuous on (−∞,c1], [c1,c2],..., [cn 1,cn], [cn, +∞) and
f(x) has jumps at ck, k = 1,...,n :
f(ck + 0) f(ck 0) = hk.
Denote by f the regular functional corresponding to f(x) and by g
the regular functional corresponding to f (x). Prove that
df
dx
= g +
n
k=1
hkδ(x ck),
where
df
dx
is the derivative of f in the distribution sense and δ(x−ck)
is the delta-function at ck, i.e.,
δ(x ck)(ϕ) = ϕ(ck), k = 1,...,n.
2. Find the distribution derivatives of the following regular functionals
in
R1:
a)
d2
dx2
|x|, b)
d2
dx2
| cos x|, c)
d2
dx2
sin |x|.
3. Find the limits of the following sequences of distributions in R1:
a) cos nx, b)
n10
sin nx, c)
sin nx
x
.
4. Prove that any trigonometric series
∑∞
n=−∞
cneinx
such that |cn|
C1|n|r + C2, converges to a distribution f D .
5. Denote by T (x) a 2π-periodic function such that
T (x) =
x(2π x)

for 0 x 2π.
a) Using the Fourier series
T (x) =
π
6

1

n=0
1
n2
einx,
prove that

n=−∞
δ(x 2πn) =
1


n=−∞
einx.
This formula is called the Poisson summation formula.
b) Find
∑∞
n=1
n sin nx.
Previous Page Next Page