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)

4π

for 0 x 2π.

a) Using the Fourier series

T (x) =

π

6

−

1

2π

n=0

1

n2

einx,

prove that

∞

n=−∞

δ(x − 2πn) =

1

2π

∞

n=−∞

einx.

This formula is called the Poisson summation formula.

b) Find

∑∞

n=1

n sin nx.