30 I. Theory of Distributions
1. Let f(x), x ∈
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
= g +
hkδ(x − ck),
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
| cos x|, c)
3. Find the limits of the following sequences of distributions in R1:
a) cos nx, b)
sin nx, c)
4. Prove that any trigonometric series
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) =
δ(x − 2πn) =
This formula is called the Poisson summation formula.
n sin nx.