A self-similar measure on
R
n
{{\mathbf {R}}^n}
was defined by Hutchinson to be a probability measure satisfying
(
∗
)
({\ast })
\[
μ
=
∑
j
=
1
m
a
j
μ
∘
S
j
−
1
\mu = \sum \limits _{j = 1}^m {{a_j}\mu \circ S_j^{ - 1}}
\]
, where
S
j
x
=
ρ
j
R
j
x
+
b
j
{S_j}x = {\rho _j}{R_j}x + {b_j}
is a contractive similarity
(
0
>
ρ
j
>
1
,
R
j
(0 > {\rho _j} > 1,{R_j}
orthogonal) and the weights
a
j
{a_j}
satisfy
0
>
a
j
>
1
,
∑
j
=
1
m
a
j
=
1
0 > {a_j} > 1,\sum \nolimits _{j = 1}^m {{a_j} = 1}
. By analogy, we define a self-similar distribution by the same identity
(
∗
)
( {\ast } )
but allowing the weights
a
j
{a_j}
to be arbitrary complex numbers. We give necessary and sufficient conditions for the existence of a solution to
(
∗
)
( {\ast } )
among distributions of compact support, and show that the space of such solutions is always finite dimensional. If
F
F
denotes the Fourier transformation of a self-similar distribution of compact support, let
\[
H
(
R
)
=
1
R
n
−
β
∫
|
x
|
≤
R
|
F
(
x
)
|
2
d
x
,
H(R) = \frac {1}{{{R^{n - \beta }}}}\int _{|x| \leq R} {|F(x){|^2}dx,}
\]
where
β
\beta
is defined by the equation
∑
j
=
1
m
ρ
j
−
β
|
a
j
|
2
=
1
\sum \nolimits _{j = 1}^m {\rho _j^{ - \beta }|{a_j}{|^2} = 1}
. If
ρ
j
ν
j
=
ρ
\rho _j^{{\nu _j}} = \rho
for some fixed
ρ
\rho
and
ν
j
{\nu _j}
positive integers we say the
{
ρ
j
}
\{ {\rho _j}\}
are exponentially commensurable. In this case we prove (under some additional hypotheses) that
H
(
R
)
H(R)
is asymptotic (in a suitable sense) to a bounded function
H
~
(
R
)
\tilde H(R)
that is bounded away from zero and periodic in the sense that
H
~
(
ρ
R
)
=
H
~
(
R
)
\tilde H(\rho R) = \tilde H(R)
for all
R
>
0
R > 0
. If the
{
ρ
j
}
\{ {\rho _j}\}
are exponentially incommensurable then
lim
R
→
∞
H
(
R
)
{\lim _{R \to \infty }}H(R)
exists and is nonzero.