Abstract
Abstract
Let
{
(
N
j
,
B
j
,
L
j
)
:
1
⩽
j
⩽
m
}
be finitely many Hadamard triples in
R
. Given a sequence of positive integers
{
n
k
}
k
=
1
∞
and
ω
=
(
ω
k
)
k
=
1
∞
∈
{
1
,
2
,
…
,
m
}
N
, let
μ
ω
,
{
n
k
}
be the infinite convolution given by
μ
ω
,
n
k
=
δ
N
ω
1
−
n
1
B
ω
1
∗
δ
N
ω
1
−
n
1
N
ω
2
−
n
2
B
ω
2
∗
⋯
∗
δ
N
ω
1
−
n
1
N
ω
2
−
n
2
⋯
N
ω
k
−
n
k
B
ω
k
∗
⋯
.
In order to study the spectrality of
μ
ω
,
{
n
k
}
, we first show the spectrality of general infinite convolutions generated by Hadamard triples under the equi-positivity condition. Then by using the integral periodic zero set of Fourier transform we show that if
g
c
d
(
B
j
−
B
j
)
=
1
for
1
⩽
j
⩽
m
, then all infinite convolutions
μ
ω
,
{
n
k
}
are spectral measures. This implies that we may find a subset
Λ
ω
,
{
n
k
}
⊆
R
such that
{
e
λ
(
x
)
=
e
2
π
i
λ
x
:
λ
∈
Λ
ω
,
{
n
k
}
}
forms an orthonormal basis for
L
2
(
μ
ω
,
{
n
k
}
)
.
Funder
Science and Technology Commission of Shanghai Municipality
National Natural Science Foundation of China
Fundamental Research Funds for the Central Universities
Subject
Applied Mathematics,General Physics and Astronomy,Mathematical Physics,Statistical and Nonlinear Physics