Let
R
R
be an expanding matrix with integer entries, and let
B
,
L
B,L
be finite integer digit sets so that
(
R
,
B
,
L
)
(R,B,L)
form a Hadamard triple on
R
d
{\mathbb {R}}^d
in the sense that the matrix
1
|
det
R
|
[
e
2
π
i
⟨
R
−
1
b
,
ℓ
⟩
]
ℓ
∈
L
,
b
∈
B
\begin{equation*} \frac {1}{\sqrt {|\det R|}}\left [e^{2\pi i \langle R^{-1}b,\ell \rangle }\right ]_{\ell \in L,b\in B} \end{equation*}
is unitary. We prove that the associated fractal self-affine measure
μ
=
μ
(
R
,
B
)
\mu = \mu (R,B)
obtained by an infinite convolution of atomic measures
μ
(
R
,
B
)
=
δ
R
−
1
B
∗
δ
R
−
2
B
∗
δ
R
−
3
B
∗
⋯
\begin{equation*} \mu (R,B) = \delta _{R^{-1} B}\ast \delta _{R^{-2}B}\ast \delta _{R^{-3}B}\ast \cdots \end{equation*}
is a spectral measure, i.e., it admits an orthonormal basis of exponential functions in
L
2
(
μ
)
L^2(\mu )
. This settles a long-standing conjecture proposed by Jorgensen and Pedersen and studied by many other authors. Moreover, we also show that if we relax the Hadamard triple condition to an almost-Parseval-frame condition, then we obtain a sufficient condition for a self-affine measure to admit Fourier frames.