A class of spectral measures and its spectral eigenvalues

In this paper we will investigate the harmonic analysis of a class of infinite convolutions μ on $R$. A necessary and sufficient condition for the Hilbert space $L^2\left(\mu \right)$ has an orthonormal basis of exponential functions is given. Moreover, we give a complete characterization on the spectral eigenvalues of the spectral measure μ, that is, to find all real numbers p which corresponds to a discrete set Λ such that the sets $\{e^{2\pi i\lambda x}:\lambda \in \Lambda \}$ and $\{e^{2\pi ip\lambda x}:\lambda \in \Lambda \}$ are both orthonormal bases for $L^2\left(\mu \right)$.