Existence of Exponential Orthonormal Bases for Infinite Convolutions on $${{\mathbb {R}}}^n$$

In this paper we investigate the harmonic analysis of infinite convolutions generated by admissible pairs on Euclidean space \({{\mathbb {R}}}^n\). Our main results give several sufficient conditions so that the infinite convolution \(\mu \) to be a spectral measure, that is, its Hilbert space \(L^2(\mu )\) admits a family of orthonormal basis of exponentials. As a concrete application, we give a complete characterization on the spectral property for certain infinite convolution on the plane \({{\mathbb {R}}}^2\) in terms of admissible pairs.