On the Fourier orthonormal bases of a class of self-similar measures on \(\mathbb{R}^n\) (Q6085852)

The authors consider self-similar measures \(\mu_{M,D}\) generated by iterated function systems consisting of maps of the form \(\phi_d:\mathbb{R}^n\to \mathbb{R}^n\), \(x\mapsto M^{-1}(x+d)\), where \(M\) is an expanding real \(n\times n\) matrix and \(d\in D\), for some finite digit set \(D\subset \mathbb{Z}^d\). Necessary and sufficient conditions on \(D\) are derived so that \(L^2(\mu_{M,D})\) admits an infinite orthogonal set of exponential functions. Moreover, necessary and sufficient conditions for \(L^2(\mu_{M,D})\) to have a Fourier basis are given. Examples illustrating the results are also provided.