\(M\)-positive sets in \(\mathbb{R}^d\) (Q2133852)

The authors study an orthogonal Walsh-type basis for the space \(L_2(U),\) where \(U\) is a compact self-similar set in \(\mathbb R^d\) playing the role of the unit interval on the half-line. The characteristic functions of such self-similar sets are known to be the scaling functions used for constructing wavelets in \(L_2(\mathbb R^d).\) The authors define Walsh type functions on a set of \(M\)-positive vectors and show that they form an orthogonal basis for the space \(L_2(U).\)