Local polynomial property and linear independence of refinable distributions (Q1347777)

Let \(M\geq 2\) be an integer and let \(\{d_j\}_{j=0}^{M-1}\) be a sequence such that \(\sum_{j=0}^{M-1} d_j = M\) and the number of nonzero \(d_j\)'s is at least 2 but at most \(M-1\). Let \(Q(z)= \sum_{j=0}^{M-1} d_j z^j/M\). Define the functions \(\phi_N\;(N \geq 0)\) recursively by \[ \widehat{\phi}_0 (\xi) = \prod_{j=0}^\infty Q\big(e^{-i2^{-j}\xi}\big) \quad \text{ and } \;\phi_{N+1} = \chi_{[0, 1]} \star \phi_N, \] where \(f \star g\) denotes the convolution of \(f\) and \(g\). In this paper under review, the authors show that for any \(N \geq 1\), \(\phi_N\) has the local polynomial property, global linear independence and local linear independence.