Characterization of continuous, four-coefficient scaling functions via matrix spectral radius

We characterize the existence of continuous solutions of a four-coefficient dilation equation in terms of the usual spectral radius of a matrix. The c riteria for the existence of such a solution can be very quickly examined. As a result we give an affirmative answer to a conjecture raised by Colella and Heil in 1992. Moreover, using our criteria we nd the smoothest compact ly supported four-coefficient orthogonal scaling function and thus the smoo thest compactly supported orthonormal wavelet generated by this scaling fun ction.