Dilation equations with finitely many nonzero coefficients have been extensively studied because of their connection with compactly supported wavelets . However, some properties required in applications are not compatible with the compactness of the support. For example, \textit{X.-G. Xia} [J. Fourier Anal. Appl. 1, No. 2, 193-199 (1994; MR 96i:42026)] has shown that the only orthonormal and compactly supported scaling function with the sampling property is the characteristic function of the interval \([0,1)\). In [IEEE Trans.\ Signal Process. 41, No. 12, 3524-3535 (1993; Zbl 0841.94022)], \textit{X.-G. Xia} and \textit{Z. Zhang} constructed an orthonormal scaling function with exponential decay coefficients and the sampling property. The paper under review studies the properties of such scaling functions, and in particular the relationship between the exponential decay property of the coefficients in the dilation equation and the exponential decay of the scaling function.