On the parametrization of the coefficients of dilation equations for compactly supported wavelets (Q1310618)

The coefficients \(c_ 0,c_ 1,\dots,c_ N\), \(N\) odd, of the so-called dilation equation for compactly supported wavelets are real numbers which satisfy the \((N+3)/2\) equations \(\sum c_ k = 2\), \(\sum c^ 2_ k = 2\), \(\sum c_ kc_{k+2l} = 0\) for \(l\neq 0\) with \(c_ k = 0\) for \(k < 0\) and \(k > N\). The authors give a different characterization of the admissible coefficients in terms of the polynomial \(2p_ N(z) = \sum c_ kz^ k\). A recursion formula is derived which connects the coefficient vector for \(N\) to the vector for \(N-2\). Finally an explicit parametric representation of the coefficients is given in terms of real parameters \(\alpha_ i\), \(i = 1,\dots,(N-1)/2\).