Let \(\{x_n\}\) be a frame for a Hilbert space \(H\). A dual for \(\{x_n\}\) is a frame \(\{x_n^*\}\) for which \(x= \sum \langle x,x_n\rangle x_n^*= \sum \langle x,x_n^*\rangle x_n\) for all \(x\in H\). More generally, a pseudo-dual of \(\{x_n\}\) is any family \(\{x_n^*\}\) for which \(\langle x,y\rangle = \sum \langle x,x_n^*\rangle \langle x_n,y\rangle\) for all \(x,y\in H\). If the pseudo-dual is a Bessel sequence (i.e., it satisfies the upper frame condition), then it is automatically a dual frame. The paper gives a construction of a wavelet frame and an associated pseudo-dual which is not a Bessel sequence. Via unbounded operators satisfying special conditions on the size of the domain, a construction of a class of pseudo-duals is given for general frames (of course, this class reduces to a single function in some special cases).