Wavelets have been successfully used for nonparametric function estimation, but a major challenge in density and hazard estimation is that the function must be nonnegative. As there does not exist a nonnegative scaling function for an orthonormal wavelet basis, we develop a quasi-continuous nonnegative "wavelet" basis from Daubechies wavelets with good approximation properties. We use this basis to develop a Bayesian nonparametric approach to density estimation and estimation of the hazard function with randomly right censored data. The proposed method is compared with other wavelet methods for hazard estimation and is also illustrated on the Stanford heart transplant data.