In this paper we propose a framework, called mixtures of truncated basis functions (MoTBFs), for representing general hybrid Bayesian networks. The proposed framework generalizes both the mixture of truncated exponentials (MTEs) framework and the mixture of polynomials (MoPs) framework. Similar to MTEs and MoPs, MoTBFs are defined so that the potentials are closed under combination and marginalization, which ensures that inference in MoTBF networks can be performed efficiently using the Shafer-Shenoy architecture. Based on a generalized Fourier series approximation, we devise a method for efficiently pproximating an arbitrary density function using the MoTBF framework. The translation method is more flexible than existing MTE or MoP-based methods, and it supports an online/anytime tradeoff between the accuracy and the complexity of the approximation. Experimental results show that the approximations obtained are either comparable or significantly better than the approximations obtained using existing methods.