Based on the approximation theory of Fourier-series working in square integrable space, a Fourier neuronal network was constructed by using activation functions of the complex exponential form. Then a weightsdirectdetermination method was derived to decide the neuralnetwork weights immediately, which remedied the weaknesses of conventional BP neural networks such as small convergence rate, easily converging to local minimum and possibly lengthy or oscillatory learning process. A hidden-neurons-growing algorithm was presented to adjust the neural-network structure adaptively. Theoretical analysis and simulation results substantiate further that the presented Fourier neural network and algorithm could have good properties of high-precision learning, noise-suppressing and discontinuous-function approximating.