An approximation method is proposed for a class of band-limited waveforms whose Fourier spectra F(ω) are confined in the interval -ω1 ≦ ω ≦ ω1. The side lobes of such waveforms are also small in the range of 0 < ω0 < ∥ω∥ < ω1. First, the approximation waveform is obtained from summation of the product of an interpolating function and the sampling value f(tk) at the sampling point tk (k = 0 ∼ K). One objective of the proposed method is to find an interpolating function that will minimize the envelope of a family of approximation waveforms, which is used as a measure of error associated with the approximation. The optimum interpolating function, also satisfies the condition of discrete orthogonality. The proposed approximation technique also piovides convergence to the actual waveform with respect to a set of sampling points determined by the conventional sampling theorem. The approximation error is evaluated and weighted in the frequency domain using Fourier series expansion. The lower and upper bounds of approximation error in the frequency domain are derived and their relationships to the original waveform are established.