In practical simultaneous information and energy transmission (SIET), the exact energy harvesting function is usually unavailable because an energy harvesting circuit is nonlinear and nonideal. In this work, we consider a SIET problem where the harvesting function is accessible only at experimentally-taken sample points and study how close we can design SIET to the optimal system with such sampled knowledge. Assuming that the harvesting function is of bounded variation that may have discontinuities, we separately consider two settings where samples are taken without and with additive noise. For these settings, we propose to design a SIET system as if a wavelet-reconstructed harvesting function is the true one and study its asymptotic performance loss of energy and information delivery from the true optimal one. Specifically, for noiseless samples, it is shown that designing SIET as if the wavelet-reconstructed harvesting function is the truth incurs asymptotically vanishing energy and information delivery loss with the number of samples. For noisy samples, we propose to reconstruct wavelet coefficients via soft-thresholding estimation. Then, we not only obtain similar asymptotic losses to the noiseless case but also show that the energy loss by wavelets is asymptotically optimal up to a logarithmic factor.