In epilepsy diagnosis or epileptic seizure detection, much effort has been focused on finding effective combination of feature extraction and classification methods. In this paper, we develop a wavelet-based sparse functional linear model for representation of EEG signals. The aim of this modeling approach is to capture discriminative random components of EEG signals using wavelet variances. To achieve this goal, a forward search algorithm is proposed for determination of an appropriate wavelet decomposition level. Two EEG databases from University of Bonn and University of Freiburg are used for illustration of applicability of the proposed method to both epilepsy diagnosis and epileptic seizure detection problems. For this data considered, we show that wavelet-based sparse functional linear model with a simple classifier such as 1-NN classification method leads to higher classification results than those obtained using other complicated methods such as support vector machine. This approach produces a 100% classification accuracy for various classification tasks using the EEG database from University of Bonn, and outperforms many other state-of-the-art techniques. The proposed classification scheme leads to 99% overall classification accuracy for the EEG data from University of Freiburg.