Modern medical science demands sophisticated signal representation methods in order to cope with the increasing amount of data. Important criteria for these methods are mainly low computational and storage costs, whereas the underlying mathematical model should still be interpretable and meaningful for the data analyst. One of the most promising models fulfilling these criteria is based on Hermite functions, however having some important limitations for specific biomedical wave shapes. We extend this model by using weighted Hermite functions and develop a gradient based constrained optimization method to adapt the system for different types of signals. In order to demonstrate the potential of our approach, we consider the problem of electrocardiogram signal compression. The experiments on the MIT/BIH arrhythmia database show a significant improvement compared to the former works using classical Hermite functions.