Some enhancements to the approximation of one-variable functions with respect to an orthogonal basis are considered. A two-step approximation scheme is presented here. In the first step, a constant bias is extracted from the approximated function, while in the second, the function with extracted bias is approximated in a usual way. Later, these two components are added together. First of all we prove that a constant bias extracted from the function decreases the error. We demonstrate how to calculate that bias. Secondly, in a minor contribution, we show how to choose basis from a selected set of orthonormal functions to achieve minimum error. Finally we prove that loss of orthonormality due to truncation of the argument range of the basis functions does not effect the overall error of approximation and the expansion coefficients' correctness. We show how this feature can be used. Our attention is focused on Hermite orthonormal functions. An application of the obtained results to ECG data compression is presented.