In this paper we derive an orthonormal basis of wavelet scaling functions for L2 ([0, 1]) motivated by the need for such a basis in the field of generalized sampling. A special property of this basis is that it includes carefully constructed boundary functions and it can be constructed with arbitrary smoothness. This construction makes assumptions about the signal outside the interval unnecessary. Furthermore, we provide a Python package implementing this wavelet decomposition. Wavelets defined on a bounded interval are widely used for signal analysis, compression, and for numerical solution of differential equations. We show that for many cases using the basis that we derive results in smaller error than the commonly used alternative.