In general, the accuracy of any calculation based on wavelets with M vanishing moments is O(hM), where h = 2-k is the resolution at level k. However, it is known that the discretized first derivative operator has accuracy O(h2M). We present an alternative, easier proof of this result which generalizes to higher derivatives;We want to preserve the higher accuracy in the numerical solution of a periodic Sturm-Liouville problem of the form u''+pu'+qu=f by wavelet-Galerkin methods. For this purpose, we introduce new ways to calculate the wavelet decomposition and reconstruction and the product of smooth functions to arbitrarily high accuracy. These techniques are of independent interest;We illustrate the new high-accuracy methods with numerical examples, and compare them to other methods.