Discrete wavelet transforms have many applications, including those in image compression and edge detection. Transforms constructed using orthogonal filters are extremely useful in that they can easily be inverted as well as coded. We review the major properties of three well-known orthogonal filters, namely, the Haar, Daubechies, and Coiflet filters. Subsequently, we analyze the Fourier series that corresponds to each of those filters and recall some important results about the smoothness of the modulus of those Fourier series. We consider a specialized case in which the length of the discrete wavelet transform is not much longer than the length of the filter used in its construction. For this case, we prove the existence of additional degrees of freedom in the system of equations used in the construction of the aforementioned orthogonal filters. We suggest a modified Coiflet filter which takes advantage of the extra degrees of freedom by imposing further conditions on the derivative of the Fourier series.