This paper aims to clarify the relationship between discrete and continuous wavelet transforms, focusing on the integration of two separately motivated implementations: the *algorithm à trous* and Mallat's multiresolution decomposition. It is observed that both algorithms are spectrally equivalent filters for the discrete wavelet transform, with the behavior governed by a single set of filters. The *algorithm à trous*, originally designed for computational efficiency, is shown to be a nonorthonormal multiresolution algorithm where the discrete wavelet transform is exact. The paper also demonstrates that Lagrange à trous filters, commonly used in the *algorithm à trous*, are in one-to-one correspondence with the convolutional squares of Daubechies filters for orthonormal wavelets of compact support. A systematic framework for the discrete wavelet transform is provided, along with conditions under which it computes the continuous wavelet transform exactly. Filter constraints for finite energy and boundedness of the discrete transform are derived, and relevant signal processing parameters are examined. The paper concludes by discussing the trade-offs between orthonormality and resolution in wavelet transforms.This paper aims to clarify the relationship between discrete and continuous wavelet transforms, focusing on the integration of two separately motivated implementations: the *algorithm à trous* and Mallat's multiresolution decomposition. It is observed that both algorithms are spectrally equivalent filters for the discrete wavelet transform, with the behavior governed by a single set of filters. The *algorithm à trous*, originally designed for computational efficiency, is shown to be a nonorthonormal multiresolution algorithm where the discrete wavelet transform is exact. The paper also demonstrates that Lagrange à trous filters, commonly used in the *algorithm à trous*, are in one-to-one correspondence with the convolutional squares of Daubechies filters for orthonormal wavelets of compact support. A systematic framework for the discrete wavelet transform is provided, along with conditions under which it computes the continuous wavelet transform exactly. Filter constraints for finite energy and boundedness of the discrete transform are derived, and relevant signal processing parameters are examined. The paper concludes by discussing the trade-offs between orthonormality and resolution in wavelet transforms.