The article by Stephane Mallat explores the properties of multiresolution approximations in the context of $L^2(\mathbb{R})$. A multiresolution approximation is defined as a sequence of embedded vector spaces $(V_j)_{j \in \mathbb{Z}}$ that approximate functions in $L^2(\mathbb{R})$. The author proves that such an approximation is characterized by a $2\pi$-periodic function, which is further described. From any multiresolution approximation, a function $\psi(x)$ can be derived, known as a wavelet, such that $(\sqrt{2} \psi(2^j x - k))_{(j, k) \in \mathbb{Z}^2}$ forms an orthonormal basis of $L^2(\mathbb{R})$. This provides a new approach to understanding and computing wavelet orthonormal bases. Additionally, the article characterizes the asymptotic decay rate of multiresolution approximation errors for functions in Sobolev spaces $H^s$. The study includes detailed proofs and examples, such as the use of cubic splines to illustrate the construction of multiresolution approximations and wavelets.The article by Stephane Mallat explores the properties of multiresolution approximations in the context of $L^2(\mathbb{R})$. A multiresolution approximation is defined as a sequence of embedded vector spaces $(V_j)_{j \in \mathbb{Z}}$ that approximate functions in $L^2(\mathbb{R})$. The author proves that such an approximation is characterized by a $2\pi$-periodic function, which is further described. From any multiresolution approximation, a function $\psi(x)$ can be derived, known as a wavelet, such that $(\sqrt{2} \psi(2^j x - k))_{(j, k) \in \mathbb{Z}^2}$ forms an orthonormal basis of $L^2(\mathbb{R})$. This provides a new approach to understanding and computing wavelet orthonormal bases. Additionally, the article characterizes the asymptotic decay rate of multiresolution approximation errors for functions in Sobolev spaces $H^s$. The study includes detailed proofs and examples, such as the use of cubic splines to illustrate the construction of multiresolution approximations and wavelets.