This paper introduces the concept of multiresolution approximations and wavelet orthonormal bases in the space $ L^2(\mathbb{R}) $. A multiresolution approximation is a sequence of nested subspaces that approximate functions in $ L^2(\mathbb{R}) $. The paper shows that such approximations are characterized by a $ 2\pi $-periodic function, and from any multiresolution approximation, one can derive a wavelet function $ \psi(x) $ such that $ (\sqrt{2^j}\psi(2^jx - k))_{(k,j)\in\mathbb{Z}^2} $ forms an orthonormal basis of $ L^2(\mathbb{R}) $. This provides a new method for understanding and computing wavelet orthonormal bases. The paper also characterizes the asymptotic decay rate of multiresolution approximation errors for functions in Sobolev spaces $ H^s $. It discusses the properties of the function $ \phi(x) $, which is used to construct the orthonormal basis for each resolution level. The Fourier transform of $ \phi(x) $ is characterized by a $ 2\pi $-periodic function $ H(\omega) $, and the paper provides conditions on $ H(\omega) $ to ensure that $ \phi(x) $ generates a regular multiresolution approximation. The paper further describes an algorithm to find a wavelet $ \psi(x) $ such that $ (\sqrt{2^j}\psi(2^jx - k))_{(k,j)\in\mathbb{Z}^2} $ forms an orthonormal basis of $ L^2(\mathbb{R}) $. It also discusses the smoothness and decay properties of wavelets, showing that the regularity of $ \psi(x) $ determines the smoothness of the resulting wavelet orthonormal basis. The paper concludes with a theorem that characterizes the approximation error for functions in Sobolev spaces $ H^s $, showing that the error decays at a rate determined by the smoothness of the function. The paper also references various applications of wavelet orthonormal bases in mathematics, physics, and signal processing.This paper introduces the concept of multiresolution approximations and wavelet orthonormal bases in the space $ L^2(\mathbb{R}) $. A multiresolution approximation is a sequence of nested subspaces that approximate functions in $ L^2(\mathbb{R}) $. The paper shows that such approximations are characterized by a $ 2\pi $-periodic function, and from any multiresolution approximation, one can derive a wavelet function $ \psi(x) $ such that $ (\sqrt{2^j}\psi(2^jx - k))_{(k,j)\in\mathbb{Z}^2} $ forms an orthonormal basis of $ L^2(\mathbb{R}) $. This provides a new method for understanding and computing wavelet orthonormal bases. The paper also characterizes the asymptotic decay rate of multiresolution approximation errors for functions in Sobolev spaces $ H^s $. It discusses the properties of the function $ \phi(x) $, which is used to construct the orthonormal basis for each resolution level. The Fourier transform of $ \phi(x) $ is characterized by a $ 2\pi $-periodic function $ H(\omega) $, and the paper provides conditions on $ H(\omega) $ to ensure that $ \phi(x) $ generates a regular multiresolution approximation. The paper further describes an algorithm to find a wavelet $ \psi(x) $ such that $ (\sqrt{2^j}\psi(2^jx - k))_{(k,j)\in\mathbb{Z}^2} $ forms an orthonormal basis of $ L^2(\mathbb{R}) $. It also discusses the smoothness and decay properties of wavelets, showing that the regularity of $ \psi(x) $ determines the smoothness of the resulting wavelet orthonormal basis. The paper concludes with a theorem that characterizes the approximation error for functions in Sobolev spaces $ H^s $, showing that the error decays at a rate determined by the smoothness of the function. The paper also references various applications of wavelet orthonormal bases in mathematics, physics, and signal processing.