Multiresolution Approximations and Wavelet Orthonormal Bases of $L^2(R)$

@article{198909, abstract = {A multiresolution approximation is a sequence of embedded vector spaces \$(V\_j)\_{j\in Z}\$ for approximating \$L^2(R)\$ functions. We study the properties of a multiresolution approximation and prove that it is characterized by a \$2\pi\$-periodic function which is further described. From any multiresolution approximation, we can derive a function \$\psi(x)\$ called a wavelet such that \$(\sqrt{2^j}\pi(2^jx - k))\_{(k,j)\in Z^2}\$ is an orthonormal basis of \$L^2(R)\$. This provides a new approach for understanding and computing wavelet orthonormal bases. Finally, we characterize the asymptotic decay rate of multiresolution approximation errors for functions in a Sobolev space \$H^s\$.}, author = {Mallat, Stephane G. }, citeulike-article-id = {973779}, journal = {Transactions of the American Mathematical Society}, keywords = {approximation, bases, orthonormal, theory, wavelets}, month = {sep}, number = {1}, pages = {69--87}, posted-at = {2006-12-04 19:13:19}, priority = {2}, publisher = {American Mathematical Society}, title = {Multiresolution Approximations and Wavelet Orthonormal Bases of \$L^2(R)\$}, url = {http://links.jstor.org/sici?sici=0002-9947\%28198909\%29315\%3A1\%3C69\%3AMAAWOB\%3E2.0.CO\%3B2-B}, volume = {315}, year = {1989} }

TY - JOUR ID - 198909 L3 - citeulike-article-id:973779 TI - Multiresolution Approximations and Wavelet Orthonormal Bases of $L^2(R)$ JF - Transactions of the American Mathematical Society VL - 315 IS - 1 SP - 69 EP - 87 PB - American Mathematical Society N2 - A multiresolution approximation is a sequence of embedded vector spaces $(V_j)_{j\in Z}$ for approximating $L^2(R)$ functions. We study the properties of a multiresolution approximation and prove that it is characterized by a $2\pi$-periodic function which is further described. From any multiresolution approximation, we can derive a function $\psi(x)$ called a wavelet such that $(\sqrt{2^j}\pi(2^jx - k))_{(k,j)\in Z^2}$ is an orthonormal basis of $L^2(R)$. This provides a new approach for understanding and computing wavelet orthonormal bases. Finally, we characterize the asymptotic decay rate of multiresolution approximation errors for functions in a Sobolev space $H^s$. KW - approximation KW - bases KW - orthonormal KW - theory KW - wavelets AU - Mallat, Stephane PY - 1989/sep// UR - http://links.jstor.org/sici?sici=0002-9947%28198909%29315%3A1%3C69%3AMAAWOB%3E2.0.CO%3B2-B ER -