Multiresolution approximations and wavelet orthonormal bases of 𝐿²(𝑅)

A multiresolution approximation is a sequence of embedded vector spaces ( V j ) j ∈ z {({{\mathbf {V}}_j})_{j \in {\text {z}}}} for approximating L 2 ( R ) {{\mathbf {L}}^2}({\mathbf {R}}) functions. We study the properties of a multiresolution approximation and prove that it is characterized by a 2 π 2\pi -periodic function which is further described. From any multiresolution approximation, we can derive a function ψ ( x ) \psi (x) called a wavelet such that ( 2 j ψ ( 2 j x − k ) ) ( k , j ) ∈ z 2 {(\sqrt {{2^j}} \psi ({2^j}x - k))_{(k,j) \in {{\text {z}}^2}}} is an orthonormal basis of L 2 ( R ) {{\mathbf {L}}^2}({\mathbf {R}})