The continuous wavelet transform and window functions

Abstract

We define a window function ψ \psi as an element of L 2 ( R n ) L^2(\mathbb R^n) satisfying certain boundedness properties with respect to the L 2 ( R n ) L^2(\mathbb R^n) norm and prove that it satisfies the admissibility condition if and only if the integral of ψ ( x 1 , x 2 , ⋯ , x n ) \psi (x_1,x_2,\cdots ,x_n) with respect to each of the variables x 1 , x 2 , ⋯ , x n x_1,x_2,\cdots ,x_n along the real line is zero. We also prove that each of the window functions is an element of L 1 ( R n ) L^1(\mathbb R^n) . A function ψ ∈ L 2 ( R n ) \psi \in L^2(\mathbb R^n) satisfying the admissibility condition is a wavelet. We define the wavelet transform of f ∈ L 2 ( R n ) f\in L^2(\mathbb R^n) (which is a window function) with respect to the wavelet ψ ∈ L 2 ( R n ) \psi \in L^2(\mathbb R^n) and prove an inversion formula interpreting convergence in L 2 ( R n ) L^2(\mathbb R^n) . It is also proved that at a point of continuity of f f the convergence of our wavelet inversion formula is in a pointwise sense.