The continuous wavelet transform and window functions

Abstract:

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