The continuous wavelet transform and window functions

Abstract

We define a window function ψ as an element of L2(ℝn) satisfying certain boundedness properties with respect to the L2(ℝn) norm and prove that it satisfies the admissibility condition if and only if the integral of ψ(x1, x2, · · · , xn) with respect to each of the variables x1, x2, · · · , xn along the real line is zero. We also prove that each of the window functions is an element of L1(ℝn). A function ψ ∈ L2(ℝn) satisfying the admissibility condition is a wavelet. We define the wavelet transform of f ∈ L2(ℝn) (which is a window function) with respect to the wavelet ψ ∈ L2(ℝn) and prove an inversion formula interpreting convergence in L2(ℝn). It is also proved that at a point of continuity of f the convergence of our wavelet inversion formula is in a pointwise sense. © 2015 American Mathematical Society.