CONVOLUTION THEOREMS FOR WAVELET TRANSFORM ON TEMPERED DISTRIBUTIONS AND THEIR EXTENSION TO TEMPERED BOEHMIANS

Abstract

We define a new convolution ⊗ : 𝒮'(ℝ × ℝ+) × 𝒟(ℝ) → 𝒮'(ℝ × ℝ+) and derive the convolution theorems for wavelet transform and dual wavelet transform in the context of tempered distributions. By using the new convolution, we construct a Boehmian space containing the tempered distributions on ℝ × ℝ+. Applying the convolution theorems in the context of tempered distributions, we also extend the wavelet transform and dual wavelet transform between the tempered Boehmian space and the new Boehmian space as linear continuous maps with respect to δ-convergence and Δ-convergence, satisfying the convolution theorems.