Continuous wavelet transform of Schwartz distributions in 𝒟′(ℝ𝑛), 𝑛 ≤ 1

Abstract

Abstract In this paper, we extend the continuous wavelet transform to Schwartz distributions in D ′ ⁢ ( R n ) \mathcal{D}^{\prime}(\mathbb{R}^{n}) , n ≥ 1 n\geq 1 , and derive the corresponding wavelet inversion formula (valid modulo a constant distribution) interpreting convergence in the weak distributional sense. The kernel of our wavelet transform is an element ψ ⁢ ( x ) \psi(x) of D ⁢ ( R n ) \mathcal{D}(\mathbb{R}^{n}) , n ≥ 1 n\geq 1 , which, when integrated along each of the real axes X 1 , X 2 , X 3 , … , X n X_{1},X_{2},X_{3},\ldots,X_{n} vanishes, but none of its moments ∫ R n ψ ⁢ ( x ) ⁢ x m ⁢ d x \int_{\mathbb{R}^{n}}\psi(x)x^{m}\,dx is zero; here x m = x 1 m 1 ⁢ x 2 m 2 ⁢ … ⁢ x n m n x^{m}=x_{1}^{{m_{1}}}\,x_{2}^{{m_{2}}}\ldots x_{n}^{{m_{n}}} , d ⁢ x = d ⁢ x 1 ⁢ d ⁢ x 2 ⁢ … ⁢ d ⁢ x n dx=dx_{1}\,dx_{2}\ldots dx_{n} and m = ( m 1 , m 2 , … , m n ) m=(m_{1},m_{2},\ldots,m_{n}) and each of m 1 , m 2 , … , m n m_{1},m_{2},\ldots,m_{n} is at least 1. The set of such kernel will be denoted by D m ⁢ ( R n ) \mathcal{D}_{m}(\mathbb{R}^{n}) . But the uniqueness theorem for our wavelet inversion formula is valid for the space D F ′ ⁢ ( R n ) \mathcal{D}_{F}^{\prime}(\mathbb{R}^{n}) obtained by filtering (deleting) (i) all non-zero constant distributions from the space D ′ ⁢ ( R n ) \mathcal{D}^{\prime}(\mathbb{R}^{n}) , (ii) all non-zero constants that appear with a distribution as a union as for example for x 1 2 + x 2 2 + ⋯ ⁢ x n 2 1 + x 1 2 + x 2 2 + ⋯ ⁢ x n 2 = 1 - 1 1 + x 1 2 + x 2 2 + ⋯ ⁢ x n 2 \frac{x_{1}^{2}+x_{2}^{2}+\cdots x_{n}^{2}}{1+x_{1}^{2}+x_{2}^{2}+\cdots x_{n}^{2}}=1-\frac{1}{1+x_{1}^{2}+x_{2}^{2}+\cdots x_{n}^{2}} , 1 is deleted and - 1 1 + x 1 2 + x 2 2 + ⋯ ⁢ x n 2 \frac{-1}{1+x_{1}^{2}+x_{2}^{2}+\cdots x_{n}^{2}} is retained.