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

Abstract In this paper, we extend the continuous wavelet transform to Schwartz distributions in D′⁢(Rn) \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⁢(Rn) \mathcal{D}(\mathbb{R}^{n}) , n≥1 n\geq 1 , which, when integrated along each of the real axes X1,X2,X3,…,Xn X_{1},X_{2},X_{3},\ldots,X_{n} vanishes, but none of its moments ∫Rnψ⁢(x)⁢xm⁢dx \int_{\mathbb{R}^{n}}\psi(x)x^{m}\,dx is zero; here xm=x1m1⁢x2m2⁢…⁢xnmn x^{m}=x_{1}^{{m_{1}}}\,x_{2}^{{m_{2}}}\ldots x_{n}^{{m_{n}}} , d⁢x=d⁢x1⁢d⁢x2⁢…⁢d⁢xn dx=dx_{1}\,dx_{2}\ldots dx_{n} and m=(m1,m2,…,mn) m=(m_{1},m_{2},\ldots,m_{n}) and each of m1,m2,…,mn m_{1},m_{2},\ldots,m_{n} is at least 1. The set of such kernel will be denoted by Dm⁢(Rn) \mathcal{D}_{m}(\mathbb{R}^{n}) . But the uniqueness theorem for our wavelet inversion formula is valid for the space DF′⁢(Rn) \mathcal{D}_{F}^{\prime}(\mathbb{R}^{n}) obtained by filtering (deleting) (i) all non-zero constant distributions from the space D′⁢(Rn) \mathcal{D}^{\prime}(\mathbb{R}^{n}) , (ii) all non-zero constants that appear with a distribution as a union as for example for x12+x22+⋯⁢xn21+x12+x22+⋯⁢xn2=1-11+x12+x2

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