English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

In this paper, we extend the continuous wavelet transform to Schwartz distributions inD′⁢(Rn)\mathcal{D}^{\prime}(\mathbb{R}^{n}),n≥1n\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 elementψ⁢(x)\psi(x)ofD⁢(Rn)\mathcal{D}(\mathbb{R}^{n}),n≥1n\geq 1, which, when integrated along each of the real axesX1,X2,X3,…,XnX_{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}\,dxis zero; herexm=x1m1⁢x2m2⁢…⁢xnmnx^{m}=x_{1}^{{m_{1}}}\,x_{2}^{{m_{2}}}\ldots x_{n}^{{m_{n}}},d⁢x=d⁢x1⁢d⁢x2⁢…⁢d⁢xndx=dx_{1}\,dx_{2}\ldots dx_{n}andm=(m1,m2,…,mn)m=(m_{1},m_{2},\ldots,m_{n})and each ofm1,m2,…,mnm_{1},m_{2},\ldots,m_{n}is at least 1. The set of such kernel will be denoted byDm⁢(Rn)\mathcal{D}_{m}(\mathbb{R}^{n}). But the uniqueness theorem for our wavelet inversion formula is valid for the spaceDF′⁢(Rn)\mathcal{D}_{F}^{\prime}(\mathbb{R}^{n})obtained by filtering (deleting) (i) all non-zero constant distributions from the spaceD′⁢(Rn)\mathcal{D}^{\prime}(\mathbb{R}^{n}), (ii) all non-zero constants that appear with a distribution as a union as for example forx12+x22+⋯⁢xn21+x12+x22+⋯⁢xn2=1-11+x12+x22+⋯⁢xn2\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-11+x12+x22+⋯⁢xn2\frac{-1}{1+x_{1}^{2}+x_{2}^{2}+\cdots x_{n}^{2}}is retained.

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