Fig. 10

Discrete \(\Sigma \)-tensor wavelet function \(\tilde{\boldsymbol{\Psi }}_{\tau _{j}}^{5}\) of the second kind on the unit sphere \(\Omega \) for different values of the discrete scale parameter \(j \in \mathbb{Z}\) on the unit sphere \(\Omega \). The left figures show the Frobenius norms of the tensor wavelets \(\tilde{\boldsymbol{\Psi }}_{\tau _{j}}^{5}(x,y)\) for a fixed value \(y \in \Omega \) and variable \(x \in \Omega \). The right figures show the scalar value \(\varepsilon ^{r}(x) \cdot \left (\tilde{\boldsymbol{\Psi }}_{\tau _{j}}^{5}(x,y)\varepsilon ^{r}(x)\right )\), where \(\varepsilon ^{r}(x)\) is the radial unit vector at the point x. This “radial projection” of the tensor kernel is suitable to show the wavelet character of the \(\Sigma \)-tensor wavelet functions