Stokes Problem, Layer Potentials and Regularizations, and Multiscale Applications

Abstract

This contribution represents a continuation of a series of papers by the authors concerned with the multiscale solution of boundary-value problems corresponding to elliptic differential equations such as Laplace equation (Freeden and Mayer, Appl Comput Harmonic Anal 14:195–237, 2003; Acta Geod Geophys Hung 41:55–86, 2006), Helmholtz equation (Freeden et al., Numer Funct Anal Optim 24:747–782, 2003; Ilyasov, A tree algorithm for Helmholtz potential wavelets on non-smooth surfaces: theoretical background and application to seismic data postprocessing. PhD thesis, Geomathematics Group, Department of Mathematics, University of Kaiserslautern, 2011), Cauchy-Navier equation (Abeyratne, Cauchy-Navier wavelet solvers and their application in deformation analysis. PhD thesis, Geomathematics Group, University of Kaiserslautern, 2003; Abeyratne et al., J Appl Math 12:605–645, 2003), and Maxwell equations (Freeden and Mayer, Int J Wavelets Multiresolut Inf Process 5:417–449, 2007). The essential idea is to transform the differential equation into an integral equation by standard surface layer potentials and to use certain regularizations of the kernels of the layer potentials as scaling kernel functions. In this context, the distance of a parallel surface to the boundary acts as the scale parameter. The scaling kernel functions are defined as restrictions of the kernel values of layer potentials to the parallel surface. Wavelet kernel functions in scale discrete case are canonically obtained as the difference between two consecutive scaling functions. The solution process is formulated in such a way that an approximation of a boundary function on a regular surface simultaneously yields the solution of the boundary-value problem itself. In the case of Stokes flow – as discussed here – the kernels are of vectorial/tensorial nature, satisfying the differential equation in each variable. Stokes flow leads to significant applications of geomathematical relevance (e.g., in oceanography, meteorology).

Appendices

Appendices

We conclude our work with a list of appendices.

Regular Surfaces

First, the main geometrical reference object discussed in this thesis, i.e., a regular surface, is defined, and certain properties are explained in more detail (see also Müller 1969; Freeden and Gerhards 2013):

Definition 13.

A subset $\Sigma \subset \mathbb{R}^{3}$ is called a regular surface in $\mathbb{R}^{3}$ if the following properties are fulfilled: 1. $\Sigma$ is a closed and compact surface free of double points. 2. $\Sigma$ divides the Euclidean space $\mathbb{R}^{3}$ into the bounded inner region $\Sigma _{\mathrm{int}}$ and the unbounded outer region $\Sigma _{\mathrm{ext}}$ with $\mathbb{R}^{3} = \Sigma _{\mathrm{int}}\,\dot{ \cup }\, \Sigma \,\dot{ \cup }\, \Sigma _{\mathrm{ext}}$. 3. The origin is in $\Sigma _{\mathrm{int}}$, $0 \in \Sigma _{\mathrm{int}}$. 4. $\Sigma$ is locally of class C⁽²⁾.

The fourth property means that, for each point $x \in \Sigma$, there exists a neighborhood $U(x) \subset \mathbb{R}^{3}$ of x such that $\Sigma \cap U(x)$ can be mapped bijectively onto an open subset $V \subset \mathbb{R}^{2}$ and that this mapping is twice continuously differentiable. The fourth property of Definition 13 is equivalent to the existence of a continuously differentiable unit normal field ν on $\Sigma$ pointing, by definition, into the outer space $\Sigma _{\mathrm{ext}}$.

Examples of a regular surface are the sphere $\Omega _{R}$ with radius R > 0, the ellipsoid, and as geoscientifically relevant example the real (regular) Earth’s surface (obtained by modern GPS technology).

Definition 14.

Let $\nu: \Sigma \rightarrow \mathbb{R}^{3}$ denote the unit normal field on $\Sigma$. Then the set

$$\displaystyle{ \Sigma (\tau ) =\{ x \in \mathbb{R}^{3}\vert x = y +\tau \nu (y),\,y \in \Sigma \} }$$

(384)

generates a parallel surface which is exterior to $\Sigma$ for τ > 0 and interior for τ < 0.

It is well known (see, e.g., Müller 1969; Freeden and Gerhards 2013) that if $\left \vert \tau \right \vert$ is sufficiently small, then the regularity of $\Sigma$ implies the regularity of $\Sigma (\tau )$. According to our regularity assumptions, imposed on $\Sigma$, the functions

$$\displaystyle{ (x,y)\mapsto \frac{\left \vert \nu (x) -\nu (y)\right \vert } {\left \vert x - y\right \vert },\quad (x,y) \in \Sigma \times \Sigma,\,x\not =y, }$$

(385)

and

$$\displaystyle{ (x,y)\mapsto \frac{\left \vert \nu (x) \cdot (x - y)\right \vert } {\left \vert x - y\right \vert ^{2}},\quad (x,y) \in \Sigma \times \Sigma,\,x\not =y, }$$

(386)

are bounded. Hence, there exists a constant M > 0 such that, for all $x,y \in \Sigma$,

$$\displaystyle\begin{array}{rcl} \left \vert \nu (x) -\nu (y)\right \vert \leq M\left \vert x - y\right \vert \,,& &{}\end{array}$$

(387)

$$\displaystyle\begin{array}{rcl} \left \vert \nu (x) \cdot (x - y)\right \vert \leq M\left \vert x - y\right \vert ^{2}\,.& &{}\end{array}$$

(388)

Moreover, it is easy to see that

$$\displaystyle{ \inf _{x,y\in \Sigma }\left \vert x +\tau \nu (x) - (y +\sigma \nu (y))\right \vert = \left \vert \tau -\sigma \right \vert }$$

(389)

provided that $\left \vert \tau \right \vert$ and $\left \vert \sigma \right \vert$ are sufficiently small.

In order to separate members of the class $c(\Sigma )$ of continuous vector fields on $\Sigma$ into their tangential and normal parts with respect to a regular surface, we introduce the projection operators p _nor and p _tan by

$$\displaystyle\begin{array}{rcl} p_{\mathrm{nor}}f(x)& =& (f(x) \cdot \nu (x))\nu (x),\quad x \in \Sigma,f \in \mathrm{ c}(\Sigma ),{}\end{array}$$

(390)

$$\displaystyle\begin{array}{rcl} p_{\mathrm{tan}}f(x)& =& f(x) - p_{\mathrm{nor}}f(x),\quad x \in \Sigma,f \in \mathrm{ c}(\Sigma )\,.{}\end{array}$$

(391)

Hence, the corresponding subspaces of $\mathrm{c}(\Sigma )$ are given by

$$\displaystyle\begin{array}{rcl} \mathrm{c}_{\mathrm{nor}}(\Sigma )& =& \{f \in \mathrm{ c}(\Sigma )\vert f = p_{\mathrm{nor}}f\},{}\end{array}$$

(392)

$$\displaystyle\begin{array}{rcl} \mathrm{c}_{\mathrm{tan}}(\Sigma )& =& \{f \in \mathrm{ c}(\Sigma )\vert f = p_{\mathrm{tan}}f\}.{}\end{array}$$

(393)

The spaces $\mathrm{c}_{\mathrm{nor}}^{(p)}(\Sigma )$ and $\mathrm{c}_{\mathrm{tan}}^{(p)}(\Sigma )$, 0 ≤ p ≤ ∞ are definable in the same fashion.

The set of vector fields $f: \Sigma \rightarrow \mathbb{R}$ which are measurable and for which

$$\displaystyle{ \left \Vert f\right \Vert _{l^{p}(\Sigma )} = \left (\int _{\Sigma }\left \vert f(x)\right \vert ^{p}\,\mathrm{d}\omega (x)\right )^{\frac{1} {p} } < \infty, }$$

(394)

is denoted by $l^{p}(\Sigma )$, where dω(x) denotes the surface element on $\Sigma$ (note that in the case of $\Sigma = \Omega _{R}$ with radius R > 0, we write dω _R (x) instead of $\mathrm{d}\omega _{\Omega _{R}}(x)$ and dω instead of dω ₁ in the case R = 1).

The definition of the normal and the tangential operator can be extended in canonical way to vector fields in $l^{2}(\Sigma )$ by a density argument. Hence, we define

$$\displaystyle\begin{array}{rcl} l_{\mathrm{nor}}^{2}(\Sigma )& =& \{f \in l^{2}(\Sigma )\vert f = p_{\mathrm{ nor}}f\},{}\end{array}$$

(395)

$$\displaystyle\begin{array}{rcl} l_{\mathrm{tan}}^{2}(\Sigma )& =& \{f \in l^{2}(\Sigma )\vert f = p_{\mathrm{ tan}}f\}.{}\end{array}$$

(396)

Clearly, we have the orthogonal decomposition

$$\displaystyle{ l^{2}(\Sigma ) = l_{\mathrm{ nor}}^{2}(\Sigma ) \oplus l_{\mathrm{ tan}}^{2}(\Sigma )\,. }$$

(397)

Kernel Functions

When we introduce layer potentials with respect to a regular surface, scalar- and tensor-valued kernel functions defined on the regular surface are of particular importance. Thus, they are discussed in the following.

Definition 15.

Let $\Sigma$ be a regular surface. A bivariate scalar kernel function $K: \mathbb{R}^{3} \times \mathbb{R}^{3} \rightarrow \mathbb{R}$ is called weakly continuous if K is continuous for all $x,y \in \Sigma$ with x ≠ y, and there exist positive constants M and 0 < α ≤ 2 such that, for all $x,y \in \Sigma$, x ≠ y, we have

$$\displaystyle{ \left \vert K(x,y)\right \vert \leq M \frac{1} {\left \vert x - y\right \vert ^{2-\alpha }}\,. }$$

(398)

The pair $\left < \mathrm{C}^{(0)}(\Sigma ),\mathrm{C}^{(0)}(\Sigma )\right >$ with $\Sigma$ being a regular surface, together with the $\mathop{\L }^{2}(\Sigma )$-inner product, is a dual system. Thus, the first requirements of the theorem of Fredholm (see, e.g., Kress 1989; Heuser 1992) is fulfilled. To finally apply this theorem, we need compact operators on the space $\mathrm{C}^{(0)}(\Sigma )$.

Theorem 22.

Let $\Sigma$ be a regular surface and let the integral operator $A:\mathrm{ C}^{(0)}(\Sigma ) \rightarrow \mathrm{ C}^{(0)}(\Sigma )$ be given by

$$\displaystyle{ (AF)(x) =\int _{\Sigma }K(x,y)F(y)\,\mathrm{d}\omega (y),\quad x \in \Sigma, }$$

(399)

where the kernel K is continuous or weakly singular. Then the operator A is compact on $\mathrm{C}^{(0)}(\Sigma )$ .

For a proof of this theorem, the reader is referred to, e.g., Kupradze (1965) and Kress (1989).

Theorem 23.

Let $\Sigma$ be a regular surface. Assume the kernel K to be weakly continuous with constant α. Furthermore, let us assume that there exists an $N \in \mathbb{N}$ and a constant M > 0 such that

$$\displaystyle{ \left \vert K(x_{1},y) - K(x_{2},y)\right \vert \leq M\sum _{j=1}^{N} \frac{\left \vert x_{1} - x_{2}\right \vert ^{j}} {\left \vert x_{1} - y\right \vert ^{2+j-\alpha }} }$$

(400)

for all $x_{1},x_{2} \in \mathbb{R}^{3}$ and $y \in \Sigma$ with $2\left \vert x_{1} - x_{2}\right \vert \leq \left \vert x_{1} - y\right \vert$ . Then the scalar potential $U: \mathbb{R}^{3} \rightarrow \mathbb{R}$ formally defined by

$$\displaystyle{ U(x) =\int _{\Sigma }K(x,y)F(y)\,\mathrm{d}\omega (y),\quad x \in \mathbb{R}^{3}, }$$

(401)

with layer density $F \in \mathrm{ C}^{(0)}(\Sigma )$ belongs to the Hölder space $\mathrm{C}^{(0,\beta )}(\mathbb{R}^{3})$ for all 0 < β ≤α if 0 < α < 1, for all 0 < β < 1 if α = 1, and for all 0 < β ≤ 1 if 1 < α < 2.

Theorem 24.

Let $\Sigma$ be a regular surface and let $x_{0} \in \Sigma$ . Assume the kernel K to be continuous for all $x \in D_{\tau _{0}}$ , $y \in \Sigma$ , x≠y with $D_{\tau _{0}}$ given by

$$\displaystyle{ D_{\tau _{0}} =\{ y = x +\tau \nu (x)\,\vert \,x \in \Sigma,\,\left \vert \tau \right \vert \leq \left \vert \tau _{0}\right \vert \}, }$$

(402)

and assume that there exists a constant C > 0 such that for all $x \in D_{\tau _{0}}$ , $y \in \Sigma$ , x≠y, we have

$$\displaystyle{ \left \vert K(x,y)\right \vert \leq \frac{C} {\left \vert x - y\right \vert ^{2}}\,. }$$

(403)

Furthermore, let us assume that there exists an $N \in \mathbb{N}$ such that

$$\displaystyle{ \left \vert K(x_{1},y) - K(x_{2},y)\right \vert \leq C\sum _{j=1}^{N} \frac{\left \vert x_{1} - x_{2}\right \vert ^{j}} {\left \vert x_{1} - y\right \vert ^{2+j}} }$$

(404)

for all $x_{1},x_{2} \in D_{\tau _{0}}$ , $y \in \Sigma$ , x≠y with $2\left \vert x_{1} - x_{2}\right \vert \leq \left \vert x_{1} - y\right \vert$ , and that

$$\displaystyle{ \left \vert \int _{\Sigma \setminus (B_{r}(z)\cap \Sigma )}K(x,y)\,\mathrm{d}\omega (y)\right \vert \leq C }$$

(405)

for all $z \in \Sigma$ , $x \in D_{\tau _{0}}$ and for all 0 < r < R, where R is chosen sufficiently small such that $B_{R}(z) \cap \Sigma$ is still connected. We formally define, for $F \in \mathrm{ C}^{(0,\alpha )}(\Sigma )$ ,

$$\displaystyle{ U(x) =\int _{\Sigma }K(x,y)\big(F(y) - F(z)\big)\,\mathrm{d}\omega (y),\quad x \in D_{\tau _{0}}\,. }$$

(406)

Then the potential U is continuous and belongs to the Hölder space $\mathrm{C}^{(0,\alpha )}(D_{\tau _{0}})$ .

The pair $\left < \mathrm{c}^{(0)}(\Sigma ),\mathrm{c}^{(0)}(\Sigma )\right >$ together with the $\mathop{\l }^{2}(\Sigma )$-inner product is a dual system. In order to use the theorem of Fredholm, we finally need compact operators on the space $\mathrm{c}^{(0)}(\Sigma )$. It is clear that Definition 15, Theorems 22, and 23 can canonically be extended to the case of a tensor kernel function k.

Definition 16.

Let $\Sigma$ be a regular surface. A tensorial kernel function $\mathbf{k}: \mathbb{R}^{3} \times \mathbb{R}^{3} \rightarrow \mathbb{R}^{3\times 3}$ is said to be weakly continuous if k is defined and continuous for all $x,y \in \Sigma$ with x ≠ y, and there exist positive constants M and 0 < α ≤ 2 such that for all $x,y \in \Sigma$, x ≠ y, we have

$$\displaystyle{ \left \vert \mathbf{k}(x,y)\right \vert \leq M \frac{1} {\left \vert x - y\right \vert ^{2-\alpha }}. }$$

(407)

Corollary 9.

Let $\Sigma$ be a regular surface.

• 1. Let the integral operator $A:\mathrm{ c}^{(0)}(\Sigma ) \rightarrow \mathrm{ c}^{(0)}(\Sigma )$ be given by

  $$\displaystyle{ (Af)(x) =\int _{\Sigma }\mathbf{k}(x,y)f(y)\,\mathrm{d}\omega (y),\quad x \in \Sigma, }$$

  (408)

  where the tensor kernel k is continuous or weakly singular. Then the operator A is compact on $\mathrm{c}^{(0)}(\Sigma )$ .

• 2. Let us assume that the tensor kernel k be weakly continuous with constant α. Furthermore, let us assume that there exists $N \in \mathbb{N}$ and a constant M > 0 such that

  $$\displaystyle{ \left \vert \mathbf{k}(x_{1},y) -\mathbf{k}(x_{2},y)\right \vert \leq M\sum _{j=1}^{N} \frac{\left \vert x_{1} - x_{2}\right \vert ^{j}} {\left \vert x_{1} - y\right \vert ^{2+j-\alpha }} }$$

  (409)

  for all $x_{1},x_{2} \in \mathbb{R}^{3}$ and $y \in \Sigma$ with $2\left \vert x_{1} - x_{2}\right \vert \leq \left \vert x_{1} - y\right \vert$ . Then the vector potential $u: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ defined by

  $$\displaystyle{ u(x) =\int _{\Sigma }\mathbf{k}(x,y)f(y)\,\mathrm{d}\omega (y),\quad x \in \mathbb{R}^{3}, }$$

  (410)

  with layer density $f \in \mathrm{ c}^{(0)}(\Sigma )$ is an element of the Hölder space $\mathrm{c}^{(0,\beta )}(\mathbb{R}^{3})$ with the same relations between β and α as given in Theorem  23 .

• 3. Let us assume that the tensor kernel k be continuous for all $x \in D_{\tau _{0}}$ , $y \in \Sigma$ , x≠y, where $D_{\tau _{0}}$ is defined in Theorem  24 , and let us assume that there exists a constant C > 0 such that for all $x \in D_{\tau _{0}}$ , $y \in \Sigma$ , x≠y, we have

  $$\displaystyle{ \left \vert \mathbf{k}(x,y)\right \vert \leq \frac{C} {\left \vert x - y\right \vert ^{2}}\,. }$$

  (411)

  Furthermore, assume that there exists $N \in \mathbb{N}$ such that

  $$\displaystyle{ \left \vert \mathbf{k}(x_{1},y) -\mathbf{k}(x_{2},y)\right \vert \leq C\sum _{j=1}^{N} \frac{\left \vert x_{1} - x_{2}\right \vert ^{j}} {\left \vert x_{1} - y\right \vert ^{2+j}} }$$

  (412)

  for all $x_{1},x_{2} \in D_{\tau _{0}}$ , $y \in \Sigma$ , x≠y with $2\left \vert x_{1} - x_{2}\right \vert \leq \left \vert x_{1} - y\right \vert$ , and that

  $$\displaystyle{ \left \vert \int _{\Sigma \setminus (B_{r}(z)\cap \Sigma )}\mathbf{k}(x,y)\,\mathrm{d}\omega (y)\right \vert \leq C }$$

  (413)

  for all $z \in \Sigma$ , $x \in D_{\tau _{0}}$ and for all 0 < r < R. We define, for $f \in \mathrm{ c}^{(0,\alpha )}(\Sigma )$ , the vector potential u by

  $$\displaystyle{ u(x) =\int _{\Sigma }\mathbf{k}(x,y)\big(f(y) - f(z)\big)\,\mathrm{d}\omega (y),\quad x \in D_{\tau _{0}}\,. }$$

  (414)

  Then the vector potential u is continuous and belongs to the space $\mathrm{c}^{(0,\alpha )}(D_{\tau _{0}})$ .

Scaling Functions and Wavelets

In the following, we present some helpful auxiliary material. In particular, we are interested in the explicit representations of the $\Sigma$-tensor scaling functions and wavelets presented in Sect. 4.1 as well as those of the second kind as introduced in Sect. 4.4. Furthermore, we give some graphical illustrations of the tensor scaling functions and wavelets.

4.1 Scaling Functions

Tensorial scaling functions on regular surfaces have been introduced in Corollary 7 and Definition 5. Their explicit representations are, for the tensorial case, given by

$$\displaystyle{ \boldsymbol{\Phi }_{\pm \tau }^{1}(x,y) = -\frac{1} {8\pi }\left ( \frac{\mathbf{i}} {\left \vert x \pm \tau \nu (x) - y\right \vert } + \frac{(x \pm \tau \nu (x) - y) \otimes (x \pm \tau \nu (x) - y)} {\left \vert x \pm \tau \nu (x) - y\right \vert ^{3}} \right )\,, }$$

(415)

$$\displaystyle{ \boldsymbol{\Phi }_{\pm \tau }^{2}(x,y) = \frac{3} {2\pi }\left (\frac{(x \pm \tau \nu (x) - y) \otimes (x \pm \tau \nu (x) - y)} {\left \vert x \pm \tau \nu (x) - y\right \vert ^{5}} \left ((x \pm \tau \nu (x) - y) \cdot \nu (y)\right )\right. }$$

(416)

$$\displaystyle{ \qquad + \left.\frac{(x - y) \otimes (x - y)} {\left \vert x - y\right \vert ^{5}} \left ((x - y) \cdot \nu (y)\right )\right )\,, }$$

(417)

$$\displaystyle{ \boldsymbol{\Phi }_{\pm \tau }^{3}(x,y) = \frac{3} {2\pi }\left (\frac{(x \pm \tau \nu (x) - y) \otimes (x \pm \tau \nu (x) - y)} {\left \vert x \pm \tau \nu (x) - y\right \vert ^{5}} \left ((x \pm \tau \nu (x) - y) \cdot \nu (x)\right )\right. }$$

(418)

$$\displaystyle{ \qquad + \left.\frac{(x - y) \otimes (x - y)} {\left \vert x - y\right \vert ^{5}} \left ((x - y) \cdot \nu (x)\right )\right )\,, }$$

(419)

$$\displaystyle{ \boldsymbol{\Phi }_{\tau }^{4}(x,y) = -\frac{1} {8\pi }\left (\left ( \frac{\mathbf{i}} {\left \vert x +\tau \nu (x) - y\right \vert } + \frac{(x +\tau \nu (x) - y) \otimes (x +\tau \nu (x) - y)} {\left \vert x +\tau \nu (x) - y\right \vert ^{3}} \right )\right. }$$

(420)

$$\displaystyle{ \qquad \left.-\left ( \frac{\mathbf{i}} {\left \vert x -\tau \nu (x) - y\right \vert } + \frac{(x -\tau \nu (x) - y) \otimes (x -\tau \nu (x) - y)} {\left \vert x -\tau \nu (x) - y\right \vert ^{3}} \right )\right )\,, }$$

(421)

$$\displaystyle{ \boldsymbol{\Phi }_{\tau }^{5}(x,y) = \frac{3} {4\pi }\left (\left (\frac{(x +\tau \nu (x) - y) \otimes (x +\tau \nu (x) - y)} {\left \vert x +\tau \nu (x) - y\right \vert ^{5}} \left ((x +\tau \nu (x) - y) \cdot \nu (y)\right )\right )\right. }$$

(422)

$$\displaystyle{ \qquad \left.-\left (\frac{(x -\tau \nu (x) - y) \otimes (x -\tau \nu (x) - y)} {\left \vert x -\tau \nu (x) - y\right \vert ^{5}} \left ((x -\tau \nu (x) - y) \cdot \nu (y)\right )\right )\right ), }$$

(423)

$$\displaystyle{ \boldsymbol{\Phi }_{\tau }^{6}(x,y) = \frac{3} {4\pi }\left (\left (\frac{(x +\tau \nu (x) - y) \otimes (x +\tau \nu (x) - y)} {\left \vert x +\tau \nu (x) - y\right \vert ^{5}} \left ((x +\tau \nu (x) - y) \cdot \nu (x)\right )\right )\right. }$$

(424)

$$\displaystyle{ \qquad \left.-\left (\frac{(x -\tau \nu (x) - y) \otimes (x -\tau \nu (x) - y)} {\left \vert x -\tau \nu (x) - y\right \vert ^{5}} \left ((x -\tau \nu (x) - y) \cdot \nu (x)\right )\right )\right ), }$$

(425)

where τ > 0 is the scale parameter and $x,y \in \Sigma$. Graphical illustrations of the $\Sigma$-tensor scaling function of type i = 5, $\boldsymbol{\Phi }_{\tau }^{5}$, for different values of the scale parameter τ can be found in Fig. 7.

Fig. 7

$\Sigma$-tensor scaling function $\boldsymbol{\Phi }_{\tau }^{5}$ on the unit sphere $\Omega$ for different values of the scale parameter τ. The left figures show the Frobenius norms of the tensor scaling functions $\boldsymbol{\Phi }_{\tau }^{5}(x,y)$ for a fixed value $y \in \Omega$ and variable $x \in \Omega$. The right figures show a sectional cut of the left one along the equator

4.2 Wavelet Functions

The $\Sigma$-tensor wavelet functions corresponding to the $\Sigma$-tensor scaling functions have been, for the weight function α(τ) = τ ⁻¹, defined in Definition 6 by

$$\displaystyle{ \boldsymbol{\Psi }_{\tau }^{i}(x,y) = -\tau \frac{\mathrm{d}} {\mathrm{d}\tau }\boldsymbol{\Phi }_{\tau }^{i}(x,y),\qquad x,y \in \Sigma \,. }$$

(426)

Their explicit representations can be calculated to be

$$\displaystyle\begin{array}{rcl} & & \boldsymbol{\Psi }_{\pm \tau }^{2}(x,y) \\ & & \qquad = -\tau \frac{\mathrm{d}} {\mathrm{d}\tau }\left (2\mathbf{k}(x \pm \tau \nu (x),y) - 2\mathbf{k}(x,y)\right ), \\ & & \qquad = -\frac{3\tau } {4\pi }\left [5\left (\frac{(x \pm \tau \nu (x) - y) \otimes (x \pm \tau \nu (x) - y)} {\left \vert x \pm \tau \nu (x) - y\right \vert ^{7}} \left ((x \pm \tau \nu (x) - y) \cdot \nu (y)\right )^{2}\right )\right. \\ & & \qquad -\left (\frac{(x \pm \tau \nu (x) - y) \otimes \nu (x) +\nu (x) \otimes (x \pm \tau \nu (x) - y)} {\left \vert x \pm \tau \nu (x) - y\right \vert ^{5}} \left ((x \pm \tau \nu (x) - y) \cdot \nu (y)\right )\right ) \\ & & \qquad -\left (\frac{(x \pm \tau \nu (x) - y) \otimes (x \pm \tau \nu (x) - y)} {\left \vert x \pm \tau \nu (x) - y\right \vert ^{5}} \left (\nu (x) \cdot \nu (y)\right )\right ) \\ & & \qquad + 5\left (\frac{(x - y) \otimes (x - y)} {\left \vert x - y\right \vert ^{7}} \left ((x - y) \cdot \nu (y)\right )^{2}\right ) \\ & & \qquad -\left (\frac{(x - y) \otimes \nu (x) +\nu (x) \otimes (x - y)} {\left \vert x - y\right \vert ^{5}} \left ((x - y) \cdot \nu (y)\right )\right ) \\ & & \qquad -\left.\left (\frac{(x - y) \otimes (x - y)} {\left \vert x - y\right \vert ^{5}} \left (\nu (x) \cdot \nu (y)\right )\right )\right ]\,, {}\end{array}$$

(427)

$$\displaystyle\begin{array}{rcl} & & \boldsymbol{\Psi }_{\tau }^{3}(x,y) \\ & & \qquad = -\frac{3\tau } {4\pi }\left [5\left (\frac{(x \pm \tau \nu (x) - y) \otimes (x \pm \tau \nu (x) - y)} {\left \vert x \pm \tau \nu (x) - y\right \vert ^{7}} \left ((x \pm \tau \nu (x) - y) \cdot \nu (x)\right )^{2}\right )\right. \\ & & \qquad -\left (\frac{(x \pm \tau \nu (x) - y) \otimes \nu (x) +\nu (x) \otimes (x \pm \tau \nu (x) - y)} {\left \vert x \pm \tau \nu (x) - y\right \vert ^{5}} \left ((x \pm \tau \nu (x) - y) \cdot \nu (x)\right )\right ) \\ & & \qquad -\left (\frac{(x \pm \tau \nu (x) - y) \otimes (x \pm \tau \nu (x) - y)} {\left \vert x \pm \tau \nu (x) - y\right \vert ^{5}} \right ) \\ & & \qquad + 5\left (\frac{(x - y) \otimes (x -\tau \nu (x) - y)} {\left \vert x - y\right \vert ^{7}} \left ((x - y) \cdot \nu (x)\right )^{2}\right ) \\ & & \qquad -\left (\frac{(x - y) \otimes \nu (x) +\nu (x) \otimes (x - y)} {\left \vert x - y\right \vert ^{5}} \left ((x - y) \cdot \nu (x)\right )\right ) \\ & & \qquad -\left.\left (\frac{(x - y) \otimes (x - y)} {\left \vert x - y\right \vert ^{5}} \right )\right ]\,, {}\end{array}$$

(428)

$$\displaystyle\begin{array}{rcl} & & \boldsymbol{\Psi }_{\tau }^{5}(x,y) \\ & & \qquad = -\tau \frac{\mathrm{d}} {\mathrm{d}\tau }\left (2\mathbf{k}(x +\tau \nu (x),y) - 2\mathbf{k}(x -\tau \nu (x),y)\right ), \\ & & \qquad = -\frac{3\tau } {4\pi }\left [5\left (\frac{(x +\tau \nu (x) - y) \otimes (x +\tau \nu (x) - y)} {\left \vert x +\tau \nu (x) - y\right \vert ^{7}} \left ((x +\tau \nu (x) - y) \cdot \nu (y)\right )^{2}\right )\right. \\ & & \qquad -\left (\frac{(x +\tau \nu (x) - y) \otimes \nu (x) +\nu (x) \otimes (x +\tau \nu (x) - y)} {\left \vert x +\tau \nu (x) - y\right \vert ^{5}} \left ((x +\tau \nu (x) - y) \cdot \nu (y)\right )\right ) \\ & & \qquad -\left (\frac{(x +\tau \nu (x) - y) \otimes (x +\tau \nu (x) - y)} {\left \vert x +\tau \nu (x) - y\right \vert ^{5}} \left (\nu (x) \cdot \nu (y)\right )\right ) \\ & & \qquad + 5\left (\frac{(x -\tau \nu (x) - y) \otimes (x -\tau \nu (x) - y)} {\left \vert x -\tau \nu (x) - y\right \vert ^{7}} \left ((x -\tau \nu (x) - y) \cdot \nu (y)\right )^{2}\right ) \\ & & \qquad -\left (\frac{(x -\tau \nu (x) - y) \otimes \nu (x) +\nu (x) \otimes (x -\tau \nu (x) - y)} {\left \vert x -\tau \nu (x) - y\right \vert ^{5}} \left ((x -\tau \nu (x) - y) \cdot \nu (y)\right )\right ) \\ & & \qquad -\left.\left (\frac{(x -\tau \nu (x) - y) \otimes (x -\tau \nu (x) - y)} {\left \vert x -\tau \nu (x) - y\right \vert ^{5}} \left (\nu (x) \cdot \nu (y)\right )\right )\right ]\,, {}\end{array}$$

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$$\displaystyle\begin{array}{rcl} & & \boldsymbol{\Psi }_{\tau }^{6}(x,y) \\ & & \qquad = -\frac{3\tau } {4\pi }\left [5\left (\frac{(x +\tau \nu (x) - y) \otimes (x +\tau \nu (x) - y)} {\left \vert x +\tau \nu (x) - y\right \vert ^{7}} \left ((x +\tau \nu (x) - y) \cdot \nu (x)\right )^{2}\right )\right. \\ & & \qquad -\left (\frac{(x +\tau \nu (x) - y) \otimes \nu (x) +\nu (x) \otimes (x +\tau \nu (x) - y)} {\left \vert x +\tau \nu (x) - y\right \vert ^{5}} \left ((x +\tau \nu (x) - y) \cdot \nu (x)\right )\right ) \\ & & \qquad -\left (\frac{(x +\tau \nu (x) - y) \otimes (x +\tau \nu (x) - y)} {\left \vert x +\tau \nu (x) - y\right \vert ^{5}} \right ) \\ & & \qquad + 5\left (\frac{(x -\tau \nu (x) - y) \otimes (x -\tau \nu (x) - y)} {\left \vert x -\tau \nu (x) - y\right \vert ^{7}} \left ((x -\tau \nu (x) - y) \cdot \nu (x)\right )^{2}\right ) \\ & & \qquad -\left (\frac{(x -\tau \nu (x) - y) \otimes \nu (x) +\nu (x) \otimes (x -\tau \nu (x) - y)} {\left \vert x -\tau \nu (x) - y\right \vert ^{5}} \left ((x -\tau \nu (x) - y) \cdot \nu (x)\right )\right ) \\ & & \qquad -\left.\left (\frac{(x -\tau \nu (x) - y) \otimes (x -\tau \nu (x) - y)} {\left \vert x -\tau \nu (x) - y\right \vert ^{5}} \right )\right ]\,, {}\end{array}$$

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for $x,y \in \Sigma$ and τ > 0. In Sect. 4.3, we introduced a scale discretization which led to scale discrete $\Sigma$-tensor wavelet functions of type i. They are given, for $i = 1,\ldots,6$, by

$$\displaystyle{ \boldsymbol{\Psi }_{\tau _{j}}^{i}(x,y) = \boldsymbol{\Phi }_{\tau _{ j+1}}^{i}(x,y) -\boldsymbol{\Phi }_{\tau _{ j}}^{i}(x,y),\qquad x,y \in \Sigma \,, }$$

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where the sequence $\{\tau _{j}\}_{j\in \mathbb{Z}}$ is a discretization of the scale interval (0, ∞). The graphical illustrations of the scale discrete $\Sigma$-tensor scaling function of type i = 5, $\boldsymbol{\Psi }_{\tau _{ j}}^{5}$, can be found in Fig. 8.

Fig. 8

Discrete $\Sigma$-tensor wavelet function $\boldsymbol{\Psi }_{\tau _{j}}^{5}$ on the unit sphere $\Omega$ for different values of the discrete scale parameter $j \in \mathbb{Z}$. The left figures show the Frobenius norms of the tensor kernel $\boldsymbol{\Psi }_{\tau _{ j}}^{5}(x,y)$ for a fixed value $y \in \Omega$ and variable $x \in \Omega$. The right figures show the scalar value $\varepsilon ^{r}(x) \cdot \left (\boldsymbol{\Psi }_{\tau _{j}}^{5}(x,y)\varepsilon ^{r}(x)\right )$, 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

4.3 Scaling Functions of the Second Kind

$\Sigma$-tensor scaling functions of the second kind have been defined in Definition 10. For τ > 0, they are defined by

$$\displaystyle{ \tilde{\boldsymbol{\Phi }}_{\tau }^{i}(x,y) = \Phi _{\tau }^{i}(x,y)\,\mathbf{i}\,,\qquad x,y \in \Sigma,\quad \,i = 2,3,5,6, }$$

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where the scalar kernels $\Phi _{\tau }^{i}$, i = 2, 3, 5, 6 are given by

$$\displaystyle{ \Phi _{\pm \tau }^{2}(x,y) = \frac{1} {2\pi }\left (\frac{(x \pm \tau \nu (x) - y) \cdot \nu (y)} {\left \vert x \pm \tau \nu (x) - y\right \vert ^{3}} -\frac{(x - y) \cdot \nu (y)} {\left \vert x - y\right \vert ^{3}} \right )\,, }$$

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$$\displaystyle{ \Phi _{\pm \tau }^{3}(x,y) = \frac{1} {2\pi }\left (\frac{(x \pm \tau \nu (x) - y) \cdot \nu (x)} {\left \vert x \pm \tau \nu (x) - y\right \vert ^{3}} -\frac{(x - y) \cdot \nu (x)} {\left \vert x - y\right \vert ^{3}} \right )\,, }$$

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$$\displaystyle{ \Phi _{\tau }^{5}(x,y) = \frac{1} {4\pi }\left (\frac{(x +\tau \nu (x) - y) \cdot \nu (y)} {\left \vert x +\tau \nu (x) - y\right \vert ^{3}} -\frac{(x -\tau \nu (x) - y) \cdot \nu (y)} {\left \vert x -\tau \nu (x) - y\right \vert ^{3}} \right )\,, }$$

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$$\displaystyle{ \Phi _{\tau }^{6}(x,y) = \frac{1} {4\pi }\left (\frac{(x +\tau \nu (x) - y) \cdot \nu (x)} {\left \vert x +\tau \nu (x) - y\right \vert ^{3}} -\frac{(x -\tau \nu (x) - y) \cdot \nu (x)} {\left \vert x -\tau \nu (x) - y\right \vert ^{3}} \right )\,, }$$

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for $x,y \in \Sigma$. Graphical illustrations of the $\Sigma$-tensor scaling function of the second kind of type i = 5, $\tilde{\boldsymbol{\Phi }}_{\tau }^{5}$, for different values of the scale parameter $\tau$ can be found in Fig. 9.

Fig. 9

$\Sigma$-tensor scaling function of the second kind $\tilde{\boldsymbol{\Phi }}_{\tau }^{5}$ on the unit sphere $\Omega$ for different values of the scale parameter τ. The left figures show the Frobenius norms of the tensor scaling functions $\tilde{\boldsymbol{\Phi }}_{\tau _{j}}^{5}(x,y)$ for a fixed value $y \in \Omega$ and variable $x \in \Omega$. The right figures show a sectional cut of the left one along the equator

4.4 Wavelet Functions of the Second Kind

The $\Sigma$-tensor wavelet functions of the second kind corresponding to the $\Sigma$-tensor scaling functions of the second kind presented in Appendix C.3 have been, for the weight function α(τ) = τ ⁻¹, τ > 0 and i = 2, 3, 5, 6, defined in Definition 11 by

$$\displaystyle{ \boldsymbol{\Psi }_{\tau }^{i}(x,y) = -\tau \frac{\mathrm{d}} {\mathrm{d}\tau }\boldsymbol{\Phi }_{\tau }^{i}(x,y),\qquad x,y \in \Sigma \,. }$$

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They can be calculated explicitly using the representations of the tensor scaling functions of the second kind given in Definition 10. It easily follows that

$$\displaystyle{ \tilde{\boldsymbol{\Psi }}_{\tau }^{i}(x,y) = \Psi _{\tau }^{i}(x,y)\,\mathbf{i}\,,\qquad x,y \in \Sigma,\quad \,i = 2,3,5,6, }$$

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with the scalar kernels $\Psi _{\tau }^{i}: \Sigma \times \Sigma \rightarrow \mathbb{R}$ given by

$$\displaystyle{ \Psi _{\tau }^{i}(x,y) = -\frac{1} {\alpha (\tau )} \frac{\mathrm{d}} {\mathrm{d}\tau }\Phi _{\tau }^{i}(x,y),\qquad x,y \in \Sigma, }$$

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and the scalar kernels $\Phi _{\tau }^{i}$, i = 2, 3, 5, 6, given in Appendix C. Explicit representations of the scalar wavelet functions $\Psi _{\tau }^{i}$ can be calculated to be (see also Freeden and Mayer 2003)

$$\displaystyle\begin{array}{rcl} \Psi _{\pm \tau }^{2}(x,y)& =& \frac{-\tau } {2\pi } \frac{\nu (x) \cdot \nu (y)} {\vert x \pm \tau \nu (x) - y\vert ^{3}} \\ & & \quad + \frac{3\tau } {2\pi } \frac{((x \pm \tau \nu (x) - y) \cdot \nu (x))((x \pm \tau \nu (x) - y) \cdot \nu (y))} {\vert x \pm \tau \nu (x) - y\vert ^{5}} \,,{}\end{array}$$

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$$\displaystyle{ \Psi _{\pm \tau }^{3}(x,y) = \frac{-\tau } {2\pi } \frac{1} {\vert x \pm \tau \nu (x) - y\vert ^{3}} + \frac{3\tau } {2\pi } \frac{((x \pm \tau \nu (x) - y) \cdot \nu (x))^{2}} {\vert x \pm \tau \nu (x) - y\vert ^{5}} \,, }$$

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$$\displaystyle\begin{array}{rcl} \Psi _{\tau }^{5}(x,y)& =& \frac{-\tau } {4\pi } \left ( \frac{\nu (x) \cdot \nu (y)} {\vert x +\tau \nu (x) - y\vert ^{3}} + \frac{\nu (x) \cdot \nu (y)} {\vert x -\tau \nu (x) - y\vert ^{3}}\right ) \\ & & +\ \frac{3\tau } {4\pi }\left (\frac{((x +\tau \nu (x) - y) \cdot \nu (x))((x +\tau \nu (x) - y) \cdot \nu (y))} {\vert s +\tau \nu (x) - y\vert ^{5}} \right. \\ & & +\ \left.\frac{((x -\tau \nu (x) - y) \cdot \nu (x))((x -\tau \nu (x) - y) \cdot \nu (y))} {\vert x -\tau \nu (x) - y\vert ^{5}} \right )\,,{}\end{array}$$

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$$\displaystyle\begin{array}{rcl} \Psi _{\tau }^{6}(x,y)& =& \frac{-\tau } {4\pi } \left ( \frac{1} {\vert x +\tau \nu (x) - y\vert ^{3}} + \frac{1} {\vert x -\tau \nu (x) - y\vert ^{3}}\right ) \\ & & \qquad +\ \frac{3\tau } {4\pi }\left (\frac{((x +\tau \nu (x) - y) \cdot \nu (x))^{2}} {\vert x +\tau \nu (x) - y\vert ^{5}} + \frac{((x -\tau \nu (x) - y) \cdot \nu (x))^{2}} {\vert x -\tau \nu (x) - y\vert ^{5}} \right )\,,{}\end{array}$$

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for τ > 0 and $x,y \in \Sigma$.

In Sect. 4.4, we have also defined scale discrete $\Sigma$-tensor wavelet functions of the second kind of type i. They are given, for $i = 1,\ldots,6$, by

$$\displaystyle{ \tilde{\boldsymbol{\Psi }}_{\tau _{j}}^{i}(x,y) =\tilde{\boldsymbol{ \Phi }}_{\tau _{ j+1}}^{i}(x,y) -\tilde{\boldsymbol{ \Phi }}_{\tau _{ j}}^{i}(x,y),\qquad x,y \in \Sigma \,, }$$

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where the sequence $\{\tau _{j}\}_{j\in \mathbb{Z}}$ is a discretization of the scale interval (0, ∞). Some graphical illustrations of the scale discrete $\Sigma$-tensor scaling function of the second kind of type i = 5, $\tilde{\boldsymbol{\Psi }}_{\tau _{j}}^{5}$, can be found in Fig. 10.

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