1 Introduction

In our mathematical models, we often assume bandlimitedness for physical or computational reasons, yet also wish that our solutions be localized, with respect to their energy, inside a finite spatial region. Since these are mutually exclusive conditions [31], we need to resort to bandlimited functions with an optimal spatial localization. The goal of the so-called spatial concentration problem [22, p. 75] is to find such functions, and since its first thorough investigation by Slepian, Landau and Pollak in one and multiple Cartesian dimensions [14, 15, 30, 32], it has been revisited many times, including solutions for spherical and planar regions of arbitrary shape [1, 10, 26, 28, 29].

Each individual concentration problem gives rise to an orthogonal set of well-localized functions, which now we refer to under the common name Slepian functions. They have enjoyed increasing popularity in applications involving signal processing, function representation and approximation or the solution of inverse problems. In particular, the scalar spherical Slepian functions have been utilized, for instance, in geodesy and geophysics [1, 11, 26–29], cosmology [5, 6], computer science [16] and mathematics [19].

While they have been widely applied in the last two decades, it was not until recently that the theory of vector Slepian functions began to mature. The first successful construction of bandlimited vector fields, localized to a spherical cap, was reported in the context of biomedical science [18], followed by an application in physical optics [13]. Recently, a more general treatment of the vector spherical concentration problem has been published [23, 24], however, the question on the existence of a commuting differential operator for the spherical cap has been left unresolved. This question is important for the following reason. Slepian functions of a particular problem are eigenfunctions of the so-called concentration operator associated with the spatial region of interest, an integral operator exhibiting a peculiar step-like eigenvalue spectrum. This property makes the direct calculation of its eigenfunctions numerically difficult [4].

A particularly important result of Slepian and his colleagues was the introduction of a Sturm–Liouville differential operator that commutes with the corresponding concentration operator, hence they share a common set of eigenfunctions. Since the differential operator has a simple spectrum with more evenly distributed eigenvalues, it allows a more stable and accurate numerical computation of the eigenfunctions [4]. Two decades after Slepian and his colleagues’ seminal papers, Grünbaum, Longhi and Perlstadt found such a commuting differential operator for the scalar concentration problem within a spherical cap as well [10]. However, we are not aware of a similar proposal for the vectorial problem.

In this paper, we construct a differential operator commuting with the concentration operator of the vector case. This constitutes our main result. We restrict ourselves to tangential vector fields, since the concentration problem of the radial component is equivalent to the scalar concentration problem on the sphere [24], which has already been studied extensively [26, 28]. The key functions in our investigations are the novel mixed vector spherical harmonics \(\varvec{\mathbf {Q}}_{lm}^{\pm }(\theta , \phi )\) which enable us to reduce the vectorial problem to a scalar one involving the special functions \(F_{lm}\) of Sheppard and Török [25]. After that, the problem can be solved in an analogous way to its scalar counterpart [26, 28].

We note that we only consider spatially localized, bandlimited fields here. The symmetric problem of spectrally concentrated, spacelimited functions can be derived by exploiting the analogy to previously published results [24, 28].

2 Preliminaries: Associated Special Functions

In preparation for the concentration problem, we give a detailed survey on the essential properties of important special functions used in our investigations. We start by proving several fundamental relations for the functions \(F_{lm}\) of Sheppard and Török [25], which are needed for subsequent proofs. Next we introduce the mixed vector spherical harmonics and discuss their orthogonality properties. Finally we show how to expand an arbitrary tangential vector field in terms of the mixed vector spherical harmonics and define bandlimitedness in this context, so that we can use these new vector fields as basis functions in the treatment of the concentration problem in Sect. 3.

2.1 Special Functions \(F_{lm}\) of Sheppard and Török

Following Sheppard and Török [25], first we define the functions

$$\begin{aligned} F_{lm}(x) :=\frac{1}{\sqrt{l(l \!+\! 1)}} \left[ \sqrt{1 \!-\! x^2} \frac{\mathrm{d }\!^{}U_{lm}(x)}{\mathrm{d }\!x^{}} \!-\! \frac{m}{\sqrt{1 \!-\! x^2}} U_{lm}(x) \right] , \quad l \ge 1, \!-\!m \le l \le m,\nonumber \\ \end{aligned}$$
(1)

where \(U_{lm}\) are the normalized associated Legendre functions of integer degree \(l\) and order \(m\). The classical (unnormalized) associated Legendre functions \(P_l^m\) are thus normalized as [3, p. 757]

$$\begin{aligned} U_{lm}(x) :=c_{lm} P_{l}^{m}(x), \quad l \ge 0, -l \le m \le l, \end{aligned}$$
(2)

where

$$\begin{aligned} c_{lm} :=\sqrt{\frac{2l + 1}{2} \frac{(l - m)!}{(l + m)!}}. \end{aligned}$$
(3)

We can express the conditions for the integer indices \(l\) and \(m\) in definition (1) alternatively as \(-\infty < m < \infty \) and \(l \ge \ell _{m}\), where the minimal degree \(\ell _{m}\) for fixed \(m\) is

$$\begin{aligned} \ell _{m} :=\max (1, {|m |}). \end{aligned}$$
(4)

The functions \(F_{lm}\) are orthonormal with respect to the usual \(L^2\)-inner product for fixed \(m\) [25], i.e.

$$\begin{aligned} \int \limits _{-1}^{1} F_{lm}(x) F_{l'm}(x) \mathrm{d }\!x = \delta _{ll'}. \end{aligned}$$
(5)

This normalization differs by a constant factor from that used by Sheppard and Török [25].

Sometimes it is inconvenient that expression (1) is singular at \(x=\pm 1\) because of the factor \((1-x^2)^{-1/2}\). However, a singularity-free form can also be obtained by using two lesser-known recurrence relations [8, 17, 24]:

$$\begin{aligned} -\sqrt{1 - x^2}\, \frac{\mathrm{d }\!^{}U_{lm}(x)}{\mathrm{d }\!x^{}} = a_{lm}^{+} U_{l,\,m+1}(x) + a_{lm}^{-} U_{l,\,m-1}(x), \end{aligned}$$
(6a)

where

$$\begin{aligned} a_{lm}^{\pm } :=\pm \frac{\sqrt{(l \mp m)(l \pm m + 1)}}{2}, \end{aligned}$$
(6b)

and

$$\begin{aligned} \frac{m U_{lm}(x)}{\sqrt{1-x^2}} = b_{lm}^{+} U_{l-1,\,m+1}(x) + b_{lm}^{-} U_{l-1,\,m-1}(x), \end{aligned}$$
(7a)

where

$$\begin{aligned} b_{lm}^{\pm } :=- \sqrt{\frac{2l + 1}{2l - 1}} \frac{\sqrt{(l \mp m)(l \mp m - 1)}}{2}. \end{aligned}$$
(7b)

We note that everywhere in this paper, where the \(\pm \) or \(\mp \) signs occur, either the upper or the lower one has to be used consistently in the whole expression. The singularity-free form then reads

$$\begin{aligned} F_{lm}(x) \!=\! \frac{\!-\!1}{\sqrt{l(l\!+\!1)}} \left[ a_{lm}^{+} U_{l,\,m\!+\!1}(x) \!+\!a_{lm}^{-} U_{l,\,m\!-\!1}(x) \!+\! b_{lm}^{+} U_{l-1,\,m+1}(x) \!+\! b_{lm}^{-} U_{l-1,\,m\!-\!1}(x) \right] .\nonumber \\ \end{aligned}$$
(8)

It is straightforward to show using the symmetry relation [3, p. 743]

$$\begin{aligned} U_{l,\,-m}(x) = (-1)^{m} U_{lm}(x) \end{aligned}$$
(9)

that for the special case \(m=0\), the equivalence \(F_{l,\,0}(x) = -U_{l,\,1}(x)\) holds. Moreover, by using symmetry relation (9) and parity relation [3, p. 746]

$$\begin{aligned} U_{lm}(-x) = (-1)^{l+m} U_{lm}(x) \end{aligned}$$
(10)

we can obtain a symmetry relation for \(F_{lm}\), too:

$$\begin{aligned} F_{l,\,-m}(-x) = (-1)^{l+1} F_{lm}(x). \end{aligned}$$
(11)

This symmetry is apparent in Fig. 1 where a small subset of \(F_{lm}\) (\(l \le 3\)) is depicted.

Fig. 1
figure 1

The functions \(F_{lm}(x)\) for \(l \le 3\). For \({|m |} > 0\), the black and gray curves correspond to the function of positive and negative values of \(m\), respectively

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We have found the following recurrence relations for \(F_{lm}\), which simplify subsequent proofs:

$$\begin{aligned} \left[ x - \frac{m}{l(l+1)} \right] F_{lm}(x)&= \zeta _{lm} F_{l-1,\, m}(x) + \zeta _{l+1,\, m} F_{l+1, \,m}(x), \end{aligned}$$
(12)
$$\begin{aligned} (1 - x^2) \frac{\mathrm{d }\!^{}F_{lm}(x)}{\mathrm{d }\!x^{}}&= -l\left( x - \frac{m}{l^2} \right) F_{lm}(x) + (2l + 1) \zeta _{lm} F_{l-1,\, m}(x),\end{aligned}$$
(13)
$$\begin{aligned} (1 - x^2) \frac{\mathrm{d }\!^{}F_{lm}(x)}{\mathrm{d }\!x^{}}&= (l + 1)\left[ x - \frac{m}{(l+1)^2} \right] F_{lm}(x) - (2l + 1) \zeta _{l+1,\, m} F_{l+1,\, m}(x),\nonumber \\ \end{aligned}$$
(14)

where

$$\begin{aligned} \zeta _{lm} :=\frac{1}{l} \sqrt{\frac{(l+1)(l-1)(l+m)(l-m)}{(2l+1)(2l-1)}}. \end{aligned}$$
(15)

A proof for (12) and (13) is provided in Appendix 1a and 1b, respectively. Relation (14) is straightforward to prove by combining (12) and (13).

Relation (12) can be used to derive a Christoffel–Darboux formula [34, p. 42] specialized for \(F_{lm}\):

$$\begin{aligned} (x - x') \sum _{l=\ell _{m}}^{L} F_{lm}(x) F_{lm}(x') = \zeta _{L+1,\,m} \bigl [ F_{L+1,\,m}(x) F_{Lm}(x') - F_{Lm}(x) F_{L+1,\,m}(x') \bigr ].\nonumber \\ \end{aligned}$$
(16)

See Appendix 1c for the details. Moreover, we can formulate an addition theorem for \(F_{lm}(x)\) as well that will be used later in Sect. 3.2:

$$\begin{aligned} \sum _{m=-l}^{l} \bigl [ F_{lm}(x) \bigr ]^2 = \frac{2l + 1}{2}. \end{aligned}$$
(17)

The proof can be found in Appendix 1d.

At the interval endpoints \(x = \pm 1\), we can explicitly calculate the values of \(F_{lm}\). Considering that \(U_{lm}\) takes the values [3, p. 746]

$$\begin{aligned} U_{lm}(\pm 1) = \left\{ \begin{array}{ll} (\pm 1)^{l} c_{l,\,0} &{} \text {if}\, m = 0, \\ 0 &{} \text {otherwise}. \end{array}\right. \end{aligned}$$
(18)

and using the singularity-free form (8), we get

$$\begin{aligned} F_{lm}(1)&= \left\{ \begin{array}{ll} c_{l,\,0} &{} \text {if}\, m=1, \\ 0 &{} \text {otherwise,} \end{array}\right. \end{aligned}$$
(19a)
$$\begin{aligned} F_{lm}(-1)&= \left\{ \begin{array}{ll} (-1)^{l-1} c_{l,\,0} &{}\text {if}\, m=-1, \\ 0 &{} \text {otherwise.} \end{array}\right. \end{aligned}$$
(19b)

We can also formulate a closed-form expression for the \(l = l_m\) case. We combine expression [3, p. 745]

$$\begin{aligned} U_{mm}(x) = (-1)^{m} c_{mm}\, (2m - 1)!!\, (1 - x^2)^{m/2}, \quad m \ge 0, \end{aligned}$$
(20)

with definition (1) of \(F_{lm}\), and thus get

$$\begin{aligned} F_{\ell _{m},\,m}(x) = \left\{ \begin{array}{ll} (-1)^{m+1} (1 + x) \Phi _{m}(x) \qquad &{}\text {if}\, m > 0,\\ \frac{\sqrt{3}}{2} \sqrt{1 - x^2} \qquad &{}\text {if}\, m = 0, \\ (1 - x) \Phi _{{|m |}}(x) \qquad &{}\text {if}\, m < 0, \end{array}\right. \end{aligned}$$
(21)

where

$$\begin{aligned} \Phi _{m}(x) :=\sqrt{ \frac{m}{m + 1} }\, c_{mm}\,(2m-1)!!\, (1 - x^2)^{(m - 1)/2}, \end{aligned}$$
(22)

and \((2m - 1)!! = (2m - 1)(2m - 3) \cdots (1)\) is the double factorial. This expression together with recurrence relation (12) provides a stable and efficient method to evaluate \(F_{lm}(x)\) numerically. By setting \(F_{\ell _{m}-1,\,m} = 0\) and starting with the closed-form expression (21), recurrence relation (12) can be used repeatedly in the upward direction until one obtains \(F_{lm}(x)\).

Finally, we turn to the differential equation of the functions \(F_{lm}\). They satisfy the Sturm–Liouville equation

$$\begin{aligned} \frac{\mathrm{d }\!^{}}{\mathrm{d }\!x^{}} \left[ (1 - x^2) \frac{\mathrm{d }\!^{}u(x)}{\mathrm{d }\!x^{}} \right] - \frac{m^2 - 2mx + 1}{1 - x^2} u(x) = -l(l + 1) u(x), \end{aligned}$$
(23)

as proven in Appendix 1e. The operator on the left-hand side is closely related to the (surface) vector Laplace–Beltrami operator [33] on the unit sphere \(\Omega :=\left\{ (\theta , \phi ): 0 \le \theta \le \pi ,\ 0 \le \phi < 2\pi \right\} \).

Let \(\varvec{\mathbf {w}}(\theta , \phi ) :=\varvec{\mathbf {v}}(\theta ) \exp (\mathrm {i}m \phi )\) be a tangential separable vector field, where \(\mathrm {i}\) is the imaginary unit, \(\varvec{\mathbf {v}}(\theta ) :=v_{\theta }(\theta ) {\hat{\varvec{\mathbf {\theta }}}} + v_{\phi }(\theta ) {\hat{\varvec{\mathbf {\phi }}}}\), and \(\hat{\varvec{\mathbf {\theta }}}\) and \(\hat{\varvec{\mathbf {\phi }}}\) are unit vectors in the \(\theta \)- and \(\phi \)-directions, respectively, as shown in Fig. 2. When considering the action of the vector Laplace–Beltrami operator on \(\varvec{\mathbf {w}}\), we obtain the fixed-order vector Laplace–Beltrami operator

$$\begin{aligned} \nabla _{\Omega ,\,m}^2\, \varvec{\mathbf {v}} :=\left[ \nabla _{\Omega ,\,m}^2 v_{\theta } \!-\! \frac{v_{\theta }}{\sin ^2\theta } \!-\! 2\mathrm {i}m \frac{\cos \theta }{\sin ^2\theta } v_{\phi } \right] \hat{\varvec{\mathbf {\theta }}} \!+\! \left[ \nabla _{\Omega ,\,m}^2 v_{\phi } \!-\! \frac{v_{\phi }}{\sin ^2\theta }\, \!+\! 2\mathrm {i}m \frac{\cos \theta }{\sin ^2\theta } v_{\theta } \right] \hat{\varvec{\mathbf {\phi }}}\nonumber \\ \end{aligned}$$
(24)
Fig. 2
figure 2

Sketch of the unit sphere \(\Omega \) showing Cartesian unit vectors \(\hat{\varvec{\mathbf {x}}}\), \(\hat{\varvec{\mathbf {y}}}\), \(\hat{\varvec{\mathbf {z}}}\), and the polar and azimuthal unit vectors \(\hat{\varvec{\mathbf {\theta }}}\) and \(\hat{\varvec{\mathbf {\phi }}}\), respectively

Full size image

where

$$\begin{aligned} \nabla _{\Omega ,\,m}^2 :=\frac{1}{\sin \theta } \frac{\mathrm{d }\!^{}}{\mathrm{d }\!\theta ^{}} \left( \sin \theta \frac{\mathrm{d }\!^{}}{\mathrm{d }\!\theta ^{}} \right) - \frac{m^2}{\sin ^2\theta } \end{aligned}$$
(25)

is the fixed-order (surface) scalar Laplace–Beltrami operator. We can diagonalize \(\nabla _{\Omega ,\,m}^2\, \varvec{\mathbf {v}}\) by introducing the tangential basis vectors

$$\begin{aligned} \hat{\varvec{\mathbf {\tau }}}_{\pm } :=\frac{1}{\sqrt{2}} \bigl ( \hat{\varvec{\mathbf {\theta }}} \pm \mathrm {i}\, \hat{\varvec{\mathbf {\phi }}} \bigr ), \end{aligned}$$
(26)

which are orthogonal with respect to the complex dot product

$$\begin{aligned} \hat{\varvec{\mathbf {\tau }}}^{*}_{\pm } \cdot \hat{\varvec{\mathbf {\tau }}}_{\mp } = 0 \end{aligned}$$
(27)

where the asterisk denotes the complex conjugate.

Since \(\nabla _{\Omega ,\,m}^2 = \nabla _{\Omega ,\,-m}^2\), in this new basis, we can write (24) as

$$\begin{aligned} \nabla _{\Omega ,\,m}^2\, \varvec{\mathbf {v}} = \bigl ( \Delta _{\Omega ,m} v_{+} \bigr ) \hat{\varvec{\mathbf {\tau }}}_{+} + \bigl ( \Delta _{\Omega ,-m} v_{-} \bigl ) \hat{\varvec{\mathbf {\tau }}}_{-}, \end{aligned}$$
(28)

where \(\varvec{\mathbf {v}} = v_{+}(\theta ,\, \phi ) \hat{\varvec{\mathbf {\tau }}}_{+} + v_{-}(\theta , \phi ) \hat{\varvec{\mathbf {\tau }}}_{-}\) and

$$\begin{aligned} \Delta _{\Omega ,\,m} :=\nabla _{\Omega ,\,m}^2 - \frac{1 - 2 m \cos \theta }{\sin ^2 \theta }. \end{aligned}$$
(29)

Upon substituting \(x = \cos \theta \) in \(\Delta _{\Omega ,\,m}\), we regain the differential operator on the left-hand side of differential equation (23). Thus the functions \(F_{lm}\) are eigenfunctions of \(\Delta _{\Omega ,\,m}\):

$$\begin{aligned} \Delta _{\Omega ,\,m} F_{lm}(\cos \theta ) = -l(l + 1) F_{lm}(\cos \theta ). \end{aligned}$$
(30)

2.2 Mixed Vector Spherical Harmonics

It follows from the diagonal form (28) of the fixed-order vector Laplace–Beltrami operator and Eq. (30) that a new kind of vector spherical harmonics can be defined,

$$\begin{aligned} \varvec{\mathbf {Q}}_{lm}^{\pm }(\theta ,\,\phi ) :=F_{l,\,\pm m}(\cos \theta ) \frac{\exp (\mathrm {i}m \phi )}{\sqrt{2\pi }} \, \hat{\varvec{\mathbf {\tau }}}_{\pm }. \end{aligned}$$
(31)

We call these vector fields mixed vector spherical harmonics, because their connection to conventional vector spherical harmonics can be expressed as

$$\begin{aligned} \varvec{\mathbf {Q}}_{lm}^{\pm }(\theta ,\,\phi ) = \frac{(\pm 1)^{m+1}}{\sqrt{2}} \bigl [ \varvec{\mathbf {Y}}_{lm}(\theta ,\,\phi ) \pm \mathrm {i}\, \varvec{\mathbf {Z}}_{lm}(\theta ,\,\phi ) \bigr ]. \end{aligned}$$
(32)

The tangential vector spherical harmonics \(\varvec{\mathbf {Y}}_{lm}\) and \(\varvec{\mathbf {Z}}_{lm}\) are defined as [20]

$$\begin{aligned} \varvec{\mathbf {Y}}_{lm}(\theta ,\,\phi )&:=\frac{\mathrm {i}}{\sqrt{l(l+1)}} \left[ \frac{1}{\sin \theta }\frac{\partial ^{}Y_{lm}(\theta ,\,\phi )}{\partial \phi ^{}} \hat{\varvec{\mathbf {\theta }}} - \frac{\partial ^{}Y_{lm}(\theta ,\,\phi )}{\partial \theta ^{}} \hat{\varvec{\mathbf {\phi }}} \right] , \end{aligned}$$
(33a)
$$\begin{aligned} \varvec{\mathbf {Z}}_{lm}(\theta ,\,\phi )&:=\frac{\mathrm {i}}{\sqrt{l(l + 1)}} \left[ \frac{\partial ^{}Y_{lm}(\theta ,\,\phi )}{\partial \theta ^{}} \hat{\varvec{\mathbf {\theta }}} + \frac{1}{\sin \theta } \frac{\partial ^{}Y_{lm}(\theta ,\,\phi )}{\partial \phi ^{}} \hat{\varvec{\mathbf {\phi }}} \right] , \end{aligned}$$
(33b)

where

$$\begin{aligned} Y_{lm}(\theta ,\,\phi ) :=U_{lm}(\cos \theta ) \frac{\exp (\mathrm {i}m \phi )}{\sqrt{2\pi }} \end{aligned}$$
(34)

are the scalar spherical harmonics. We note that a comprehensive discussion of scalar and vector spherical harmonics, although with notations different from ours, can be found in Ref. [9]. Expressions (32) can be verified by substituting definitions (33) and (34) and using definition (1) of \(F_{lm}\) and definition (26) of the unit vectors \(\hat{\varvec{\mathbf {\tau }}}_{\pm }\).

Definition (31) and the orthogonality of \(\hat{\varvec{\mathbf {\tau }}}_{\pm }\) implies that, unlike \(\varvec{\mathbf {Y}}_{lm}\) and \(\varvec{\mathbf {Z}}_{lm}\), the functions \(\varvec{\mathbf {Q}}_{lm}^{\pm }\) exhibit local (vector) orthogonality, regardless of their degree and order, i.e.

$$\begin{aligned} \varvec{\mathbf {Q}}_{lm}^{\pm \, *}(\theta , \phi ) \cdot \varvec{\mathbf {Q}}_{l'm'}^{\mp }(\theta , \phi ) = 0. \end{aligned}$$
(35)

This equation together with definition (31) can be used to prove the orthonormality relations

$$\begin{aligned} \int \limits _{\Omega } \varvec{\mathbf {Q}}_{lm}^{\pm \, *}(\theta , \phi ) \cdot \varvec{\mathbf {Q}}_{l'm'}^{\pm }(\theta , \phi )\, \mathrm{d }\!\Omega&= \delta _{ll'} \delta _{mm'}, \end{aligned}$$
(36a)
$$\begin{aligned} \int \limits _{\Omega } \varvec{\mathbf {Q}}_{lm}^{\pm \, *}(\theta , \phi ) \cdot \varvec{\mathbf {Q}}_{l'm'}^{\mp }(\theta , \phi )\, \mathrm{d }\!\Omega&= 0, \end{aligned}$$
(36b)

where \(\int _{\Omega } \dots \mathrm{d }\!\Omega :=\int _{0}^{2\pi } \int _{0}^{\pi } \dots \sin \theta \mathrm{d }\!\theta \mathrm{d }\!\phi \).

The values of \(\varvec{\mathbf {Q}}_{lm}^{\pm }\) at the \(\theta \)-coordinate singularities deserve extra attention, since \(\hat{\varvec{\mathbf {\tau }}}_{\pm }\) are not well defined there. We can circumvent this problem by expressing \(\varvec{\mathbf {Q}}_{lm}^{\pm }\) using Cartesian basis vectors (see Fig. 2). Using Eq. (19) for \(F_{lm}(\pm 1)\), we have

$$\begin{aligned} \varvec{\mathbf {Q}}_{lm}^{\pm }(\theta =0, \phi )&= \left\{ \begin{array}{ll} \frac{1}{2 \sqrt{\pi }} c_{l,\,0} (\hat{\varvec{\mathbf {x}}} \pm \mathrm {i}\hat{\varvec{\mathbf {y}}}) &{} \text {if}\, m = \pm 1, \\ 0 &{} \text {otherwise}, \end{array}\right. \end{aligned}$$
(37a)
$$\begin{aligned} \varvec{\mathbf {Q}}_{lm}^{\pm }(\theta =\pi , \phi )&= \left\{ \begin{array}{ll} \frac{(-1)^{l}}{2 \sqrt{\pi }} c_{l,\,0} (\hat{\varvec{\mathbf {x}}} \mp \mathrm {i}\hat{\varvec{\mathbf {y}}}) &{} \text {if}\, m = \mp 1, \\ 0 &{} \text {otherwise}. \end{array}\right. \end{aligned}$$
(37b)

Like \(\varvec{\mathbf {Y}}_{lm}\) and \(\varvec{\mathbf {Z}}_{lm}\) [7], the mixed vector spherical harmonics \(\varvec{\mathbf {Q}}_{lm}^{\pm }\) also form a complete basis of the Hilbert space of square-integrable tangential vector fields defined over \(\Omega \). Hence we can expand an arbitrary tangential vector field \(\varvec{\mathbf {v}}\) in terms of \(\varvec{\mathbf {Q}}_{lm}^{\pm }\) as

$$\begin{aligned} \varvec{\mathbf {v}}(\theta , \phi ) = \sum _{l=1}^{\infty } \sum _{m=-l}^{l} \left[ v_{lm}^{+} \varvec{\mathbf {Q}}_{lm}^{+}(\theta , \phi ) + v_{lm}^{-} \varvec{\mathbf {Q}}_{lm}^{-}(\theta , \phi ) \right] , \end{aligned}$$
(38)

where the expansion coefficients \(v_{lm}^{\pm }\) can be calculated as

$$\begin{aligned} v_{lm}^{\pm } :=\int \limits _{\Omega } \varvec{\mathbf {Q}}_{lm}^{\pm \, *}(\theta , \phi ) \cdot \varvec{\mathbf {v}}(\theta , \phi ) \, \mathrm{d }\!\Omega . \end{aligned}$$
(39)

If \(v_{lm}^{\pm } = 0\) for \(L < l < \infty \) and some \(L > 0\), we call \(\varvec{\mathbf {v}}\) bandlimited. The limit \(L\) is the maximal degree of functions \(\varvec{\mathbf {Q}}_{lm}^{\pm }\) that contribute to the expansion (38). Therefore the subspace \(S_{L}\) of bandlimited vector fields is finite dimensional, its dimension is equal to the number of terms in (38):

$$\begin{aligned} \dim S_{L} = 2 \sum _{l=1}^{L} (2l + 1) = 2 \bigl [ (L + 1)^2 - 1 \bigr ] = 2L(L + 2). \end{aligned}$$
(40)

3 The Concentration Problem of Tangential Vector Fields within an Axisymmetrical Spherical Cap

Having defined all important special functions, we now turn our attention to the main topic of the paper. After a brief discussion of the concentration problem in terms of the mixed vector spherical harmonics, we introduce a reduced scalar problem which can be solved analogously to the theory of scalar spherical Slepian functions [10, 26, 28].

In Sect. 3.2, we analyze the eigenvalue spectrum of the concentration operator and give an illustration on the scalar eigenfunctions. Finally, we propose a fast and numerically stable way to calculate the eigenfunctions by using a differential operator that commutes with the scalar concentration operator obtained previously.

3.1 Formulation of the Vector Problem and its Reduction to a Scalar One

We consider the variational problem of finding a bandlimited, tangential vector field \(\varvec{\mathbf {G}}(\theta , \phi ) \in S_{L}\) that maximizes the fractional energy contained within an axisymmetric spherical cap \(C\):

$$\begin{aligned} \max _{\varvec{\mathbf {G}}}\ \frac{\int _{C} {|\varvec{\mathbf {G}}(\theta , \phi ) |}^2 \mathrm{d }\!\Omega }{\int _{\Omega } {|\varvec{\mathbf {G}}(\theta , \phi ) |}^2 \mathrm{d }\!\Omega } = \max _{\varvec{\mathbf {G}}}\ \frac{\int _{C} \varvec{\mathbf {G}}^*(\theta , \phi ) \cdot \varvec{\mathbf {G}}(\theta , \phi ) \mathrm{d }\!\Omega }{\int _{\Omega } \varvec{\mathbf {G}}^{*}(\theta , \phi ) \cdot \varvec{\mathbf {G}}(\theta , \phi ) \mathrm{d }\!\Omega }. \end{aligned}$$
(41)

Without loss of generality, we can center our spherical cap at \(\theta =0\), as seen in Fig. 3:

$$\begin{aligned} C = \bigl \{ (\theta , \phi ): 0 \le \theta \le \Theta , 0 \le \phi < 2\pi \bigr \}, \end{aligned}$$
(42)

where \(\Theta > 0\) is assumed.

Fig. 3
figure 3

Sketch of the spherical cap \(C\)

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To solve the vectorial problem (41), we adapt the method of the scalar case [28] and turn (41) into a Rayleigh–Ritz matrix variational problem [12, p. 176]. We can achieve this by expanding \(\varvec{\mathbf {G}}\) in terms of \(\varvec{\mathbf {Q}}_{lm}^{\pm }\):

$$\begin{aligned} \varvec{\mathbf {G}}(\theta , \phi )&= \sum _{l=1}^{L} \sum _{m=-l}^{l} \left[ g_{lm}^{+} \varvec{\mathbf {Q}}_{lm}^{+}(\theta , \phi ) \!+\! g_{lm}^{-} \varvec{\mathbf {Q}}_{lm}^{-}(\theta , \phi ) \right] \nonumber \\&= \sum _{m=-L}^{L} \sum _{l=\ell _{m}}^{L} \left[ g_{lm}^{+} \varvec{\mathbf {Q}}_{lm}^{+}(\theta , \phi ) \!+\! g_{lm}^{-} \varvec{\mathbf {Q}}_{lm}^{-}(\theta , \phi ) \right] . \end{aligned}$$
(43)

In the second step, we have interchanged the order of double summation to facilitate the transition to the matrix formulation of (41) later. A visual comparison of the two summation schemes is given in Fig. 4.

Fig. 4
figure 4

Order of double summation in a \(\sum _{l=1}^{L} \sum _{m=-l}^{l}\) and b \(\sum _{m=-L}^{L} \sum _{l=\ell _{m}}^{L}\) for \(L=3\), where \(\ell _{m} = \max (1, {|m |})\). The filled circles represent terms with corresponding indices \(l\) and \(m\)

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Next we insert (43) into (41), interchange the order of summation and integration and use orthonormality relations (35) and (36). Hence the numerator can be written as

$$\begin{aligned} \int \limits _{C} \varvec{\mathbf {G}}^*(\theta , \phi ) \cdot \varvec{\mathbf {G}}(\theta , \phi ) \mathrm{d }\!\Omega \!&= \! \sum _{m=-L}^{L} \sum _{l=\ell _{m}}^{L} g_{lm}^{+\,*} \sum _{m'=-L}^{L} \sum _{l'=\ell _{m'}}^{L} \left[ \int \limits _{C} \varvec{\mathbf {Q}}_{lm}^{+\,*}(\theta , \phi ) \cdot \varvec{\mathbf {Q}}_{l'm'}^{+}(\theta , \phi ) \mathrm{d }\!\Omega \right] g_{l'm'}^{+} \nonumber \\&\!+\! \sum _{m=-L}^{L} \sum _{l=\ell _{m}}^{L} g_{lm}^{-\,*} \sum _{m'=-L}^{L} \sum _{l'=\ell _{m'}}^{L} \left[ \int \limits _{C} \varvec{\mathbf {Q}}_{lm}^{-\,*}(\theta , \phi ) \cdot \varvec{\mathbf {Q}}_{l'm'}^{-}(\theta , \phi ) \mathrm{d }\!\Omega \right] g_{l'm'}^{-},\nonumber \\ \end{aligned}$$
(44)

while the denominator becomes

$$\begin{aligned} \int \limits _{\Omega } \varvec{\mathbf {G}}^*(\theta , \phi ) \cdot \varvec{\mathbf {G}}(\theta , \phi ) \mathrm{d }\!\Omega = \sum _{m=-L}^{L} \sum _{l=\ell _{m}}^{L} \left( g_{lm}^{+\,*} g_{lm}^{+} + g_{lm}^{-\,*} g_{lm}^{-} \right) . \end{aligned}$$
(45)

The integrals on the right-hand side of (44) can be expressed as

$$\begin{aligned} \int \limits _{C} \varvec{\mathbf {Q}}_{lm}^{\pm \,*}(\theta , \phi ) \cdot \varvec{\mathbf {Q}}_{l'm'}^{\pm }(\theta , \phi ) \mathrm{d }\!\Omega&= \frac{1}{2\pi }\int \limits _{0}^{2\pi } \exp \bigl [ \mathrm {i}(m' - m) \phi \bigr ] \mathrm{d }\!\phi \nonumber \\&\quad \times \int \limits _{0}^{\Theta } F_{l,\,\pm m}(\cos \theta ) F_{l',\,\pm m}(\cos \theta ) \sin \theta \mathrm{d }\!\theta . \end{aligned}$$
(46)

Since \((2\pi )^{-1} \int _{0}^{2\pi } \exp [\mathrm {i}(m' - m) \phi ] \mathrm{d }\!\phi = \delta _{mm'}\), Eq. (46) further simplifies to

$$\begin{aligned} \int \limits _{C} \varvec{\mathbf {Q}}_{lm}^{\pm \,*}(\theta ,\,\phi ) \cdot \varvec{\mathbf {Q}}_{l'm'}^{\pm }(\theta ,\,\phi ) \mathrm{d }\!\Omega = \delta _{mm'} K_{\pm m,\,ll'}, \end{aligned}$$
(47)

where

$$\begin{aligned} K_{m,ll'} :=\int \limits _{0}^{\Theta } F_{lm}(\cos \theta ) F_{l'm}(\cos \theta ) \sin \theta \mathrm{d }\!\theta = \int \limits _{\cos \Theta }^{1} F_{lm}(x) F_{l'm}(x) \mathrm{d }\!x. \end{aligned}$$
(48)

The integrand of \(K_{m,\,ll'}\) is a polynomial of degree \(l + l'\), hence it can exactly be integrated numerically, for instance, by a Gauss–Legendre formula of \(\lceil (l + l' + 1)/2 \rceil \) nodes.

Taking (47) into account, (41) can be rewritten as

$$\begin{aligned}&\max _{\{g_{lm}^{\pm }\}}\ \left\{ \ \left[ \sum _{m=-L}^{L} \left( \sum _{l=\ell _{m}}^{L} g_{lm}^{+\,*} \sum _{l'=\ell _{m}}^{L} K_{m,ll'} g_{l'm}^{+} \right) \!+\! \sum _{m=-L}^{L} \left( \sum _{l=\ell _{m}}^{L} g_{lm}^{-\,*} \sum _{l'=\ell _{m}}^{L} K_{-m,ll'} g_{l'm}^{-} \right) \right] \right. \nonumber \\&\left. \quad \!\times \! \ \left[ \sum _{m=-L}^{L} \sum _{l=\ell _{m}}^{L} \left( g_{lm}^{+\,*} g_{lm}^{+} + g_{lm}^{-\,*} g_{lm}^{-} \right) \right] ^{-1} \ \right\} . \end{aligned}$$
(49)

To express (49) in matrix formalism, we construct a column vector \(\mathsf {g}\) of \(2L(L + 2)\) elements as

$$\begin{aligned} \mathsf {g} :=\left[ g_{L,-L}^{+}; g_{L-1, -L+1}^{+}, g_{L, -L+1}^{+}; \ldots ; g_{L, L}^{+}; g_{L,-L}^{-}; g_{L-1, -L+1}^{-}, g_{L, -L+1}^{-}; \ldots ; g_{L, L}^{-} \right] ^{\mathrm {T}}.\nonumber \\ \end{aligned}$$
(50)

where the expansion coefficients \(g_{lm}^{\pm }\) are enumerated according to the scheme of Fig. 4b. In addition, we introduce the \(2L(L+2) \times 2L(L+2)\) block-diagonal matrix

$$\begin{aligned} \mathsf {K} :=\left[ \begin{array}{ll} \mathsf {K}^{+} &{} \mathsf {0} \\ \mathsf {0} &{} \mathsf {K}^{-} \end{array}\right] , \end{aligned}$$
(51)

where the \(L(L+2) \times L(L+2)\) blocks \(\mathsf {K}^{\pm }\) are themselves block-diagonal:

$$\begin{aligned} \mathsf {K}^{+} :={{\mathrm{diag}}}\left[ \mathsf {K}_{-L}; \mathsf {K}_{-L+1}; \ldots ; \mathsf {K}_{L}; \right] , \qquad \mathsf {K}^{-} :={{\mathrm{diag}}}\left[ \mathsf {K}_{L}; \mathsf {K}_{L-1}; \ldots ; \mathsf {K}_{-L} \right] .\nonumber \\ \end{aligned}$$
(52)

The elementary blocks

$$\begin{aligned} \mathsf {K}_{m} :=\left[ \begin{array}{lll} K_{m,\ell _{m}\ell _{m}} &{} \cdots &{} K_{m,\ell _{m}L} \\ \vdots &{} \ddots &{} \vdots \\ K_{m,L\ell _{m}} &{} \cdots &{} K_{m,LL} \end{array}\right] \end{aligned}$$
(53)

correspond to different orders \(m\) and have an order-dependent size of \((L - \ell _{m} + 1) \times (L - \ell _{m} + 1)\). Note that \(\mathsf {K}^{+}\) contains the same blocks \(\mathsf {K}_{m}\) as \(\mathsf {K}^{-}\), however, the order of the blocks is reversed owing to (47).

We can thus use these constructions to transform problem (49) into the Rayleigh–Ritz matrix variational problem

$$\begin{aligned} \max _{\mathsf {g}}\ \frac{\mathsf {g}^{\dag }\, \mathsf {K}\, \mathsf {g}}{\mathsf {g}^{\dag } \, \mathsf {g}}, \end{aligned}$$
(54)

where the dagger sign denotes the conjugate transpose. Equivalently, we have to find the eigenvector \(\mathsf {g}\) of the eigenvalue problem [12, p. 176]

$$\begin{aligned} \mathsf {K} \mathsf {g} = \eta \, \mathsf {g} \end{aligned}$$
(55)

with the maximal eigenvalue \(\eta \). However, rather than solving the large \(2L(L + 2) \times 2L(L + 2)\) eigenvalue problem (55), the block-diagonal structure of \(\mathsf {K}\) allows us to solve a series of smaller \((L - \ell _{m} + 1) \times (L - \ell _{m} + 1)\) problems instead,

$$\begin{aligned} \mathsf {K}_{m} \mathsf {g}_{m} = \eta _{m} \mathsf {g}_{m}, \quad -L \le m \le L, \end{aligned}$$
(56)

one for each order \(m\).

From (48) follows that \(K_{m,\,ll'} = K_{m,\,l'l}\), implying that the matrices \(\mathsf {K}_{m}\) are symmetric. Hence their eigenvalues are always real. For a given \(m\), we rank-order the \((L - \ell _{m} + 1)\) distinct eigenvalues \(\eta _{mn}\) as \(1 > \eta _{m,\,1} > \eta _{m,\,2} > \cdots > \eta _{m,\,L - \ell _{m} + 1} > 0\). The associated eigenvectors \(\mathsf {g}_{mn}\) can be chosen to be real and orthonormal:

$$\begin{aligned} \mathsf {g}_{mn}^{\mathrm {T}}\mathsf {g}_{mn'} = \delta _{nn'}, \qquad \mathsf {g}_{mn}^{\mathrm {T}}\mathsf {K}_{m} \mathsf {g}_{mn'} = \eta _{mn} \delta _{nn'}, \qquad 1 \le n,\,n' \le (L - \ell _{m} + 1).\nonumber \\ \end{aligned}$$
(57)

Here we have distinguished between the different eigenvalues and the corresponding eigenvectors by the use of the additional index \(n\) (or \(n'\)). However, we drop this additional index for brevity when we refer to any of the \((L - \ell _{m} + 1)\) eigenvalues or eigenvectors.

Let us denote the elements of an eigenvector \(\mathsf {g}_{m}\) simply by \(g_{lm}\). The definitions (51), (52) and (53) imply that eigenfunctions contain either \(\hat{\varvec{\mathbf {\tau }}}_{+}\) or \(\hat{\varvec{\mathbf {\tau }}}_{-}\), they are composed of mixed vector spherical harmonics of a single order \(m\) only and \(g_{lm}^{\pm } = g_{l,\,\pm m}\). Hence we can considerably simplify our general expansion (43) and write instead

$$\begin{aligned} \varvec{\mathbf {G}}_{m}^{\pm }(\theta , \phi ) = \sum _{l=\ell _{m}}^{L} g_{l,\,\pm m} \varvec{\mathbf {Q}}_{lm}^{\pm }(\theta , \phi ). \end{aligned}$$
(58)

Upon substituting definition (31) of \(\varvec{\mathbf {Q}}_{lm}^{\pm }\) into Eq. (58), we obtain

$$\begin{aligned} \varvec{\mathbf {G}}_{m}^{\pm }(\theta , \phi ) = G_{\pm m}(\cos \theta ) \frac{\exp (\mathrm {i}m \phi )}{\sqrt{2\pi }}\, \hat{\varvec{\mathbf {\tau }}}_{\pm }, \end{aligned}$$
(59)

where

$$\begin{aligned} G_{m}(x) :=\sum _{l=\ell _{m}}^{L} g_{lm} F_{lm}(x) \end{aligned}$$
(60)

are real functions.

In this way, we managed to reduce the vectorial concentration problem within a spherical cap to equivalent one-dimensional, scalar concentration problems of various orders \(m\). The key idea in this simplification was the choice (31) for our basis functions. The scalar concentration problem for a fixed order \(m\) can be formulated as

$$\begin{aligned} \max _{G_{m}} \frac{\int _{\cos \Theta }^{1} \left[ G_{m}(x) \right] ^2 \mathrm{d }\!x}{ \int _{-1}^{1} \left[ G_{m}(x) \right] ^2 \mathrm{d }\!x}, \end{aligned}$$
(61)

where \(G_{m}\) is a bandlimited scalar function belonging to the subspace spanned by \(F_{lm}\). The corresponding Rayleigh–Ritz matrix variational problem is

$$\begin{aligned} \max _{\mathsf {g}_{m}} \frac{\mathsf {g}_{m}^{\mathrm {T}}\mathsf {K}_{m} \mathsf {g}_{m}}{\mathsf {g}_{m}^{\mathrm {T}}\mathsf {g}_{m}}. \end{aligned}$$
(62)

Instead of the eigenvalue Eq. (56) specifying eigenvectors \(\mathsf {g}_{m}\), we can directly formulate an eigenvalue equation in terms of the functions \(G_{m}\), too. Therefore we first express Eq. (56) component-wise as

$$\begin{aligned} \sum _{l'=\ell _{m}}^{L} K_{m,\,ll'} g_{l'm} = \eta _{m} g_{lm}, \quad \ell _{m} \le l \le L. \end{aligned}$$
(63)

Now we multiply both sides by \(F_{lm}(x)\) and sum over \(l\):

$$\begin{aligned} \sum _{l=\ell _{m}}^{L} \sum _{l'=\ell _{m}}^{L} K_{m,\,ll'} g_{l'm} F_{lm}(x) = \eta _{m} \sum _{l=\ell _{m}}^{L} g_{lm} F_{lm}(x). \end{aligned}$$
(64)

The left-hand side can be rewritten as

$$\begin{aligned} \begin{aligned} \sum _{l=\ell _{m}}^{L} \sum _{l'=\ell _{m}}^{L} K_{m,\,ll'} g_{l'm} F_{lm}(x)&= \sum _{l=\ell _{m}}^{L} \sum _{l'=\ell _{m}}^{L} \left[ \int \limits _{\cos \Theta }^{1} F_{lm}(x') F_{l'm}(x') \mathrm{d }\!x' \right] g_{l'm} F_{lm}(x) \\&= \int \limits _{\cos \Theta }^{1} \left\{ \left[ \, \sum _{l=\ell _{m}}^{L} F_{lm}(x) F_{lm}(x') \right] \sum _{l'=\ell _{m}}^{L} g_{l'm} F_{l'm}(x') \right\} \mathrm{d }\!x'. \end{aligned} \end{aligned}$$

This way, we obtain a Fredholm integral equation of the second kind for \(G_{m}\),

$$\begin{aligned} \int \limits _{\cos \Theta }^{1} \mathcal {K}_{m}(x,\,x')\, G_{m}(x') \mathrm{d }\!x' = \eta _{m} G_{m}(x), \quad -1 \le x \le 1, \end{aligned}$$
(65)

where the kernel function \(\mathcal {K}_{m}\) is defined as

$$\begin{aligned} \mathcal {K}_{m}(x,\,x') :=\sum _{l=\ell _{m}}^{L} F_{lm}(x) F_{lm}(x'). \end{aligned}$$
(66)

It follows from the orthogonality relations (57) of the eigenvectors that the scalar eigenfunctions \(G_{m}\) are doubly orthogonal:

$$\begin{aligned} \int \limits _{-1}^{1} G_{mn}(x) G_{mn'}(x) \mathrm{d }\!x&= \delta _{nn'}, \end{aligned}$$
(67a)
$$\begin{aligned} \int \limits _{\cos \Theta }^{1} G_{mn}(x) G_{mn'}(x) \mathrm{d }\!x&= \eta _{mn} \delta _{nn'}. \end{aligned}$$
(67b)

The vectorial eigenfunctions \(\varvec{\mathbf {G}}_{m}^{\pm }\) inherit this property as well:

$$\begin{aligned} \int \limits _{\Omega } \varvec{\mathbf {G}}_{mn}^{\pm \, *}(\theta ,\,\phi ) \cdot \varvec{\mathbf {G}}_{m'n'}^{\pm }(\theta ,\,\phi ) \, \mathrm{d }\!\Omega&\!=\!&\delta _{mm'} \delta _{nn'}, \quad \int \limits _{\Omega } \varvec{\mathbf {G}}_{mn}^{\pm \, *}(\theta ,\,\phi ) \cdot \varvec{\mathbf {G}}_{m'n'}^{\mp }(\theta ,\,\phi ) \, \mathrm{d }\!\Omega \!=\! 0, \nonumber \\ \end{aligned}$$
(68a)
$$\begin{aligned} \int _{C} \varvec{\mathbf {G}}_{mn}^{\pm \, *}(\theta ,\,\phi ) \cdot \varvec{\mathbf {G}}_{m'n'}^{\pm }(\theta ,\,\phi ) \, \mathrm{d }\!\Omega&\!=\!&\eta _{mn} \delta _{mm'} \delta _{nn'}, \quad \int \limits _{C} \varvec{\mathbf {G}}_{mn}^{\pm \, *}(\theta ,\,\phi ) \cdot \varvec{\mathbf {G}}_{m'n'}^{\mp }(\theta ,\,\phi ) \, \mathrm{d }\!\Omega \!=\! 0.\nonumber \\ \end{aligned}$$
(68b)

3.2 The Eigenvalue Spectrum and its Peculiarity

The eigenvalue spectrum of Slepian-type concentration problems [29, 30, 32] exhibits a characteristic step-like shape, and the present case is no exception. Figure 5 shows rank-ordered spectra including \(\eta _{mn}\) for all orders \(m\). They correspond to \(\Theta = 30^\circ , 60^\circ , 90^\circ \) and the maximal degree was chosen \(L=18\).

Fig. 5
figure 5

Rank-ordered eigenvalue spectrum including the eigenvalues \(\eta _{m}\) of all \(\mathsf {K}_{m}\), \(-L \le m \le L\) for a \(\Theta = 30^{\circ }\), b \(\Theta = 60^{\circ }\), c \(\Theta = 90^{\circ }\), and \(L = 18\). The vertical gridlines mark the corresponding Shannon numbers \(N\) of (70)

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The majority of the eigenvalues for each case is either close to one or zero, corresponding to well-concentrated and poorly concentrated eigenfunctions, respectively. As an illustration, in Fig. 6 we have plotted a small number of scalar eigenfunctions \(G_{mn}\), corresponding to different parts of the eigenvalue spectrum.

Fig. 6
figure 6

Four scalar eigenfunctions \(G_{mn}(\cos \theta )\), \(n=1,3,5,7\) of each order \(-2 \le m \le 2\). The maximal degree is \(L=18\) and \(\Theta =60^\circ \). The black and gray curves mark contributions of \(G_{mn}\) to the interior of the spherical cap (\(0 \le \theta \le 60^\circ \)) and the rest of the sphere (\(60^\circ < \theta \le 180^\circ \)), respectively. Labels show the eigenvalues \(\eta _{mn}\) which express the quality of concentration within \(C\)

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Strictly speaking, the solution of the concentration problem (41) is the pair of vectorial eigenfunctions which corresponds to the maximally concentrated \(G_{m}\). However, having solved the equivalent eigenvalue problem (55), we have gained a whole set of well-concentrated, orthogonal pairs of eigenfunctions \(\varvec{\mathbf {G}}_{m}^{\pm }\). How many pairs belong to this set? To answer this question, we first define the partial Shannon number [28]

$$\begin{aligned} N_{m} :={{\mathrm{Tr}}}\mathsf {K}_{m} = \sum _{n=1}^{L-\ell _{m}+1} \eta _{mn} = \int \limits _{\cos \Theta }^{1} \mathcal {K}_{m}(x, x) \mathrm{d }\!x, \end{aligned}$$
(69)

which gives the approximate number of reasonably well-concentrated (\(\eta \ge 0.5\)) scalar eigenfunctions for a given maximal degree \(L\) and order \(m\). Summing over all possible values of \(m\), we obtain the (total) Shannon number

$$\begin{aligned} N :=\sum _{m=-L}^{L} N_{m} = \sum _{m=-L}^{L} \sum _{n=1}^{L-\ell _{m}+1} \eta _{mn} = \int \limits _{\cos \Theta }^{1} \sum _{m=-L}^{L} \mathcal {K}_{m}(x, x) \mathrm{d }\!x = L(L+2) \frac{A_{C}}{4\pi },\nonumber \\ \end{aligned}$$
(70)

where \(A_{C} = 2\pi (1 - \cos \Theta )\) is the area of the spherical cap \(C\). In the last equality we substituted definition (66), interchanged the order of double summation and used addition theorem (17).

Hence there are \(N\) pairs of orthogonal vectorial eigenfunctions which are suitable for approximating bandlimited, tangential vector fields localized to \(C\). Equivalently, the use of this basis reduces the number of degrees of freedom from \(\dim S_{L} = 2L(L+2)\) to \(2N\).

3.3 Toward an Efficient Numerical Solution: The Commuting Differential Operator and its Eigenvalue Problem

In Sect. 3.1, we obtained the expansion coefficients \(g_{lm}\) by solving eigenvalue Eq. (56) directly. However, while it is theoretically possible to calculate \(g_{lm}\) this way, the accumulation of the eigenvalues \(\eta \) at one and zero, as seen in Fig. 5, makes the numerical solution of (56) ill-conditioned [4]. In order to circumvent this problem, we set out to construct another matrix with a simple spectrum to supply the expansion coefficients \(g_{lm}\).

Therefore, we first return to the Fredholm eigenvalue Eq. (65). We wish to find a Sturm–Liouville differential operator \(\mathcal {J}_{m}\) that commutes with the concentration (integral) operator on the left-hand side of (65):

$$\begin{aligned} \int \limits _{\cos \Theta }^{1} \mathcal {K}_{m}(x,\,x') \mathcal {J}_{m}' u(x') \mathrm{d }\!x' \!=\! \mathcal {J}_{m} \int \limits _{\cos \Theta }^{1} \mathcal {K}_{m}(x,\,x') u(x') \mathrm{d }\!x' \!=\! \int \limits _{\cos \Theta }^{1} \mathcal {J}_{m} \mathcal {K}_{m}(x,\,x') u(x') \mathrm{d }\!x'\nonumber \\ \end{aligned}$$
(71)

for any square-integrable bandlimited function \(u\), so that the two operators share a common set of eigenfunctions [3, pp. 314]. It is known from the Sturm–Liouville theory that \(\mathcal {J}_{m}\) has a simple spectrum of distinct eigenvalues with an accumulation point in infinity [21, p. 724]. If such a differential operator \(\mathcal {J}_{m}\) can be found, its matrix representation can be used to obtain the expansion coefficients \(g_{lm}\) (hence the eigenfunctions) in a numerically stable way.

The same approach was taken by Grünbaum et al. for the concentration problem of scalar functions within \(C\) [10]. They proposed the differential operator

$$\begin{aligned} \mathcal {G}_{m} :=(\cos \Theta - \cos \theta ) \nabla _{\Omega ,\,m}^2 + \sin \theta \, \frac{\mathrm{d }\!^{}}{\mathrm{d }\!\theta ^{}} - L(L + 2) \cos \theta , \end{aligned}$$
(72)

where \(\nabla _{\Omega ,\,m}^2\) is the fixed-order scalar Laplace–Beltrami operator (25). This operator commutes with the concentration operator of the scalar case which contains the kernel function \(\mathcal {D}_{m}(x, x') = \sum _{l={|m |}}^{L} U_{lm}(x) U_{lm}(x')\) [28].

Based on (72), we make the following ansatz on \(\mathcal {J}_{m}\):

$$\begin{aligned} \mathcal {J}_{m} :=(\cos \Theta - \cos \theta ) \Delta _{\Omega ,\,m} + \sin \theta \frac{\mathrm{d }\!^{}}{\mathrm{d }\!\theta ^{}} - L(L + 2) \cos \theta , \end{aligned}$$
(73)

where \(\Delta _{\Omega ,\,m}\) is the fixed-order operator (29) related to the fixed-order vector Laplace–Beltrami operator (28). Changing the variable to \(x=\cos \theta \) yields

$$\begin{aligned} \mathcal {J}_{m} = (\cos \Theta - x) \Delta _{\Omega ,\,m} - (1 - x^2) \frac{\mathrm{d }\!^{}}{\mathrm{d }\!x^{}} - L(L + 2)x, \end{aligned}$$
(74)

which is equivalent to

$$\begin{aligned} \mathcal {J}_{m} = \frac{\mathrm{d }\!^{}}{\mathrm{d }\!x^{}} \left[ (\cos \Theta - x)(1 - x^2) \frac{\mathrm{d }\!^{}}{\mathrm{d }\!x^{}} \right] - L(L+2)x - (\cos \Theta - x) \frac{m^2 - 2mx + 1}{1 - x^2}.\nonumber \\ \end{aligned}$$
(75)

To prove that \(\mathcal {J}_{m}\) satisfies the commutation relation (71), we suggest following the concept of Grünbaum et al. [10]. First, one proves the identity

$$\begin{aligned} \int \limits _{\cos \Theta }^{1} u_{1}(x) \left[ \mathcal {J}_{m} u_{2}(x) \right] \mathrm{d }\!x = \int \limits _{\cos \Theta }^{1} \left[ \mathcal {J}_{m} u_{1}(x) \right] u_{2}(x) \mathrm{d }\!x, \end{aligned}$$
(76)

which holds for any two functions \(u_{1}\) and \(u_{2}\) that are non-singular at the interval endpoints (see Appendix 1f for details). Therefore, the left-hand side of the commutation relation (71) can be rewritten as

$$\begin{aligned} \int \limits _{\cos \Theta }^{1} \mathcal {K}_{m}(x,x') \mathcal {J}_{m}' u(x') \mathrm{d }\!x' = \int \limits _{\cos \Theta }^{1} \left[ \mathcal {J}_{m}' \mathcal {K}_{m}(x,x') \right] u(x') \mathrm{d }\!x'. \end{aligned}$$
(77)

Finally, one verifies that

$$\begin{aligned} \mathcal {J}_{m} \mathcal {K}_{m}(x, x') = \mathcal {J}_{m}' \mathcal {K}_{m}(x, x'). \end{aligned}$$
(78)

The proof of (78), like the proof of (76), closely resembles its counterpart from the scalar concentration problem [28]. The key steps are the same, with the main difference that the associated Legendre functions are replaced by \(F_{lm}\) together with the corresponding identities. The details can be found in Appendix 1g.

Since \(\mathcal {J}_{m}\) commutes with the integral operator of (65), the functions \(G_{m}\) are eigenfunctions of \(\mathcal {J}_{m}\), too:

$$\begin{aligned} \mathcal {J}_{m} G_{m}(x) = \chi _{m} G_{m}(x). \end{aligned}$$
(79)

Figure 7 shows the \(\chi \)-eigenvalue spectrum of all orders \(m\) for \(\Theta = 30^\circ , 60^\circ , 90^\circ \) and \(L=18\) (cf. Fig. 5). Similarly to the scalar concentration problems [28, 30, 32], the rank-ordering for \(\chi _{mn}\) is the opposite of the rank-ordering for \(\eta _{mn}\). Importantly, the \(\chi \)-spectrum does not exhibit an accumulation of eigenvalues.

Fig. 7
figure 7

Rank-ordered eigenvalue spectrum including the eigenvalues \(\chi _{m}\) of all \(\mathsf {J}_{m}\), \(-L \le m \le L\) for a \(\Theta = 30^{\circ }\), b \(\Theta = 60^{\circ }\), c \(\Theta = 90^{\circ }\), and \(L = 18\). The vertical gridlines mark the corresponding Shannon numbers \(N=24,90,180\) (see (70))

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To obtain a matrix equation similar to the component-wise eigenvalue Eq. (63) of \(\mathsf {K}_{m}\), we substitute expansion (60) of \(G_{lm}\) in terms of \(F_{lm}\) into eigenvalue Eq. (79), but this time, writing \(l'\) instead of \(l\). After that we multiply by \(F_{lm}(x)\), integrate over \(-1 \le x \le 1\), and invoke orthonormality relation (5) of \(F_{lm}\). In this way, we arrive at the equation

$$\begin{aligned} \sum _{l'=\ell _{m}}^{L} J_{m,\,ll'} g_{l'm} = \chi _{m} g_{lm}, \quad \ell _{m} \le l \le L, \end{aligned}$$
(80)

where

$$\begin{aligned} J_{m,\,ll'} :=\int \limits _{-1}^{1} F_{lm}(x) \mathcal {J}_{m} F_{l'm}(x) \mathrm{d }\!x. \end{aligned}$$
(81)

Similarly to \(\mathsf {K}_{m}\), we can arrange \(J_{m,\,ll'}\) into a matrix \(\mathsf {J}_{m}\):

$$\begin{aligned} \mathsf {J}_{m} = \left[ \begin{array}{lll} J_{m,\ell _{m}\ell _{m}} &{} \cdots &{} J_{m,\ell _{m}L} \\ \vdots &{} \ddots &{} \vdots \\ J_{m,L\ell _{m}} &{} \cdots &{} J_{m,LL} \end{array}\right] . \end{aligned}$$
(82)

However, the only non-zero matrix elements, as proven in Appendix 1h, are

$$\begin{aligned} J_{m,\,ll}&= -l(l + 1)\cos \Theta + m \left[ 1 - \frac{L(L + 2) + 1}{l(l + 1)} \right] \end{aligned}$$
(83a)
$$\begin{aligned} J_{m,\,l,\,l+1}&= J_{m,\,l+1,\,l} = \bigl [ l(l + 2) - L(L + 2) \bigr ] \zeta _{l+1,\,m}, \end{aligned}$$
(83b)

hence \(\mathsf {J}_{m}\) is real, symmetric and tridiagonal. The eigenvalue Eq. (80) can thus be written as

$$\begin{aligned} \mathsf {J}_{m} \mathsf {g}_{m} = \chi _{m} \mathsf {g}_{m}, \quad -L \le m \le L. \end{aligned}$$
(84)

We have already seen in Sect. 2.1 that \(F_{l,\,0} = U_{l,\,1}\), hence in the special case of \(m=0\), matrix \(\mathsf {J}_{0}\) is identical to the matrix of the Grünbaum operator \(\mathcal {G}_{1}\) [28].

In summary, to calculate the scalar eigenfunctions \(G_{m}\) for each order \(m\), we first construct the tridiagonal matrices \(\mathsf {J}_{m}\) using formulae (83) and then solve the corresponding eigenvalue problem (84) numerically. The resulting eigenvectors \(\mathsf {g}_{m}\) contain the expansion coefficients \(g_{lm}\), \(\ell _{m} \le l \le L\), which, substituted into expansion (60) give the eigenfunctions \(G_{m}\). The corresponding energy concentration ratio \(\eta _{m}\) can be calculated using either \(\eta _{m} = \int _{\cos \Theta }^{1} [G_{m}(x)]^2 \mathrm{d }\!x\) or \(\eta _{m} = \mathsf {g}_{m}^{\mathrm {T}}\mathsf {K}_{m} \mathsf {g}_{m}\).

Finally, we demonstrate the numerical stability of the proposed method. We calculated the eigenvectors \(\mathsf {g}_{mn}\) for \(m=1\), \(L=18\) and \(\Theta =30^{\circ }, 60^{\circ }, 90^{\circ }\) in multiple ways. First, as a reference, we used arbitrary precision arithmetic to obtain the eigenvectors of \(\mathsf {K}_{1}\) with the relative error of each coefficient \(g_{l,1}\) being less than \(10^{-23}\). Let \(\mathsf {g}_{1,n}^\text {ref}\) denote these vectors. Then we computed both \(\mathsf {K}_{1}\) and \(\mathsf {J}_{1}\) in double precision and fed them into the divide-and-conquer routines of LAPACK [2] to produce the eigenvectors again. Let \(\mathsf {g}_{1,\,n}^\text {K}\) and \(\mathsf {g}_{1,\,n}^\text {J}\) stand for these results, respectively. In addition, we furthermore assume \({||\mathsf {g}_{1,\,n}^\text {ref} ||} = {||\mathsf {g}_{1,\,n}^\text {K} ||} = {||\mathsf {g}_{1,\,n}^\text {J} ||} = 1\) where \({||\mathsf {v} ||} :=\sqrt{\mathsf {v}^{\mathrm {T}}\mathsf {v}}\).

Figure 8a–c plot the eigenvalue gaps [2, p. 104]

$$\begin{aligned} {{\mathrm{gap}}}\bigl ( \eta _{1,\,n} \bigr )&:=\min _{j \ne n} {|\eta _{1,\,n} - \eta _{1,\,j} |}, \end{aligned}$$
(85a)
$$\begin{aligned} {{\mathrm{gap}}}\bigl ( \chi _{1,\,n} \bigr )&:=\min _{j \ne n} {|\chi _{1,\,n} - \chi _{1,\,j} |}, \end{aligned}$$
(85b)

for all three values of \(\Theta \), respectively, where \(1 \le n \le L\). Figure 8d–e show the errors

$$\begin{aligned} \delta \mathsf {g}_{1,\,n}^\text {K}&:=\min \left( {||\mathsf {g}_{1,\,n}^\text {K} - \mathsf {g}_{1,\,n}^\text {ref} ||}, {||\mathsf {g}_{1,\,n}^\text {K} - \bigl ( -\mathsf {g}_{1,\,n}^\text {ref} \bigr ) ||} \right) , \end{aligned}$$
(86a)
$$\begin{aligned} \delta \mathsf {g}_{1,\,n}^\text {J}&:=\min \left( {||\mathsf {g}_{1,\,n}^\text {J} - \mathsf {g}_{1,\,n}^\text {ref} ||}, {||\mathsf {g}_{1,\,n}^\text {J} - \bigl ( -\mathsf {g}_{1,\,n}^\text {ref} \bigr ) ||} \right) \end{aligned}$$
(86b)

of the eigenvectors, where we have taken their sign ambiguity into account.

Fig. 8
figure 8

a–c Eigenvalue gap (85) for \(\mathsf {K}_{1}\) circle and \(\mathsf {J}_{1}\) triangle for \(\Theta =30^{\circ }, 60^{\circ }, 90^{\circ }\), respectively. d–f Error (86) of the eigenvectors of \(\mathsf {K}_{1}\) circle and \(\mathsf {J}_{1}\) triangle for \(\Theta =30^{\circ }, 60^{\circ }, 90^{\circ }\), respectively. The number \(\epsilon _\text {M}\) denotes the machine epsilon in double precision. The maximal degree is \(L=18\) and the vertical gridlines mark the partial Shannon numbers \(N_{1}\) of (69)

Full size image

In Fig. 8a–c, we clearly see the accumulation of eigenvalues \(\eta _{1,n}\) of \(\mathsf {K}_{1}\) for both small and large values of \(n\). The decrease in the eigenvalue gap by many orders of magnitude is accompanied by a rapid increase in the error \(\delta \mathsf {g}_{1,\,n}^\text {K}\) [2, p. 104], as seen in Fig. 8d–f. Therefore, with a naïve treatment of \(\mathsf {K}_{1}\), we failed to calculate the well-concentrated eigenfunctions accurately; precisely those that play an important role in the approximation of functions localized to \(C\).

On the contrary, Fig. 8a–c demonstrate again that the eigenvalues \(\chi _{1,\,n}\) of \(\mathsf {J}_{1}\) are well separated, hence we can expect the accuracy of eigenvectors \(\mathsf {g}_{1,\,n}^\text {J}\) to stay reasonably close to machine precision. Indeed, the error is below \(120\epsilon _\text {M}\) for all values of \(n\), as indicated by Fig. 8d–f, where \(\epsilon _\text {M} = 2^{-53} \approx 1.11 \times 10^{-16}\) denotes the machine epsilon in double precision [2, p. 79]. Considering the tridiagonal form of \(\mathsf {J}_{m}\) with the simple expressions (83) for the matrix elements, its superiority over \(\mathsf {K}_{m}\) in the calculation of eigenvectors is justified.

4 Concluding Remarks

We have formulated a scalar problem which is equivalent to the concentration problem of tangential vector fields within a spherical cap, and enables us to treat it analogously to the concentration problem of scalar functions. Hence a construction of a commuting differential operator with a simple spectrum has been made possible. This circumstance, at the same time, opens the way for computing concentrated vector fields in a fast and numerically stable way, as opposed to the direct method based on the ill-conditioned concentration matrix.

The reduction of the vector problem to an equivalent scalar one relies on a new kind of vector spherical harmonics, which we used as basis functions throughout this paper. These exhibit a simple separable form containing the functions \(F_{lm}\) of Sheppard and Török, for which we derived several novel relations. Finally, we note that these relations of \(F_{lm}\) could facilitate the development of a fast vector spherical harmonic transform, too [35].