1 Introduction

The properties of orthogonality and completeness for given function systems are complementary. For instance orthogonality of spherical harmonics is a pre-requisite for spectral analysis. Completeness on the other hand, as expressed by the addition theorem, guarantees that any square-integrable function on the sphere can be synthesized by the given set of base functions.

Due to their complementary nature both properties could and should be used together when validating algorithms for function computation. For spherical harmonics, and more particularly for Legendre-functions the two properties are briefly recapitulated fromliterature in Sect. 2.

The main objective of this contribution is to consider orthogonality and completeness for inclination functions F { lmk} (I). It is emphasized here that neither the algorithms for F { lmk} (I)- or P lm (cosθ)-calculation, nor their validation itself is object of this manuscript. Surely, since inclination functions are derived from spherical harmonics, both properties should exist. Indeed, the so-called Wagner-Gooding identities are identified as addition theorems in disguise in Sect. 3. Orthogonality of inclination functions, has not been discussed in literature. Therefore, the orthogonality property of F { lmk} (I) will be derived in Sect. 4. This allows the complementary validation of algorithms for inclination function computation.

2 Spherical Harmonics Y lm (σ)

Starting point of the following considerations are the two complementary properties of fully normalized, complex-valued spherical harmonics Y lm (σ):

Orthogonality:

$$\begin{array}{rcl} \frac{1} {4\pi }\int \nolimits \nolimits \!\!\!{\int \nolimits }_{\hspace{-8.99994pt}\sigma }{Y }_{lm}(\sigma ){Y }_{lm}^{{_\ast}}(\sigma )\mathrm{d}\sigma = {\delta }_{ ll}{\delta }_{mm}.& &\end{array}$$
(1)

Addition theorem:

$$\begin{array}{rcl} \frac{1} {2l + 1}{\sum \nolimits }_{m=-l}^{l}{Y }_{\textit{ lm}}(\sigma ){Y }_{\textit{lm}}^{{_\ast}}(\sigma ) = {P}_{ l}(\cos \psi ),& &\end{array}$$
(2)

in which \({Y }_{\textit{lm}}(\sigma ) = {Y }_{\textit{lm}}(\theta,\lambda ) = {P}_{\textit{lm}}(\cos \theta )\exp (im\lambda )\) with θ and λ the spherical co-latitude and latitude, respectively. Moreover, Y lm  ∗ (σ) refers to the complex conjugated spherical harmonic, i.e. P lm (cosθ)exp( − imλ). As a further remark on notation: it is assumed throughout this work that all functions P lm and F { lmk} are fully normalized. The more conventional overbar is not used. One exception to this notation is made for the Legendre polynomial P l in (2), and equations that derive from it, which is a non-normalized function in this context.

The link between the addition theorem and completeness of the function system is realized by the infinite sum over the degree l, leading to the Dirac function:

$${\sum \nolimits }_{l=0}^{\infty }{\sum \nolimits }_{m=-l}^{l}{Y }_{\textit{ lm}}(\sigma ){Y }_{\textit{lm}}^{{_\ast}}(\sigma ) = \delta (\psi ).$$

2.1 Testing P lm -Algorithms 1: Orthogonality

The property (1) is reduced now to an orthogonality of associated Legendre functions:

$$\begin{array}{rcl} \frac{1} {2}{\int \nolimits }_{0}^{\pi }{P}_{\textit{ lm}}(\cos \theta ){P}_{lm}(\cos \theta )\sin \theta \mathrm{d}\theta = 2(2 - {\delta }_{m,0}){\delta }_{ll}.& &\end{array}$$
(3)

Through Gauss-Legendre quadrature, e.g. Stroud and Secrest (1966), we obtain the discretized version

$$\begin{array}{rcl} {\sum \nolimits }_{i=1}^{N}{P}_{\textit{ lm}}(\cos {\theta }_{i}){P}_{lm}(\cos {\theta }_{i}){w}_{i} = 2(2 - {\delta }_{m,0}){\delta }_{ll}.& &\end{array}$$
(4)

If the right hand side is integrated in the quadrature weights w i , the equivalent matrix version reads:

$$\begin{array}{rcl}{ P}^{<Emphasis FontCategory="SansSerif">\text{ T}</Emphasis>}WP = I,& &\end{array}$$
(5)

in which the matrix P is defined for constant order m and variable degree l and co-latitude θ:

$$P = \left (\begin{array}{cccc} {P}_{mm}(\cos {\theta }_{1}) & {P}_{m+1,m}(\cos {\theta }_{1}) &\cdots & {P}_{\textit{Lm}}(\cos {\theta }_{1}) \\ {P}_{mm}(\cos {\theta }_{2}) & {P}_{m+1,m}(\cos {\theta }_{2}) &\cdots & {P}_{\textit{Lm}}(\cos {\theta }_{2})\\ \vdots & \vdots & \ddots & \vdots \\ {P}_{mm}(\cos {\theta }_{N})&{P}_{m+1,m}(\cos {\theta }_{N})&\cdots &{P}_{\textit{Lm}}(\cos {\theta }_{N}) \end{array} \right )\!.$$

As a demonstration, Fig. 1 displays the result of applying (4) for validating a typical two-point recursion for P lm -with constant order m and increasing degree l.

$$\begin{array}{rcl}{ P}_{\textit{lm}} = {W}_{\textit{lm}}\left [\cos \theta {P}_{l-1,m} - {W}_{l-1,m}^{-1}{P}_{ l-2,m}\right ]& &\end{array}$$
(6)

with \({W}_{\textit{lm}} = \sqrt{\frac{(2l+1)(2l-1)} {(l+m)(l-m)}}\). For brevity, the starting values have been omitted here.

Fig. 1
figure 1

Testing Legendre function algorithms through orthogonality. Displayed is \(\log ({\epsilon }_{\textit{lm}})\) with \({\epsilon }_{\textit{lm}}\,=\,\mathrm{d}iag({P}^{<Emphasis FontCategory="SansSerif">\text{ T}</Emphasis>}WP)\,-\,1\) up till maximum degree L = 3, 600

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It shows that this particular recursion breaks down for double-precision arithmetic beyond degree l = 2000, although (near-)zonals and (near-)sectorials behave well up till the tested maximum degree L = 3600. Again, it is not the purpose of this paper to discuss the algorithms themselves, e.g. the need for scaling in this particular recursion.

2.2 Testing P lm -Algorithms 2: Addition Theorem

Also the addition theorem (2) is reduced now to its variant for associated Legendre functions by letting λ = λ′:

$$\begin{array}{rcl} \frac{1} {2l + 1}{\sum \nolimits }_{m=0}^{l}{P}_{\textit{ lm}}(\cos \theta ){P}_{\textit{lm}}(\cos \theta ) = {P}_{l}(\cos \psi ),& &\end{array}$$
(7)

again with the caveat that the right hand side denotes an unnormalized Legendre polynomial. Particularly for θ = θ′ and, hence, ψ = 0 a useful version arises:

$$\begin{array}{rcl} \frac{1} {2l + 1}{\sum }_{m=0}^{l}{P}_{\textit{ lm}}^{2}(\cos \theta ) = {P}_{ l}(1) = 1,& &\end{array}$$
(8)

which is visualized in Fig. 2.

Fig. 2
figure 2

Numerical test using the addition theorem of Legendre functions for θ = θ′. Displayed is the discrepancy in (8)

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But also for θ≠θ′ a useful test of (7) is demonstrated in Fig. 3.

Fig. 3
figure 3

Numerical test for θ = 90 ∘  and θ′ ∈ [0, 90 ∘ ]. Displayed is the discrepancy in (7)

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3 Inclination Functions F { lmk} (I)

Inclination functions F { lmk} (I) arise naturally when a spherical harmonic Y lm (σ) is expressed in a new coordinate system, which has been rotated through three Euler angles. The spherical harmonic is transformed through representation coefficients D { lmk} (α, β, γ), e.g. Edmonds (1957):

$$\begin{array}{rcl} & & {Y }_{\textit{lm}}(\sigma ) ={ \sum \nolimits }_{k=-l}^{l}{D}_{ {\it { lmk}}}(\alpha,\beta,\gamma ){Y }_{lk}(\sigma ),\qquad \quad \end{array}$$
(9)
$$\begin{array}{rcl}\hspace{-8.0pt}\mbox{with: }& & {D}_{{\it { lmk}}}(\alpha,\beta,\gamma ) =\mathrm{ {e}}^{im\alpha }{d}_{ {\it { lmk}}}(\beta )\mathrm{{e}}^{ik\gamma }. \end{array}$$
(10)

When rotating into the orbital coordinate system, in which the orbital plane is the new equator \(\theta = \frac{\pi } {2}\) and the new x-axis continuously points toward the satellite, we have:

$$\begin{array}{rcl}{ Y }_{\textit{lm}}(u,I,\Lambda ) ={ \sum \nolimits }_{k=-l}^{l}{i}^{k-m}\mathrm{{e}}^{i(ku+m\Lambda )}{d}_{ {\it { lmk}}}(I){Y }_{lk}(0,0),& &\end{array}$$
(11)

in which the orbital elements u, I and Λ are defined (see also Fig. 4):

$$\begin{array}{rcl} u& = \omega + \nu = \mbox{ argument of latitude} & \\ \Lambda & = \Omega -<Emphasis Type="SmallCaps">\text{ gast}</Emphasis> = \mbox{ longitude of ascending node}& \\ \end{array}$$

Sneeuw (1992) defines complex-valued normalized inclination functions F { lmk} (I) as:

$$\begin{array}{rcl}{ F}_{{\it { lmk}}}(I) = {i}^{k-m}{d}_{ {\it { lmk}}}(I){P}_{lk}(0).& &\end{array}$$
(12)

Except for normalization and for being complex-valued, the notation here deviates from the conventional Kaula inclination functions F lmp (I), which uses the non-negative index p. In contrast F { lmk} (I) uses \(k = l - 2p\) which may be negative and runs in steps of two, due to the term P lk (0) which is zero for l − k odd. For further details of rotating spherical harmonics and of the definition of inclination functions it is referred to Sneeuw (1992).

Fig. 4
figure 4

Orbit configuration. (Note: θ means gast here)

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3.1 Testing F { lmk} -Algorithms 1: Addition Theorem

Two invariances of inclination functions are given in Gooding and Wagner (2008). The restricted relation

$$\begin{array}{rcl} {\sum \nolimits }_{m=0}^{l}{F}_{ {\it { lmk}}}^{2}(I) + {F}_{ lm,-k}^{2}(I) = (1 + {\delta }_{ k,0}){P}_{lk}^{2}(0),& &\end{array}$$
(13)

also known as Wagner’s conjecture, Wagner (1983), was proved by Sneeuw (1992) by:

  • Using the symmetry \({F}_{l,m,-k} = {(-1)}^{k}{F}_{l,-m,k}\).

  • Employing the definition (12).

  • Considering that the coefficients d { lmk} (I) are an orthonormal matrix representation.

Thus, we obtain \({\sum \nolimits }_{m=-l}^{l}{F}_{\mathit{lmk}}^{2}(I) = {P}_{lk}^{2}(0)\), indeed. The general invariance

$$\begin{array}{rcl} {\sum \nolimits }_{k=-l}^{l}{ \sum \nolimits }_{m=0}^{l}{F}_{ {\it { lmk}}}^{2}(I) = 2l + 1& &\end{array}$$
(14)

can then be derived by further summation over k. It basically reflects the addition theorem (8). Sneeuw (1991) used the restricted invariance successfully for testing inclination function algorithms. The use of the general invariance for testing inclination functions is demonstrated in Fig. 5, where the residuals of the invariance are given for inclinations in the domain [0 ∘ ; 180 ∘ ] and for degrees l up to 100.

Fig. 5
figure 5

Wagner-Gooding invariance: residual of (14)

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4 Orthogonality of F { lmk} (I)

Orthogonality of inclination functions has, to the author’s knowledge, so far not been discussed in literature. The property of orthogonality does not refer here to the fact that the d { lmk} -symbols in (10) and (12) are an orthonormal matrix representation. Instead, the aim here is a relation of the type:

$${\int \nolimits }_{I}{F}_{{\it { lmk}}}(I){F}_{lmk}(I)\stackrel{?}{=}{\delta }_{\ldots },$$

in which the right hand side would be an expression with Kronecker δ-symbols.

Starting point for finding such a relation is the orthogonality of SO(3) representation symbols D { lmk} (Ω) = D { lmk} (α, β, γ), e.g. Edmonds (1957):

$$\begin{array}{rcl} &{ \frac{1} {8{\pi }^{2}} \,\int\int\int }_{\Omega }{D}_{{\it { lmk}}}(\Omega ){D}_{lmk}^{{_\ast}}(\Omega )\mathrm{d}\Omega = \frac{{\delta }_{ll}{\delta }_{mm}{\delta }_{kk}} {2l+1} & \\ & \quad \Rightarrow { \frac{1} {8{\pi }^{2}} \!\,\int\int\int }_{\Omega }\!\mathrm{{e}}^{i(m-m)\alpha }\mathrm{{e}}^{i(k-k)\gamma }{d}_{{\it { lmk}}}(\beta ){d}_{lmk}^{{_\ast}}(\beta )\mathrm{d}\Omega & \\ & \quad = \frac{1} {2l+1}{\delta }_{ll}{\delta }_{mm}{\delta }_{kk}, & \\ \end{array}$$

with

$${\,\int\int\int }_{\Omega }\ldots \mathrm{d}\Omega ={ \int \nolimits }_{0}^{2\pi }{ \int \nolimits }_{0}^{\pi }{ \int \nolimits }_{0}^{2\pi }\ldots \sin \beta \mathrm{d}\alpha \mathrm{d}\beta \mathrm{d}\gamma.$$

The orthogonality of the trigonometric functions leads to δ mm′ δ kk′ , implying that we can at most hope to find an orthogonality of inclination functions of equal orders m = m′ and k = k′. We first extract the orthogonality of representation coefficients d { lmk} :

$$\begin{array}{rcl} \frac{1} {2}{\int \nolimits }_{0}^{\pi }{d}_{ {\it { lmk}}}(\beta ){d}_{lmk}(\beta )\sin \beta \mathrm{d}\beta = \frac{{(-1)}^{k-m}{\delta }_{ll}} {2l + 1}.& &\end{array}$$
(15)

Applying definition (12) yields the sought for relation:

$$\begin{array}{rcl} \frac{1} {2}{\int \nolimits }_{0}^{\pi }{F}_{ {\it { lmk}}}(I){F}_{lmk}(I)\sin I\mathrm{d}I = \frac{1} {2l + 1}{P}_{lk}^{2}(0){\delta }_{ ll}.& &\end{array}$$
(16)

4.1 Testing F { lmk} -Algorithms 2: Orthogonality

The relation (16) can now be used to test algorithms for inclination function computation. Or, vice versa, we can now numerically test the orthogonality itself. In analogy to the discretization (4) we also apply Gauss-Legendre quadrature to numerically evaluate (16):

$${\sum \nolimits }_{i=1}^{N}{F}_{ {\it { lmk}}}({I}_{i}){F}_{lmk}({I}_{i}){w}_{i} = \frac{1} {2l + 1}{P}_{lk}^{2}(0){\delta }_{ ll}.$$

the right hand side can be accomodated into the Gauss-Legendre weights, such that we arrive at the matrix equivalent F T WF = I, in which the matrix F is defined for constant orders m and k and variable degree l and inclination I:

$$F = \left (\begin{array}{cccc} {F}_{{\it { mmk}}}({I}_{1}) & {F}_{m+2,m,k}({I}_{1}) &\cdots & {F}_{{\it { Lmk}}}({I}_{1}) \\ {F}_{{\it { mmk}}}({I}_{2}) & {F}_{m+2,m,k}({I}_{2}) &\cdots & {F}_{{\it { Lmk}}}({I}_{2})\\ \vdots & \vdots & \ddots & \vdots \\ {F}_{{\it { mmk}}}({I}_{N})&{F}_{m+2,m,k}({I}_{N})&\cdots &{F}_{{\it { Lmk}}}({I}_{N}) \end{array} \right ).$$

Figure 6 indeed suggests a unit matrix as result. The off-diagonal entries arenumerically zero. The question whether this is – up to rounding errors – a unit matrix is answered by Fig. 7, which show the diagonal entries minus 1. Both figures successfully demonstrate the validity of the orthogonality relation (16).

Fig. 6
figure 6

log(F T WF) with \(m = k = 10\) and l, l′ from 10 to 100

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Fig. 7
figure 7

Diagonal of matrix (F T WF) minus 1

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5 Conclusion

The properties of completeness and orthogonality – two fundamental properties of function systems – are suggested here for testing algorithms for calculating such function systems. They are complementary properties and, hence, should be used in tandem. One property may have more diagnostic capability in the spectral domain (l, m, k), whereas the other may identify strengths and weaknesses in the spatial domain (θ, I).

This principle is followed here for Legendre functions P lm (cosθ) and for inclination functions F { lmk} (I). The completeness property for Legendre functions is represented by the addition theorem. For inclination functions the Wagner-Gooding invariances play such a role. The orthogonality property for inclination functions, not known in literature so far, was derived here.