Spherical Functions of Mathematical Geosciences: A Scalar, Vectorial, and Tensorial Setup

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ISNI Reuter Assuming that Hn is harmonic, i. Every term in 3. Hence, the left hand side of 3. Hn is assumed to be harmonic, i. Now, 3. In this notation, the matrix. Therefore, any so- lution of the linear system 3. Hence, there exists a matrix. In other words, the solution process can be performed strictly in the mod- ulus of integers. Exact computation without rounding errors is possible in integer mode by use of integer operations addition, subtraction, multiplica- tion of integers.

When the matrix c has been calculated, the homogeneous harmonic polynomials Hn,j given by 3. Finally, it should be emphasized that exact computation, i. Helpful is an arithmetic for arbitrarily long integers whose implementation on a computer system op- erates with lists so that there is no restriction on the size of the integers worked with this is a standard feature of computer algebra packages. Let us demonstrate the technique of calculating the matrix c with an example:. Example 3.

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Spherical Functions of Mathematical Geosciences. A Scalar, Vectorial, and Tensorial Setup. Authors: Freeden, Willi, Schreiner, Michael. Free Preview. Spherical Functions of Mathematical Geosciences. A Scalar, Vectorial, and Tensorial Setup. Authors; (view affiliations). Willi Freeden; Michael Schreiner. Book.

If we choose. In the same way, we generate a set of 7 linearly independent solutions of the above system the components of which are all integers, viz. The values Hn,j n,j Homn can be determined entirely by integer operations. An easy calculation gives. The basis functions obtained can be orthonormalized exactly by means of the well-known Gram-Schmidt orthonormalization process. But the disadvantage in that approach is that the linear systems of equations result in basis functions which are all in- volved in the computational work of the orthonormalization.

Later on in Section 3. We begin by introducing scalar spherical harmonics. Essential tool is the theory of homogeneous harmonic polynomials. Let Hn be a homogeneous harmonic polynomial of degree n in R3 , i. The space of all spherical harmon- ics of degree n, i. From Theorem 3. In terms of spherical harmonics, the addition theorem allows the following reformulation. The two-dimensional counterpart of the Legendre polynomial is the Cheby- shev function.

Suppose that t is an orthogonal transformation. From Lemma 3. Let " m Harm0, Pn is uniquely determined by the prop- erties:. Applying the Cauchy—Schwarz inequality to the addition theorem The- orem 3. We therefore obtain the following lemma. Table 3. Consequently, the integral 3. Moreover, we have the following result. In addition, we mention the following results involving derivatives of the Legendre polynomial.

We prove statement iii only. Now, by virtue of 3. Hohmann Since it is not hard to show that. Chapter 10 : the gravitational force f in free space i. Returning now to Lemma 3. The power series in Lemma 3. The relation between an element and its Fourier series has been the object of many investiga- tions.

We take the opportunity to base our considerations about Fourier expan- sion theory in terms of spherical harmonics on two summability methods, namely Bernstein and Abel-Poisson summability. The point of departure for the Abel-Poisson summability is Lemma 3. Observing the identity 3. Consequently, in connection 3. As an illustration see Fig. In Fig. Combining Theorem 3. Again the summability Theorem 3. Therefore, the proof of the closure is clear. Truncated spherical harmonic expansions admit the following minimum property which should be mentioned for the convenience of the reader. Therefore, the minimum of the right-hand side of the Eq.

We summarize our results in the fundamental theorem of orthogonal spher- ical harmonic expansions. The proof of Theorem 3. Davis The property i is of great importance for practical purposes. Finally, we are interested in pointwise approximation see, e. It is known from Theorem 3. If F is assumed to be Lipschitz continuous, i. Clearly, Ln is a homogeneous polynomial of degree n which is symmetric with respect to the x3 -axis. Furthermore, by use of the coordinates 2.

This, together with Lemma 2. Let Hn be a homogeneous, harmonic polynomial of degree n with the following properties:. Then Hn is uniquely determined, and Hn coincides with the Legendre har- monic Ln of degree n, i.

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We already know that Hn as homogeneous harmonic polynomial of degree n can be written in the form. This shows us that Hn is uniquely determined by the conditions i and ii , and we have. Therefore, our considerations have shown that there exists one and only one homogeneous harmonic polynomial Hn dependent only on t satisfying the conditions i and ii. This is the assertion of Theorem 3.

An outstanding result in the theory of spherical harmonics is the Funk— Hecke formula cf. Funk , E. Hecke , C. On the other hand, according to 3. Therefore, by Theorem 3. The Funk—Hecke formula establishes the close connection between the or- thogonal invariance of the sphere and the addition theorem. Next, we discuss the role played by spherical harmonics as eigenfunctions of the Beltrami operator. We start our considerations with the following lemma. But this means that K is a member of the span of Yn,1 ,.

Summarizing our results, we therefore obtain the following theorem. Conversely, every homogeneous harmonic polynomial of degree n i. If this is the case, V is said to be reducible. If there are no invariant subspaces of V other than V itself , then V is said to be irreducible. Next, we prove the following theorem. The space Harmn of spherical harmonics of order n is irreducible. This proves Theorem 3. The irreducibility of Harmn leads us to simple representations of spherical harmonics cf.

Let Zn be a member of Harmn. From the irreducibility of Harmn cf. From 3. In Lemma 3. Fortunately, it is the case that large portions of interest can be well approximated by operators that are linear and rotation-invariant. Physical devices do not transmit spherical harmonics of arbitrarily high frequency without severe attenuation. It follows from 3. A function F of the form 3. Moreover, it is clear that F is an analytic function.

The only function that is both bandlimited and space-limited is the trivial function. Now, in addition to bandlimited but non-space-limited functions, numerical analysis would like to deal with space-limited functions. Thus, there is a dilemma of seeking functions that are somehow concentrated in both space and frequency i. Its mathematical formulation is the content of Section 7.

Thus far, only a deterministic function model has been discussed. A mathematical description of these dis- crepancies has to follow the laws of probability theory in a stochastic model. Moreover, in our approach, e. Rummel and the references therein , we suppose the covariance to be known, i. We do not discuss the details of this subject.

The associated covariance kernel is isotropic and reads as follows:. The next Table 3. Maier Signal and noise spectrum intersect at a degree and order resolution set characterized by the following relations: i Signal dominates noise. By a straightforward calculation, we obtain from the Laplace representation of the Legendre polynomials see Lemma 3. Using the Rodriguez formula see 3. This leads us to the following explicit formula for any Legendre function. Some graphical impressions of Legendre functions can be found in Figs.

Furthermore, from Theorem 3. This is the desired result. An elementary calculation starting from 3. The identity 3. By virtue of the recurrence relation 3. Consequently we get, in connection with Lemma 3. Repeated application of 3. The orthonormality immediately follows from the aforementioned results. Therefore, the functions Ln,1 ,. In terms of associated Legendre harmonics, the addition theorem cf.

In other words, summing up all spherical harmonics involving associated Legendre functions via the addition theorem leads apart from a multiplica- tive factor to the orthogonal invariant Legendre kernel functions. They have n zeros. Since they divide the sphere into zones, they are also called zonal harmonics see Fig.

They divide the sphere into compart- ments in which they are alternately positive and negative, and are called tesseral harmonics. Next, we are interested in describing angular derivatives of associated Leg- endre spherical harmonics. But it also informs us that the scalar Legendre spherical harmonics are not eigenfunctions of this operator. Our considerations presented in Lemma 3.

On the other hand, the representations are lengthy and at least in the case of the operator of the longitude rather complicated to handle. Even more, singularities at the poles cannot be avoided for both surface gradient and the surface curl gradient, i. Lemoine et al. Macmillan et al. Freeden Spherical harmonics involving associated Legendre polynomials, i. We already know from Lemma 3. Fur- 2 thermore, it is well-known that. Collecting our results, we therefore obtain the following representation in terms of cartesian coordinates.

Expressed in terms of the usual polar coordinates 2. More explicitly, from the expression 3. In fact, in order to characterize the ingredients in the rep- resentations 3. Once more, the reason for the validity of 3. Earlier, in Section 3. Our investigations about Legendre harmonics in Section 3. In addition, they are computed exclusively by integer operations.

However, it turns out that the basis established by exact computation is only partially orthogonal in Harmn R3 , i. Actually, it is a compromise between the two methods presented in the preceding sections. Neverthe- less, in comparison to the exact computation explained in Section 3. By comparison with the results obtained in Section 3.

As requested by the recursion 3. From the formulas involved in the computational process, it is obvious that the partial orthogonal basis subsystems can be computed in an exact arithmetic. From each of these sets, the corresponding homogeneous harmonic polyno- mials are derived. They are written down in a schematic manner. For more theoretical details on scalar spherical harmonics, the reader may want to consult some of the following references: A. Wangerin , R. Courant, D. Hilbert O. Kellogg , E. Whittaker, G. Wat- son , C. Feshbach , J. Lense , E. Hobson , F.

John , I. Sneddon , A. Edmonds , R. Seeley , T. McRobert , E. Stein, G. Weiss , H. Hochstadt , N. Lebedev , N. Vilenkin ,W. Freeden a , W. Lemma 4. Then, we have the following properties:. Therefore, we have the following result. Remark 4. Observing the logarithmic singularity of the Green function, we see by applying the Second Green Surface Theorem that the spherical harmonics of degree n, i.

Freeden a. Keeping the linearization of 4. Theorem 4. First we are concerned with the existence of the occuring integrals. Analogous to Theorem 4. Gramsch Inserting 4. In detail,. Figures 4. An adequate answer is the construction of integral formulas that will be presented later on. The concept closely parallels the proof of Lemma 4. Observing the characteristic singularity of the Green function, we see by applying the Second Green Surface Theorem that the spherical harmonics of degree n, i.

This leads us to the formulation of the following result. In connection with Lemma 3. This shows that condition i is valid. Explicitly, written out, we have. In the identity 4. Corollary 4. In other words, the same reasoning as in Theorem 4. Kellogg Its bilinear expansion reads as follows. In other words, Corollary 4. Combining Theorem 4. From Theorem 4. Freeden, M. Schreiner , T. Fehlinger et al.

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Freeden, K. For more details the reader is referred to Chapter By virtue of the Cauchy—Schwarz inequality, we get from Theorem 4. From a table of integrals see, e. It follows that the iterated Green function is expressible by means of the dilogarithm. We give a concrete application: If a1 ,. Using the Cauchy—Schwarz inequality, we get from 4. Freeden, J. Fleck Then, there exists one and only one function S of type 4. We denote this function by SN. The integral 4. Next, we explain the intimate relationship between best approximate and spline integration.

In other words, the best approx- imation to the integral is precisely the unique approximation that is exact for spline functions. According to the classical Fredholm—Hilbert theory of linear integral equa- tions see, e. First we have. The second sum can be transformed by use of the recurrence relation Lemma 3. This enables us to establish an elementary representation of G t by integra- tion. To this end, we need certain values of G t to determine the constants of integration. In fact, we have. Hence, the representation theorem of the theory of orthogonal expansions see Theorem 3.

These properties are of basic interest in spherical spline settings corresponding to iterated Beltrami derivatives see W. Integration by parts, i. Thus, by combination of Theorem 4. The identity 4. From 4. According to our nomenclature, Harm0, The space Harm0, Let Y be an element of class Harm0, On the other hand, we know from 4. However, if we require that F is orthogonal to the null space Harm0, Freeden c. The Green functions with respect to iterated Beltrami operators have been investigated in detail by W. Freeden ; W. Free- den b , G.

Reuter in the multi-dimensional case. The integral formulas are due to W. Space regularizations of Green functions and their use in multiscale approx- imation of geodetic problems can be found in a note due to W. Wolf The explicit representation stated in Lemma 4. Gutting Related formulas in Euclidean spaces Rn are multidimensional Euler summation and cubature formulas cf. Fleck and the references therein. What is our understanding in this context?

One important aspect is the easy transition from scalar spherical harmonics to the vectorial ones.


A simple approach is to formulate the vectorial problem in terms of cartesian components. Moreover, the rotational symmetry, i.

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The layout of this chapter on vector spherical harmonics is as follows: Section 5. In Section 5.

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Section 5. The inter- relations between vector spherical harmonics and homogeneous harmonic vector polynomials are investigated in more detail in Section 5. It fol- lows in Section 5. Sec- tion 5. Vectorial counterparts of the Legendre polynomial are introduced in Section 5. Degree and order variances are discussed in Section 5.

After a deeper insight into counterparts of Legendre polynomials within the vectorial context and the degree and order variances, we consider in Section 5. Furthermore, from Lemma 2.

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Lemma 5. In connection with 2. The contour lines represent the scalar spherical harmonic Yn,j from which the vector spherical harmonics are generated. Obviously, vector spherical harmonics can be calculated from the rep- resentations of scalar spherical harmonics. Illustrations of the tangential vector spherical harmonics are given in Fig. Example 5. In what follows, we formulate the decomposition theorem in a rigor- ous sense. Our particular purpose is to show how the scalar functions Fi can be determined in an explicit way by use of the concept of the Green function with respect to the Beltrami operator.

An example is shown in Figs. Theorem 5. Furthermore, 5. Appli- cations of O 2 and O 3 to 5. Hence, the normalization conditions 5. Backus suggests calling it consoidal. The space harmn of vector spherical harmonics of degree n, therefore, 1 2 lead naturally to radial, consoidal, and toroidal subspaces harmn , harmn 3 and harmn. The spaces are mutually orthogonal. In addition, they are orthogonally invariant and irreducible.

The Helmholtz decomposition theorem Theorem 5. Next, we prove the closure and completeness of vector spherical harmonics intrinsically on the sphere note that a non-intrinsic proof follows from the arguments of Section 5. For our purpose here, we use vectorial variants of the scalar zonal Bernstein kernels. Although the approximation of func- tions by using Bernstein polynomials is one of the classical research topics and their theory is a rich one, their application within the vector theory of spherical harmonics seems to go back to W.

Essential tools are the theory of the Green function with respect to the iterated Beltrami operator and the Helmholtz decom- position theorem. Next, we are interested in the Bernstein summability of Fourier expan- sions in terms of vector spherical harmonics. To this end, we need some preparatory material more precisely, Lemma 5. Essen- tial tool of our considerations is the Green function with respect to the Beltrami operator cf.

It follows immediately from the scalar theory. Considering the o i -derivatives, we have to verify the following lemma. We have to study the convergence of the last integral. At this point, we use the recurrence relation Lemma 3. This gives us the identity.

Keeping this result in mind, we return to the integral 5. As a matter of fact, the identity 5. It is well-known that the value of our integral can be estimated as follows:.

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In addition, we get information about the speed of the convergence, i. In connection with 5. Due to Theorem 5. In what follows, similar rela- tions between vector spherical harmonics and homogeneous harmonic vec- tor polynomials are developed. Suppose that Hn is of class Harmn R3. This yields the desired result. By inverting the equations 5.

By virtue of Lemma 5. Next, we are interested in closure and completeness properties of vector spherical harmonics. It should be noted that, we avoid problems arising from the singularities of a spherical coordinate system when using cartesian coordinate representations.

Therefore, via the well-known procedure, by letting. Our purpose is to determine the vector spherical harmonics using exclu- sively exact integer arithmetic. For solutions yn of 5. Together 2. This means that, up to a constant, G is a spherical harmonic of degree n, and solutions of 5.

This i observation has immediate consequences for the spaces harmn of vector spherical harmonics. The spaces harmn of vector spherical harmonics are or- thogonally invariant and irreducible. To be more concrete, suppose i that there exists an orthogonally invariant subspace of harmn. This, however, is a contradiction to the irre- ducibility of the spaces Harmn. In what follows, we deduce some concrete consequences for vector spherical harmonics. Because of Theorem 5. Consequently, for every orthogonal transformation t we have, on the one hand, an associated matrix cj,l with.

This means that k i 5. Let t be of class O 3. Actually, it is this lemma which makes tensors of the form 5. Furthermore we 2 2. Consequently, 5. A similar argument leads to. With condition ii of Theorem 5. Hence we end up with the following representation theo- rem. Observing this setting, we are able to deduce the following identities. The following statements are true.

Assume that the function under consid- eration is a spherical harmonic Yn of class Harmn. Then, it follows from 5. Using 5. In other words, the equations 5. As an immediate consequence of Lemma 5. Remark 5. Gervens More explicitly, the following theorem is valid. From the scalar theory, we know the fundamental role of the addition the- orem of spherical harmonics. Our interest now is the formulation of a vec- torial analogue of the addition theorem involving tensorial structure.

Then, the announced vectorial analogue of i i. Our purpose is to explain how a vectorial counterpart of tensorial nature for the Legendre polynomial comes into play. In accordance with this nomenclature, 5. Altogether, this leads us to the following variant of the addition theorem for vector spherical harmonics. As auxiliary results, we verify the following identities. Suppose that n is a non-negative integer.

Chapter 3. Corollary 5. Suppose yn is a member of harmn. Next, we deal with generalizations of the Funk—Hecke formula to the vecto- rial case. Notice that the integral 5. We start with the recapitulation of some topics of representation theory needed for our studies on the Funk—Hecke formula cf. Section 3. If there exists no subspace of v other than v itself which is S-invariant, then v is said to be irreducible. But this means that the adjoint operator of Rt is given by RtT. An analogous result holds for o 3.

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Together with 5. Furthermore, harmn is irreducible. Furthermore, we have shown in Theorem 3. A generalization of these results to the vectorial case can be written down as follows:. By virtue of the addition theorem for vector spherical harmonics, we immediately obtain the following consequence. Assume that. We now come to the second variant of the vectorial Funk—Hecke formula as announced in 5. The basic ideas to handle this problem are the representation of vector spherical harmonics by means of restrictions of ho- mogeneous harmonic vector polynomials and the componentwise application of the scalar Funk—Hecke formula.

Then, we know from Lemma 5. We know from the formulas 5. Combining these results, we get the second variant of the vectorial Funk—Hecke formula. Thus, we obtain from 5. Let Pn be the Legendre polynomials of degree n, v pn the i corresponding Legendre tensors of degree n and type i, k , and pn the cor- responding Legendre vector of degree n and type i. Maier , Their degree variances are shown in Fig.

Beth The signal-to-noise relation is determined by the degree and order reso- lution set of type i. In other words, see also 5. Svensson ,W. More explicitly, we set. This consideration leads us to the introduction of the following set of vector spherical harmonics note that our approach essentially follows the ideas of the concept as introduced by A. Edmonds :.