Abstract
We propose a single one-parameter family of orthogonal harmonic functions expressed in terms of spheroidal coordinates as independent variables to construct a common orthogonal basis for the \(L_2\)-Hilbert spaces of quaternionic monogenic functions in the space exterior of a spheroidal domain (either prolate or oblate). We give recurrence relations for the elements that constitute such a basis, which are particularly easy to handle from a computational point of view. Conversion formulas among the classes of harmonic and monogenic functions associated with a spheroid of arbitrary eccentricity to those related to the Euclidean ball are derived.
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1 Introduction
The theory of monogenic (or Fueter regular) functions of a vector variable in a domain in three-dimensional Euclidean space, taking values in the space of quaternions, has a wide range of applications. A significant part of the theory of monogenic functions has been built around the study of quaternionic counterparts of holomorphic functions of one complex variable, offering a refinement of classical harmonic analysis in three and four dimensions. Said sort depends on whether monogenic functions present certain peculiarities, such as continuity, differentiability or integrability, orthogonality with respect to different inner products, and so on, throughout the domain of the variables. The study of the fundamental properties of monogenic functions is linked to harmonic functions through the Riesz and Moisil-Teodorescu systems of first-order constant coefficient partial differential equations generalizing the Cauchy–Riemann equations [5, 6, 10,11,12,13, 26,27,28,29].
The original impetus in initiating the investigation of orthogonal bases spanning the Hilbert spaces of square-integrable harmonic and monogenic functions defined in the interior of spheroidal (resp. prolate and oblate) domains of the form
and
for \(\alpha >0\) was proposed in [7], and [16, 17, 23]. These (confocal) domains become rounder as they degenerate with \(\alpha \rightarrow \infty \) (since \(\tanh \alpha \rightarrow 1\)). In [8], the spheroidal harmonics were defined following [7], with a rescaling factor that permits including the Euclidean ball to limit both the prolate and oblate cases. We refer to [9] and [20] concerning further properties of spheroidal harmonic and monogenic polynomials. These works do not include the harmonics and monogenics vanishing at infinity, which are perhaps the more fascinating classes from the point of view of a physical application. Although such types of functions have not yet been used to any great extent in mathematical physics, from the perspective of the theory of the solution of Laplace’s equation in three variables, it would be interesting to address the problem of constructing orthogonal sets of harmonic and monogenic functions defined in a region outside a spheroid, whose elements are parametrized by the shape of the corresponding spheroids. In this sense, we propose a single one-parameter family of orthogonal spheroidal harmonics to build a common orthogonal basis for the Hilbert spaces of square-integrable monogenic functions in the space exterior of a spheroidal domain of arbitrary eccentricity. To the best of our knowledge, these ideas seem to be new.
The outline of the paper is as follows. Section 3 employs the two kinds of associated Legendre functions to construct the basic external spheroidal harmonics that assume prescribed values on the boundary of the corresponding spheroids, combined into a single one-parameter family. These functions are shown to include the ordinary solid spherical, prolate, and oblate spheroidal harmonics as limiting cases. The orthogonality of the basic harmonics is taken with respect to two natural inner products, leading to the discussion of the proper external spheroidal harmonics. The main difficulties of this investigation will center on analyzing these functions. Conversion formulas that relate the coefficients of the expansions among the spheroidal and spherical harmonic systems are obtained. The basic external spheroidal monogenic functions are calculated in Sect. 4, and explicit formulas for their nonscalar parts are obtained in terms of the proper harmonics. We prove that these functions form a common orthogonal basis for the one-parameter family of \(L_2\)-Hilbert spaces of monogenic functions defined in a region outside a spheroid. Besides, we show the corresponding orthogonality of the spheroidal monogenics over the surface of the spheroids with respect to a suitable weight function.
2 Notation and Preliminaries
The majority of functions used in technical and applied mathematics originated from investigating practical problems. A relevant example is the Ferrers’ associated Legendre functions of the first and the second kinds, \(P^{m}_{n}(z)\) and \(Q^{m}_{n}(z)\), of degree n and order m, for \(z \in [-1, 1]\) and \(z \in (1, \infty )\). When n and m are nonnegative integers, these functions are defined by (cf. [14, Ch. III])
and
where
is the Legendre polynomial (or Legendre function of the first kind), and
is the Legendre function of the second kind. The relation (1) is known as Neumann’s integral formula [22, p. 24] (cf. [14, p. 63]).
We shall observe a slight difference between the variation of the index m in any of the previous definitions. Although the \(P^{m}_{n}(t)\) are only defined for nonnegative integer values of m, which are equal to or less than n, the functions \(Q^{m}_{n}(s)\) are defined for all nonnegative integer values of m.
In addition to the definitions introduced above, we further have from [14, p. 108]:
Lemma 1
Let n and m be nonnegative integers, and let \(\vert s\vert > 1\). Then
Here \(_2F_1\) is the usual notation for the classical Gaussian hypergeometric function and the Pochhammer symbol is \((a)_n = a(a + 1) \cdots (a+n-1)\) with \((a)_0 = 1\) by convention.
The associated Legendre functions of the first and the second kinds are defined for negative integer order m by
where it is supposed that \(n \ge m \ge 0\).
There is a classical formula that expresses the product of two associated Legendre functions of the second kind in terms of an associated Legendre function of the same type [3].
Proposition 1
Let \(A^{m}_{r} = (1/2+m)_{r}\). Define
For nonnegative integers \(n_1\), \(n_2\), and m, the following relation holds:
We need the following preliminary result.
Lemma 2
For all nonnegative integers n, m, \(Q^{m}_{n}(s) Q^{m}_{n+2}(s) > 0\) in \(\vert s \vert > 1\). Further, let \(c > 1\) be a fixed real constant. Then the following identity holds:
Here \(\alpha _{n_1,n_2}^{m,r}\) has the same meaning as in Proposition 1.
Proof
The statements follow by combining (4) with the identity [24, p. 123]
\(\square \)
We now consider the recurrence formulas for the associated Legendre functions of the first and the second kinds, which will be used in the forthcoming sections [14, 30].
Proposition 2
-
1.
Let \(n \ge 0\) and \(0 \le m \le n\), and let \(t \in [-1,1]\). Then
$$\begin{aligned}&(1-t^2) (P_{n+1}^m)^{\prime }(t) = (n+1+m)P_{n}^{m}(t) - (n+1)t P_{n+1}^{m}(t), \end{aligned}$$(6)$$\begin{aligned}&(n+1-m)P_{n+1}^m(t) = (2n+1)t P_{n}^m(t) - (n+m)P_{n-1}^m(t), \end{aligned}$$(7)$$\begin{aligned}&(t^2 - 1)(P_{n+1}^m)^{\prime }(t) = (1 - t^2)^{1/2} P_{n+1}^{m+1}(t) + m t P_{n+1}^m(t), \end{aligned}$$(8)$$\begin{aligned}&2mtP_{n+1}^m(t) \nonumber \\&\quad = -(1 - t^2)^{1/2} \Bigg (P_{n+1}^{m+1}(t) + (n+1+m)(n+2-m)P_{n+1}^{m-1}(t)\Bigg ), \end{aligned}$$(9)$$\begin{aligned}&(1 - t^2)^{1/2} P_{n}^{m+1}(t) = (n - m)t P_{n}^{m}(t) - (n + m)P_{n-1}^{m}(t), \end{aligned}$$(10)$$\begin{aligned}&2mP_{n+1}^m(t) = -(1 - t^2)^{1/2} \Bigg (P_{n}^{m+1}(t) + (n+m)(n+m+1)P_{n}^{m-1}(t)\Bigg ), \end{aligned}$$(11)$$\begin{aligned}&(1-t^2)^{1/2} P_{n+1}^m(t) = \frac{1}{2n+3} \, \Bigg (-P_{n+2}^{m+1}(t) + P_{n}^{m+1}(t)\Bigg ). \end{aligned}$$(12) -
2.
Let n and m be nonnegative integers, and let \(|s| > 1\). Then
$$\begin{aligned}&(1-s^2) (Q_{n+1}^m)^{\prime }(s) = (n+1+m)Q_{n}^{m}(s) - (n+1)s Q_{n+1}^{m}(s), \end{aligned}$$(13)$$\begin{aligned}&(n+1-m)Q^{m}_{n+1}(s) = (2n+1)s Q^{m}_{n}(s) - (n+m)Q^{m}_{n-1}(s), \end{aligned}$$(14)$$\begin{aligned}&(s^2 - 1) (Q_{n+1}^m)^{\prime }(s) = (s^2 - 1)^{1/2}Q_{n+1}^{m+1}(s) + msQ_{n+1}^m(s), \end{aligned}$$(15)$$\begin{aligned}&2ms Q_{n+1}^m(s) \nonumber \\&\quad = (s^2 - 1)^{1/2} \Bigg (-Q_{n+1}^{m+1}(s) + (n+1+m)(n+2-m)Q_{n+1}^{m-1}(s)\Bigg ), \end{aligned}$$(16)$$\begin{aligned}&(s^2-1)^{1/2} Q^{m+1}_{n}(s) = (n-m)s Q^{m}_{n}(s) - (n+m)Q^{m}_{n-1}(s), \end{aligned}$$(17)$$\begin{aligned}&2mQ_{n+1}^m(s) = (s^2 - 1)^{1/2} \Bigg (-Q_{n}^{m+1}(s) + (n+m)(n+m+1)Q_{n}^{m-1}(s)\Bigg ), \end{aligned}$$(18)$$\begin{aligned}&(s^2 - 1)^{1/2} Q_{n+1}^m(s) = \frac{1}{2n+3} \, \Bigg (Q_{n+2}^{m+1}(s) - Q_{n}^{m+1}(s)\Bigg ). \end{aligned}$$(19)
3 Solutions of Laplace’s Equation in Spheroidal Coordinates
This section considers the problem of finding single one-parameter families of harmonic functions applicable to the space exterior of a spheroid, with particular emphasis on those orthogonal in the \(L_2\)-Hilbert space structure. This cannot be done with models where the Euclidean ball is only treated as a degenerate case [7, 14]. It requires a separate yet utterly analogous treatment for prolate and oblate spheroids. The construction of external harmonics becomes much more complicated than internal harmonics since they contain logarithmic functions.
3.1 Basic External Spheroidal Harmonics
The starting point of the present investigation is a result previously published by the author [21]. Consider the nested family domains, bounded by coaxial spheroids scaled so that the major axis is of length 2:
(Confocal spheroids are often used, which differ by a change of scale depending on \(\mu \).) The parameter \(\mu \) denotes the eccentricity of \(\Omega _{\mu }\), which by convention is in the interval (0, 1) (prolate spheroid) or in \(i\mathbb {R}^+\) (oblate spheroid). The intermediate value \(\mu =0\) gives the Euclidean unit ball
Suppose for the moment that \(\mu \in (0,1)\). In this way, \(\Omega ^{*}_{\mu }:= \mathbb {R}^3 \setminus {\overline{\Omega }}_{\mu }\) (where \({\overline{\Omega }}_{\mu }\) denotes the closure of \(\Omega _{\mu }\)) is parametrized using prolate spheroidal coordinates \((\eta , \vartheta , \varphi )\), corresponding to the family (20), which are related to Cartesian coordinates by
with \(\eta \in [\eta _{\mu }, \infty )\), \(\vartheta \in [0,\pi ]\), and \(\varphi \in [0,2\pi )\), where the boundary value \(\eta _{\mu }\) is given by
By equations (21), direct computation shows that
and also
Hence
where
is positive. In the considerations to follow, we will often omit the argument of (23) and write \(\omega \) instead of \(\omega (\mu )\).
The oblate case is obtained by (21) via analytic continuation using \({\widetilde{\eta }} = \eta - i\pi /2\), thinking of \(\mu \in i\mathbb {R}^+\) as being boundary values of the first quadrant in the complex plane. The following terms
inside the radicals in (23) are now complex conjugates, where
equals to the product of the distances from any point on the prescribed spheroids \(\Omega _{\mu }\) to the two foci \((\pm \mu , 0, 0)\). The function defined by (24) will play an important role in the forthcoming sections.
Hence, when \(\mu \in i\mathbb {R}^+\), \(\Omega ^{*}_{\mu }\) is parametrized by
where the coordinates range over \({\widetilde{\eta }} \in [{\widetilde{\eta }}_{\mu }, \infty )\) with \((\mu /i)\sinh {\widetilde{\eta }}_\mu =1\), \(\vartheta \in [0,\pi ]\), and \(\varphi \in [0,2\pi )\).
The external harmonics to be employed in the sequel are defined as follows.
Definition 1
Let \(n \ge 0\) and \(0 \le m \le n\). For \({\textbf{x}} \in \mathbb {R}^3 {\setminus } \{{\textbf{0}}\}\), the basic external spheroidal harmonics of degree \(-(n+1)\) and order m are
where \(\Phi ^{+}_{m}(\varphi ):= \cos (m\varphi )\), \(\Phi ^{-}_{m}(\varphi ):= \sin (m\varphi )\), and for \(\mu \ne 0\),
with
The functions \(U^{-}_{n,0}[\mu ]\) vanish identically, as do all \(U^{\pm }_{n,m}[\mu ]\) for \(m > n\). Therefore when we refer to the set \(\{U^{+}_{n,m}[\mu ], \, U^{-}_{n,m}[\mu ]\}\), we always exclude the indices which apply to these trivial cases, even when we do not explicitly state \(0 \le m \le n\) for the “\(+\)” case and \(1\le m \le n\) for the “−” case.
The basic harmonics (25), except for the constant factors \(\beta _{n,m}\), and the rescaling of the \({\textbf{x}}\) variable, are the functions defined in [14, Ch. X]. The motivation behind the choice for redefining these functions is explained in Proposition 3 below.
Some examples of (25) in low degree are exhibited in Table 1.
The coefficients \(\beta _{n,m}\) in the expression (26) is for the following.
Proposition 3
For all \({\textbf{x}} \in \mathbb {R}^3 \setminus \{{\textbf{0}}\}\), the limit \(\lim _{\mu \rightarrow 0} U^{\pm }_{n,m}[\mu ]({\textbf{x}})\) exists and is given by the external solid spherical harmonics
where we employ spherical coordinates \(x_0 = \rho \cos \theta \), \(x_1 = \rho \sin \theta \cos \varphi \), and \(x_2 = \rho \sin \theta \sin \varphi \).
Proof
To prove this, we note that since the variable \(\varphi \) in (25) does not depend on the variable \(x_0\), we examine the factors \(P_n^{m}(2x_0/\omega )Q_n^{m}(\omega /(2\mu ))\) in (26) with \(\omega \) given by (23). Bearing in mind that
it follows that \(\omega = 2\vert {\textbf{x}}\vert + O(\mu ^2)\) as \(\mu \rightarrow 0\). Furthermore, we have once more from (23) that \(2x_0/\omega = x_0/\vert {\textbf{x}}\vert + O(\mu )\), so \(P_n^m(2x_0/\omega ) \rightarrow P_n^m(x_0/\vert {\textbf{x}}\vert )\) as \(\mu \rightarrow 0\). According to formula (2), direct calculation gives \(\beta _{n,m} Q^m_n(s) \simeq 1/s^{n+1}\) as \(s=\omega /2\mu \) tends to infinity, corresponding to \(\mu \rightarrow 0\) for fixed \({\textbf{x}}\). This establishes the statement. \(\square \)
By the proposition just proved, it is observed that the external solid spherical harmonics (28) are embedded in the one-parameter family of basic external spheroidal harmonics. In contrast, in treatments such as [7, 14], the external harmonics degenerate as the eccentricity of the spheroid decreases.
It is clear that, unlike \(U^{\pm }_{n,m}[0]({\textbf{x}})\), the functions \(U^{\pm }_{n,m}[\mu ]({\textbf{x}})\) are generally not homogeneous when \(\mu \ne 0\).
3.2 Further Properties of the Basic Harmonics
We study the orthogonality of the external harmonics (25) with respect to two natural inner products.
Consider the Dirichlet inner product defined by
where \(d\sigma \) denotes the area element on \(\Gamma _{\mu }:= \partial \Omega _{\mu }\), and
is the unit outward normal vector to \(\Gamma _{\mu }\) at the point \(P^{*}:= (\eta _{\mu }, \vartheta , \varphi )\), with \(\cosh \eta _{\mu } = 1/\mu \) by (22).
Since the measure on the boundary is \(d\sigma = (1-\mu ^2)^{1/2} \sin \vartheta d\vartheta d\varphi \), we can compute the outward normal derivative \(\partial /\partial {\textbf{n}}\) of g at any point using (30) to obtain
We will now show the orthogonality of the external harmonics (25) in the sense of the integral (29).
Proposition 4
Let \(\mu \) be fixed. For each \(n\ge 0\), the collection
is orthogonal in the sense of the Dirichlet integral (29) and their norms squared are equal to
where the coefficients \(\beta _{n,m}\) have the same meaning as in (27). We use the symbol: \(\delta _{m_1,m_2} = 0\) or 1, according as \(m_1 \ne m_2\), or \(m_1 = m_2\).
Proof
For the sake of simplicity in the proof, we assume that \(\mu \in (0, 1)\) because the case \(\mu \in i\mathbb {R}^{+}\) is similar. When \(m_1 \ne m_2\), we have by the orthogonality of the set \(\{\Phi _{m_1}^{+}, \Phi _{m_2}^{-}\, \vert \, m_1 \ge 0, m_2 \ge 1\}\) on \([0, 2\pi ]\),
According to (31), for \(m_1 = m_2 = m\), a direct computation shows that
The same value is obtained when we replace \(\Phi _{m}^{+}(\varphi )\) by \(\Phi _{m}^{-}(\varphi )\) throughout, \(m > 0\). Thus the statement is established. \(\square \)
Next, we assert that the basic harmonics are not necessarily orthogonal in the closed subspaces \({\text {Har}}_2(\Omega ^{*}_{\mu }) = L_2(\Omega ^{*}_{\mu }) \cap {\text {Har}}(\Omega ^{*}_{\mu })\) of \(L_2(\Omega ^{*}_{\mu })\) when \(\mu \ne 0\) with respect to the ordinary \(L_2\)-inner product:
where \(d{\textbf{x}} = dx_0 dx_1 dx_2\).
Proposition 5
The collection (32) does not form an orthogonal family of \({\text {Har}}_2(\Omega ^{*}_{\mu })\) unless \(\mu = 0\).
Proof
It is a simple matter to check that \(U^{\pm }_{n,m}[\mu ] \in L_2(\Omega ^{*}_{\mu })\). We assume again that \(\mu \in (0, 1)\). Applying the coordinates (21), gives the infinitesimal volume element \(d{\textbf{x}} = dR d\varphi \), where
It is clear that, when \(m_1 \ne m_2\), we have
Let \(m_1 = m_2 = m\). We compute
According to (7), we obtain
By substituting this value into (35), it is found that the corresponding inner product is, in particular, distinct from zero when \(n_2 - n_1 = \pm 2\). For the remainder of the proof, we consider when \(n_2 - n_1 = 2\). The other case \(n_2 - n_1 = -2\), can be treated analogously.
For \(n_2 = n_1 + 2\), a straightforward computation shows that
where
The same value is obtained when \(\Phi _{m}^{+}(\varphi )\) is replaced by \(\Phi _{m}^{-}(\varphi )\), \(m > 0\). By Lemma 2, it follows that \(\mu ^{-(2n_1+1)} I_{n_1,m}(\mu ) > 0\) for all \(n_1, m = 0, 1, \dots \) and fixed \(\mu > 0\).
For the limiting case, when \(\mu = 0\), we use Proposition 3 to show that \(\lim _{\mu \rightarrow 0} I_{n_1,m}(\mu ) / \mu ^{2n_1+1} = 0\), for all \(n_1=0, 1, \dots \) \((m \ge 0)\). Thus, \(\langle U^{+}_{n_1,m}[\mu ], U^{+}_{n_1+2,m}[\mu ] \rangle _{L_2(\Omega ^{*}_{\mu })}\) tends to zero as \(\mu \rightarrow 0\). Similarly, we can prove that \(\langle U^{-}_{n_1,m}[\mu ], U^{-}_{n_1+2,m}[\mu ] \rangle _{L_2(\Omega ^{*}_{\mu })} \rightarrow 0\) when \(\mu \) tends to zero. This establishes the statement. \(\square \)
The lack of orthogonality of the basic harmonics over the exterior of the prescribed spheroids in the usual \(L_2\) sense means that defining suitable families of orthogonal external harmonics should be handled carefully. It is always possible to use an appropriate geometric weighting factor or apply an orthogonalization process to the prescribed harmonic functions, such as the Gram–Schmidt procedure that restores orthogonality. However, this orthogonalization process may be time-consuming and unstable. We preferably discuss a constructive approach discussed in [21] and show how it will be helpful not only from a function point of view but also for fast and stable computations.
3.3 Proper External Spheroidal Harmonics
We shall now be concerned with the following functions for the actual carrying out of the construction and given definition (25).
Definition 2
Let \(U^{\pm }_{n,m}[\mu ]\) have the same meaning as in Definition 1. Let \(n \ge 0\) and \(0 \le m \le n+1\). The proper external spheroidal harmonics of degree \(-(n+3)\) and order m are
The proper harmonics \(V^{\pm }_{n,m}[\mu ]\) will play a crucial role in studying the basic external spheroidal monogenics in Sect. 4.
Following the notation already employed, we use \(V^{\pm }_{n,m}[\mu ] = V_{n,m}[\mu ] \Phi _m^\pm \) when the factors \(\Phi _m^\pm \) are not of interest.
It will be convenient before proceeding to investigate the algebraical forms of the ansatz functions \(V_{n,m}[\mu ]\). We will assume in the sequel that \(\mu \in (0,1)\) because the case \(\mu \in i\mathbb {R}^+\) is similar.
By differentiating (21),
and combining (6)–(13) and (7)–(14) with the definition (37), we are thus led to the following remarkable representation:
where the coefficients \(\beta _{n,m}\) have the same meaning as in (27).
As a consequence of (38) we notice that \(V_{n,m}[\mu ]=0\) for \(m > n+1\) since \(P_{n}^{m} = 0\) for \(m > n\).
According to (3) and (38), we deduce the following elementary identity.
Lemma 3
For each \(n \ge 0\) and \(0 \le m \le n+1\),
Define
for \(n>0\) and fixed \(\mu > 0\).
We have, for instance,
and
It can further be seen that
We shall return to these formulas in Sect. 3.5, where they will be used to prove the orthogonality of the proper harmonics (37) in the \(L_2\)-Hilbert space \({\text {Har}}_2(\Omega ^{*}_{\mu })\).
Some examples of (37) in low degree are exhibited in Tables 2 and 3.
We now proceed to establish an elementary recurrence formula for the functions \(V_{n,m}[\mu ]\) to avoid the difficulties usually associated with manipulations such as formula (38).
Proposition 6
For each \(n \ge 2\), the following recurrence relation holds:
This uses the convention \(V_{n-2,m}[\mu ] = 0\) when \(m > n-1\).
Proof
According to (7), (14), and (38), and bearing in mind that
we have
The result now follows. \(\square \)
The following “reverse Appell property”, which follows from (44), involves the derivatives of the external solid spherical harmonics defined by (28) with respect to \(x_0\):
The functions \(V^{\pm }_{n,m}[\mu ]\) are not so simply related to \(U^{\pm }_{n,m}[\mu ]\) for \(\mu \ne 0\), as we show below.
Theorem 7
Let \(n \ge 0\) and \(0 \le m \le n+1\). The coefficients \(v_{n,m,k}\) in the relation
are given by
Proof
Suppose inductively that the formula holds when n is replaced by \(n'<n\). Then
Since, by (47),
we find that the stated formula holds, completing the proof. \(\square \)
3.4 Conversions Among Spheroidal and Solid Spherical Harmonics
In this section, we determine the coefficients \(\alpha \) and \({\widetilde{\alpha }}\) of the following direct and inverse transformation formulas:
Fix a value of \(\mu \). By referring to these expansions, we shall employ the constraints that the index m is not involved in the summations, and the values of the same evenness restrict the index k as a given n. It follows from symmetry considerations that the above relations will work for the “\(+\)” and “−” cases (cosines and sines) and, strikingly, for all values of \(\mu \).
For harmonic functions outside a prolate or an oblate spheroid, the transition from the expansion in external spheroidal harmonics to that in external solid spherical harmonics (and vice-versa) is worked out in [1]. Some of these formulas are discussed thoroughly in [2]. Two of these fundamental formulas, relevant to the sequel, are reproduced in our notation below (i.e., the factor (27) has been incorporated into (48) and (49)).
Proposition 8
For \(n \ge 0\), consider the rational constants
for \(0 \le m \le n\), and let \(\alpha _{n,m,k} = {\widetilde{\alpha }}_{n,m,k} = 0\) otherwise. Then
Since \(\partial /\partial x_0\) is a linear operator, we automatically have the corresponding transformation formulas for the proper harmonics:
Corollary 1
Let \(n \ge 0\) and \(0 \le m \le n+1\). Then
Here \(\alpha _{n,m,k}\) and \({\widetilde{\alpha }}_{n,m,k}\) have the same meaning as in Proposition 8.
3.5 Orthogonality Properties of the Proper Harmonics
We begin by formulating a technical proposition, which expresses an “individual” orthogonal property of the functions \(V_{n,m}[\mu ]\) over the interval \([0, \pi ]\). We borrow from the techniques used in the earlier work [19] and extend those results for arbitrary \(\mu \) (the method also relies on Neumann’s formula (1) and identities (54) and (56) below. Hence, the proof is independent of the previous paper.)
Proposition 9
Let \(\mu \in [0,1) \cup i\mathbb {R}^{+}\) be fixed. The following orthogonality relations hold for all \(m=0,1,\dots \) and each pair (n, k) such that \(n, k \in \{m, m+1\}\),
Proof
Fix a value of \(\mu \). For the proof, let the left-hand sides of (51) be denoted by \(C^{m}_{\varepsilon _1,\varepsilon _2}(\mu )\) with \(n=m+\varepsilon _1\) and \(k=m+\varepsilon _2\). We only use pairs in the set \(\{(0,0), (0,1)\), \((1,0), (1,1)\}\).
We will assume that \(\mu \in (0, 1)\) because the case \(\mu \in i\mathbb {R}^{+}\) is similar (the intermediate case \(\mu = 0\) is trivial, and it uses (45).) We have from (38) that
Notice that the first term gives a zero-integral because of the orthogonality of the associated Legendre functions of the first kind over the interval \([0, \pi ]\). The second term also has a vanishing integral because the underlying function is odd with respect to the variable \(t=\cos \vartheta \). Similarly, it can be proved that \(C^{m}_{(0,1)}(\mu ) = 0\) for all \(m=0,1,\dots \).
We now consider the two remaining integrals, \(C^{m}_{(0,0)}(\mu )\) and \(C^{m}_{(1,1)}(\mu )\). For simplicity, we only sketch the proof for \(C^{m}_{(0,0)}(\mu )\). The other integral can be derived straightforwardly. Let us begin by computing \(C^{0}_{(0,0)}(\mu )\). As a consequence of Neumann’s formula (1), we find
Combining the explicit representation (42) for the \(V_{0,0}[\mu ]\) and (52), it follows that
For an arbitrary \(m>1\), we have once again from (42) that
Using the identity [14, p. 195]
we can rewrite the integral in (53) as follows:
Substituting this computation into \(C^{m}_{(0,0)}\) and using the relation [31, Eq. (6.17)]
for \(n=m-1\), we find \(C^{m}_{(0,0)}(\mu ) = 0\) for \(m>0\). It yields that \(C^m_{(0,0)}(\mu ) = 0\) for all \(m=0,1,\dots \). This completes the proof of the statements. \(\square \)
The orthogonality of the proper harmonics (37) over the exterior of the prescribed spheroids \(\Omega _\mu \) is given in the following theorem:
Theorem 10
Let \(\mu \) be fixed. For each \(n\ge 0\), the collection
forms an orthogonal family in \({\text {Har}}_2(\Omega ^{*}_{\mu })\) with the norms
where \(I_{n,m}(\mu )\) has the same meaning as in (36) and
Proof
We will assume that \(\mu \in (0, 1)\), because the case \(\mu \in i\mathbb {R}^{+}\) is similar. When \(m_1 \ne m_2\), we have
Using (33), we obtain
where dR has the same meaning as in (34).
Thus we need to study integrals of the form
Without loss of generality, we assume that \(n_1>n_2\) and proceed to set \(V_{n_1,m}[\mu ]\) as (38) and \(V_{n_2,m}[\mu ]\) as (46). Since
it follows that the remaining nonvanishing integrals are, respectively,
according to \(n_2-m\) being even, or
according to \(n_2-m\) being odd.
Furthermore, using (41) and (42), we are led toward the integrals of the form, as stated in Proposition 9. Hence, for \(n_1 \ne n_2\),
and also
Using the orthogonality of the system \({\Phi _{m}^{\pm }(\varphi )}\) on \([0,2\pi ]\) again, we conclude that \(\langle V^{+}_{n_1,m}[\mu ], V^{-}_{n_2,m}[\mu ]\rangle _{L_2(\Omega ^{*}_{\mu })} = 0\) when \(n_1 \ne n_2\). This establishes the orthogonality statement.
For \(n_1 = n_2 = n\) and \(0 \le m \le n\), by (44), we find
Now, combining (7) and (14), it follows that
with \(t=\cos \vartheta \) or \(\cosh \eta \) and \(R = P\) or Q.
Therefore, when \(0 \le m \le n\), we obtain
where \(I_{n,m}(\mu )\) is defined by (36). We obtain the same value when \(\Phi _{m}^{+}(\varphi )\) is replaced by \(\Phi _{m}^{-}(\varphi )\) throughout, \(0 < m \le n\).
When \(m=n+1\), by use of (43), we have
Using the identity (54), we find
Consequently, combining the relation (17) and identity (56), then (60) leads to
By the form of the \(V^{\pm }_{n,m}[\mu ]\), it follows that \(\Vert V^{+}_{n,n+1}[\mu ]\Vert _{L_2(\Omega ^{*}_{\mu })} = \Vert V^{-}_{n,n+1}[\mu ]\Vert _{L_2(\Omega ^{*}_{\mu })}\). The proof is now completed. \(\square \)
We now state and prove the orthogonality of the proper harmonics (37) over the surface of the prescribed spheroids with respect to a suitable weight function.
Theorem 11
Let \(\vert \zeta (\mu ,{\textbf{x}})\vert \) have the same meaning as in (24). For fixed \(\mu \), the collection (57) forms an orthogonal family over the surface of the spheroids \(\Omega _{\mu }\) in the sense of the scalar product
Their norms squared are equal to
Proof
For the proof, we consider the prolate case again. When \(m_1 \ne m_2\), we have
Let \(P^{*}=(\eta _{\mu }, \vartheta , \varphi )\), i.e., with \(\cosh \eta _{\mu } = 1/\mu \). For \(m_1=m_2=m\), direct calculation leads readily to
where we have used that
From this point on, the proof follows the argument used in Theorem 10 of assuming that \(n_1 > n_2\), in association with the facts
By (38), we then find in combination with Theorem 7 and Proposition 9 that
with exactly the same formula when \(\Phi _{m}^{+}(\varphi )\) is replaced by \(\Phi _{m}^{-}(\varphi )\), \(m>0\). The calculation of the norms of \(V^{\pm }_{n,m}[\mu ]\) comes from taking \(n_1 = n_2\) in (62) and using relation (14). The statement follows. \(\square \)
4 Spheroidal Monogenic Functions
In this section, we combine the results from the previous sections to construct a common orthogonal basis for the one-parameter family of monogenic \(L_2\)-Hilbert spaces applicable to the exterior of a spheroid of given eccentricity. We express the elements of such a basis in terms of the proper external harmonic functions (37). In an earlier paper [19], we treated the analogous problem for prolate spheroids. We borrow some of these techniques and fit many results into the present case. In particular, we can consider the prolate and oblate spheroidal monogenics simultaneously.
4.1 Notation
We are interested in a theory of functions from the space domain \(\Omega ^{*}_{\mu }\) to \(\mathbb {R}^3\). For this purpose, we consider the set
of (real) quaternions, where \({\textbf{i}}, {\textbf{j}}, {\textbf{k}}\) are the quaternionic imaginary units obeying the multiplication rules \({{\textbf {i}}}^2 = {{\textbf {j}}}^2 = {{\textbf {k}}}^2 = {{\textbf {i}}} {{\textbf {j}}} {{\textbf {k}}} = -1\). One usually writes \(\overline{{\textbf{x}}} = \textrm{Sc}({\textbf{x}}) - \textrm{Vec}({\textbf{x}})\) (here \(\textrm{Sc}({\textbf{x}}) = x_0\) and \(\textrm{Vec}({\textbf{x}}) = {\textbf{i}}x_1 + {\textbf{j}}x_2 + {\textbf{k}}x_3\) denote the scalar and vector parts of \({\textbf{x}}\)) and \(\vert {\textbf{x}}\vert = ({\textbf{x}} \overline{{\textbf{x}}})^{1/2} = (\overline{{\textbf{x}}} {\textbf{x}})^{1/2} = \big (\sum _{i=0}^3 x_i^2\big )^{1/2}\) for the conjugate and absolute value operations on \(\mathbb {H}\) as in [10, 11, 18, 29]. We identify the Euclidean space \(\mathbb {R}^3 = \{x = (x_0,x_1,x_2)\}\) with the real vector subspace of reduced quaternions
i.e., with vanishing \({\textbf{k}}\)-term. Although this subspace is not closed under quaternionic multiplication, it is possible to carry out a great deal of the analysis analogous to that of complex numbers [12, 13].
Consider the usual first-order differential generalized Cauchy–Riemann (or Fueter) operator
and its conjugate
on functions \({\textbf{f}} :\Omega \rightarrow \mathbb {R}^3\) on any domain \(\Omega \subseteq \mathbb {R}^3\), with \({\textbf{f}} = [{\textbf{f}}]_0 + {\textbf{i}} [{\textbf{f}}]_1 + {\textbf{j}} [{\textbf{f}}]_2\) where \([{\textbf{f}}]_i :\Omega \rightarrow \mathbb {R}\) \((i=0,1,2)\). As in [5, 12, 13], \({\textbf{f}}\) is called monogenic when \({\overline{\partial }} {\textbf{f}} = 0\). While \({\overline{\partial }}\) does not generally commute with quaternionic functions, since we are considering \(\mathbb {R}^3\)-valued functions in this paper, the definition of monogenic function does not depend on whether one applies \({\overline{\partial }}\) from the left or the right since \(-{\textbf{k}}({\overline{\partial }} {\textbf{f}}){\textbf{k}} = \overline{{\textbf{f}} {\overline{\partial }}}\). For domains \(\Omega \subseteq \mathbb {R}^3\), we write \({\mathcal {M}}(\Omega ) = \textrm{Ker} \, {\overline{\partial }} \subseteq C^1(\Omega ,\mathbb {R}^3)\) and \({\mathcal {M}}_2(\Omega ) = {\mathcal {M}}(\Omega ) \cap L_2(\Omega ,\mathbb {R}^3)\), where \(L_2(\Omega ,\mathbb {R}^3)\) denotes the \(\mathbb {R}\)-linear space of all \(\mathbb {R}^3\)-valued functions \({\textbf{f}}\), such that the real component functions \([{\textbf{f}}]_i\) are in the usual \(L_2(\Omega )\). Monogenic functions are harmonic, but not vice-versa.
4.2 Basic External Spheroidal Monogenics
In analogy to the definition (37) for harmonic functions, we define the required spheroidal monogenics to be employed for the space exterior of the prescribed spheroids (20) as follows.
Definition 3
Let \(U^{\pm }_{n,m}[\mu ]\) have the same meaning as in Definition 1. Let \(n \ge -1\) and \(0 \le m \le n+1\). The basic external spheroidal monogenics of degree \(-(n+3)\) and order m are
The statement that \({\textbf{X}}^{\pm }_{n,m}[\mu ]\) is monogenic is seen from the factorization of the Laplacian in \(\mathbb {R}^3\) by \(\Delta _3 = {\overline{\partial }} \partial \). We continue with the convention that \(m \ge 1\) when the “-” sign appears in a superscript.
In the following theorem, we express the basic monogenics in terms of their quaternionic components. (The functions \(V^{\pm }_{n,-1}[\mu ]\) defined by (39) are involved in the representation (66) for zero-order monogenic functions.)
Theorem 12
Let \(V^{\pm }_{n,m}[\mu ]\) have the same meaning as in (38). For all \(n \ge 0\), the basic external spheroidal monogenics (64) are equal to
for \(0 \le m \le n\), and
Further, they are functions in \(x_0, x_1, x_2\).
Proof
As derivatives of functions, the basic monogenics are also functions in the variables \(x_0, x_1, x_2\). To proceed with the proof, we write the operator (63) in prolate spheroidal coordinates (21):
The first line of the above expression applied to the functions \(U^{\pm }_{n+1,m}[\mu ]\) produces the scalar parts of \({\textbf{X}}^{\pm }_{n,m}[\mu ]\).
The expression (65) is trivial. Now, let \(0 \le m \le n\). For the nonscalar parts of \({\textbf{X}}^{\pm }_{n,m}[\mu ]\), we combine the identities (8)–(15) and (9)–(16) to find
Next, we use the relations (7)–(14) and (12)–(18), obtaining
Furthermore, using (10) and its counterpart (17), we arrive at
With these calculations at hand, we have
Similarly, according to (11)–(18) and (12)–(19), one can prove that
Combining the above formulas, together with the relations
one straightforwardly obtains the desired expressions for \((\partial /\partial x_1) U^{\pm }_{n+1,m}[\mu ]\) and \((\partial /\partial x_2) U^{\pm }_{n+1,m}[\mu ]\).
Now, we compute (67). By (68), we find
On the other hand, we find from (69) that
Using the expressions (70)–(73) for \(m=n+1\), the required expression (67) follows. This establishes the proof of the theorem. \(\square \)
To motivate the relevance of the expressions stated in Theorem 12, explicit recurrence rules between the basic external spheroidal monogenics are discussed below.
Proposition 13
For each \(n \ge 0\) and \(0 \le m \le n\), the following recursive formula holds:
For \(m=0\), we must take into account that
Proof
The proof is an immediate consequence of Theorem 12 by direct inspection of the relations between the quaternionic components of the basic monogenics (66)–(67). \(\square \)
From this, we easily deduce the further result:
Corollary 2
For each \(n \ge 0\) and \(0 \le m \le n+1\), the following recursive formula holds:
4.3 Conversions Among Spheroidal and Solid Spherical Monogenics
As a significant consequence of Theorem 12 and Proposition 3, together with (45), we obtain an explicit representation of the external solid spherical monogenics employing the external solid spherical harmonics.
Corollary 3
For all \({\textbf{x}} \in \mathbb {R}^3 \setminus \{{\textbf{0}}\}\), the limits \(\lim _{\mu \rightarrow 0} {\textbf{X}}^{+}_{-1,0}[\mu ]({\textbf{x}})\) and \(\lim _{\mu \rightarrow 0} {\textbf{X}}^{\pm }_{n,m}[\mu ]({\textbf{x}})\) exist and are given, respectively, by
where the \(U^{\pm }_{n,m}[0]\) have the same meaning as in (28).
This result shows that \({\textbf{X}}^{+}_{-1,0}[\mu ]\) leads to the Cauchy–Fueter kernel, except for the normalization factor \(-1/4\pi \) when \(\mu \rightarrow 0\). This observation is fundamental not only to ensure that the basic external spheroidal monogenics (64) are well-defined on the exterior of the prescribed spheroids \(\Omega _{\mu }\) but also it gives evidence of the completeness of these functions in the space \({\mathcal {M}}_2(\Omega ^{*}_{\mu })\) (see Theorem 18 below).
Given Corollary 1, it is natural to find the direct and inverse transformation formulas that permit passing from external solid spherical to spheroidal monogenics.
Theorem 14
Let \(n \ge 0\) and \(0 \le m \le n+1\). Then
Here \(\alpha _{n,m,k}\) and \({\widetilde{\alpha }}_{n,m,k}\) have the same meaning as in Proposition 8.
Proof
For simplicity in the proof, we only prove the direct transformation formula (75). The identity (76) can be established similarly. We fix n, m, \(\mu \), and the choice of sign ±. According to (66), we want to show that \({\textbf{X}}^{\pm }_{n,m}[\mu ]\) is equal to
where the quaternionic components are given by
According to (48), we obtain
and
By Corollary 1, it follows that
This justifies the assertion \({\textbf{X}}^{\pm }_{n,m}[\mu ] = A\) according to (66). Similarly, we can show that (75) holds when \(m=n+1\) in view of (67). The statement is established. \(\square \)
4.4 Orthogonality Properties of the Basic Monogenics
In this section, we prove that the basic external spheroidal monogenics (64) are orthogonal over the exterior and surface of the prescribed spheroids \(\Omega _{\mu }\).
We have
Theorem 15
Let \(\mu \) be fixed. For each \(n\ge 0\), the collection
forms an orthogonal family in \({\mathcal {M}}_2(\Omega ^{*}_{\mu })\) in the sense of the scalar inner product
Their norms squared are equal to
for \(0 \le m \le n\), and
where \(I_{n,m}(\mu )\) has the same meaning as in (36).
Proof
Combining (65) and (66), and applying the trigonometric identities
direct computation shows that
According to the definition of the integral (78) and Proposition 9, the identities \(\int _{0}^{2\pi } \Phi _{m}^{\pm }(\varphi ) d\varphi = 0\) for \(m>0\) imply that
for all \(n \in \mathbb {N}_0\).
Likewise, by the same reasoning,
for all \(n \in \mathbb {N}_0\).
Now, we compute \(\langle {\textbf{X}}^{\pm }_{n_1,m_1}[\mu ], {\textbf{X}}^{\pm }_{n_2,m_2}[\mu ] \rangle _{L_2(\Omega ^{*}_{\mu }, \, \mathbb {R}^3)}\). According to (66) and Theorem 10, we have
Thus, to verify the orthogonality of the \({\textbf{X}}^{\pm }_{n,m}[\mu ]\), it suffices to show that the vector parts of the functions \({\textbf{X}}^{\pm }_{n,m}[\mu ]\) are orthogonal.
Expanding the integrands and applying the trigonometric identities
we find
where \(p_i=n_i+2-m_i\) \((i=1,2)\).
We continue the calculations only for the prolate case. The following identities \(\int _{0}^{2\pi } \Phi ^{+}_{m_1\pm m_2}(\varphi ) d\varphi = 2\pi \delta _{m_1,m_2}\) for \(m_1,m_2>0\) imply that
where dR has the same meaning as in (34).
In consequence, using (39), we find
Now, using again the orthogonality of Theorem 10, we are left with
with \(I_{n,m}(\mu )\) defined by (36). Combining (79) and (84), we conclude that \(\langle {\textbf{X}}^{+}_{n_1,m_1}[\mu ], {\textbf{X}}^{+}_{n_2,m_2}[\mu ]\rangle _{L_2(\Omega ^{*}_{\mu }, \, \mathbb {R}^3)} = 0\) when \(n_1 \ne n_2\) or \(m_1\ne m_2\). Similarly, \(\langle {\textbf{X}}^{-}_{n_1,m_1}[\mu ], {\textbf{X}}^{-}_{n_2,m_2}[\mu ] \rangle _{L_2(\Omega ^{*}_{\mu }, \, \mathbb {R}^3)} = 0\) when \(n_1 \ne n_2\) or \(m_1 \ne m_2\). Using the orthogonality of the system \(\{\Phi _m^\pm \}\) on \([0,2\pi ]\) again, we conclude that \(\langle {\textbf{X}}^{\pm }_{n_1,m_1}[\mu ], {\textbf{X}}^{{\mp }}_{n_2,m_2}[\mu ] \rangle _{L_2(\Omega ^{*}_{\mu }, \, \mathbb {R}^3)} = 0\) when the indices do not coincide.
Now, applying the trigonometric identities
it follows from (43) and (67), in combination with Proposition 9, that
Similarly,
Furthermore, using the trigonometric identities (80)–(83) for \(m_1=n_1+1\) and \(m_2=n_2+1\), by (67) and (58), it follows that
This establishes the orthogonality statement.
The calculation of the \(L_2\)-norms of \({\textbf{X}}^{\pm }_{n,m}[\mu ]\) for \(0 \le m \le n\) comes from taking \(n_1=n_2\) and \(m_1=m_2\) in (84) and adding expression (58). By the symmetric form of the \({\textbf{X}}^{\pm }_{n,m}[\mu ]\) in (66), it follows that \(\Vert {\textbf{X}}^{+}_{n,m}[\mu ] \Vert _2 = \Vert {\textbf{X}}^{-}_{n,m}[\mu ]\Vert _2\) when \(m \ne 0\). Furthermore, taking \(n_1=n_2\) in (85) and combining (43), (55), and (56), we find
Using the relation \(Q^{n+2}_{n}(\cosh \eta ) = \sinh \eta \, Q^{n+2}_{n+1}(\cosh \eta )\), which follows from (56), the statement follows. The proof is now completed. \(\square \)
The corresponding orthogonality of the basic spheroidal monogenics over the surface of the prescribed spheroids follows immediately from Theorems 11 and 15.
Theorem 16
Let \(\vert \zeta (\mu ,{\textbf{x}})\vert \) have the same meaning as in (24). For fixed \(\mu \), the collection (77) forms an orthogonal family over the surface of the spheroids \(\,\Omega _{\mu }\) in the sense of the scalar product
4.5 An Orthogonal Basis of External Spheroidal Monogenics
The results of the preceding sections enable us to prove the following theorem about the approximation of a smooth function \({\textbf{f}}\) in \({\mathcal {M}}_2(\Omega ^{*}_{\mu })\) expanded as a linear combination of basic external spheroidal monogenics. Although the proof of this fact follows the same argument as in [19], since it is necessary to employ the definitions (25) and (64), we include the proof for the completeness of the presentation.
Theorem 17
Suppose \({\textbf{f}} \in {\mathcal {M}}_2(\Omega ^{*}_{\mu }) \cap C^1(\Gamma _{\mu }, \mathbb {R}^3)\) and let \(\mu \) be fixed. The monogenic Fourier series expansion of \({\textbf{f}}\) with respect to the collection (77),
converges to \({\textbf{f}}\) in the \(L_2\)-sense, where the (real) Fourier coefficients \(a^{\pm }_{k,m}[\mu ]\) are given by
Proof
Suppose that \({\textbf{f}} \in {\mathcal {M}}_2(\Omega ^{*}_{\mu })\). Hence, there exists a real-valued harmonic function h in \(\Omega ^{*}_{\mu }\) such that \((1/2)\partial h = {\textbf{f}}\). Moreover, the restriction of h on \(\Gamma _{\mu }\) is a twice continuously differentiable function. Now, let \(g(\vartheta , \varphi )\) be a function defined on the unit sphere, which is related to the value of \(Tr_{\Gamma _{\mu }} h\) on the prescribed spheroid by \(g(\vartheta , \varphi ) = Tr_{\Gamma _{\mu }} h(\vartheta , \varphi ) = h(\eta _\mu , \vartheta , \varphi )\), with \(\cosh \eta _{\mu } = 1/\mu \) by (22). (The trace operator \(Tr_{\Gamma _{\mu }}\) describes just the restriction onto the boundary \(\Gamma _{\mu }\).) Since \(g(\vartheta , \varphi )\) is a twice continuously differentiable function on the unit sphere, it can be expressed employing a series of surface spherical harmonics,
It was shown in [15, 25] that the above expansion is absolutely and uniformly convergent with respect to \((\vartheta , \varphi ) \in \Gamma _{0}\).
For simplicity, we again assume that \(\mu \in (0, 1)\). Extending (88) to \(\Omega ^{*}_{\mu }\) leads to a series expansion of h in terms of the basic external spheroidal harmonics (25):
Using the results of [14, pp.417–421], we find \(\left| Q^m_n(\cosh \eta )/Q^m_n(1/\mu )\right| < 1\) for all \(n, m = 0, 1, \dots \) and fixed \(\mu >0\). Under the previous circumstances, it then follows that the series (89) converges uniformly and absolutely to h in \(\Omega ^{*}_{\mu }\).
Moreover, since h is harmonic in \(\Omega ^{*}_{\mu }\) and twice continuously differentiable on the boundary \(\Gamma _{\mu }\), it yields the absolute and uniform convergence of its first derivatives in \(\Omega ^{*}_{\mu } \cup \Gamma _{\mu }\). In particular, the corresponding series expansion for the derivatives, namely
converges uniformly and absolutely to \({\textbf{f}} = (1/2)\partial h\) in \(\Omega ^{*}_{\mu } \cup \Gamma _{\mu }\). This further implies the \(L_2\)-convergence in every subset \(\Omega ^{*}_{\mu } \cap B_r\), where \(B_r\) is a ball with some radius \(r > 0\), which contains \({\overline{\Omega }}_{\mu }\). More precisely, denote by \({\textbf{S}}_{N}\) the finite sum of the first N-summands in the series (90). For any \(\epsilon > 0\), there exists a natural number \(N(\epsilon )\) such that
We have then
For the exterior domain of \(B_r\), we use another estimation. Let \(0< r_1 < r\) such that \(\Omega _{\mu } \subset B_{r_1}\). By the Cauchy integral formula for quaternionic monogenic functions [11, pp. 87-88], for all \({\textbf{y}} \in \mathbb {R}^3 \backslash B_r\), one finds
Now, the \(L_2\)-norm of the difference between \({\textbf{S}}_{N}\) and \({\textbf{f}}\) can be approximated by
To sum up, for an arbitrary small \(\epsilon > 0\), we can find a natural number \(N(\epsilon )\) such that \(\Vert {\textbf{S}}_{N(\epsilon )} - {\textbf{f}}\Vert _2 < \epsilon \). Thus, the series expansion (90) converges to \({\textbf{f}}\) in the whole domain \(\Omega ^{*}_{\mu }\) in the sense of the \(L_2\)-norm, which is the desired result. The theorem follows. \(\square \)
From this theorem, we deduce the further result:
Corollary 4
Suppose \({\textbf{f}} \in {\mathcal {M}}_2(\Omega ^{*}_{\mu })\) and let \(\mu \) be fixed. Then the restriction of \({\textbf{f}}\) in \(\Omega ^{*}_{\mu }\) can be represented by its Fourier series expansion of the form (87). Further, this series expansion converges to \({\textbf{f}}\) in the \(L_2\)-sense.
Corollary 5
Any function in the collection
can be represented by its Fourier series expansion of the form (87).
To conclude, the general result is that
Theorem 18
For fixed \(\mu \), the collection (77) forms an orthogonal basis of \({\mathcal {M}}_2(\Omega ^{*}_{\mu })\).
Proof
This result is proved by first approximating \({\textbf{f}}\) in \({\mathcal {M}}_2(\Omega ^{*}_{\mu })\) from the set (91) (which, in fact, forms an orthogonal basis of \({\mathcal {M}}_{2}(\Omega ^{*}_{0})\), see [19]), and then by the collection (77) of basic external spheroidal monogenics. \(\square \)
This theorem is the generalization of that of [19], which corresponds to the case of prolate spheroids.
5 Concluding Remarks
We introduced a single one-parameter family of basic external spheroidal harmonics that assume prescribed values on the boundary of the family of coaxial spheroidal domains \(\Omega _{\mu }\). The basic harmonics are functions in \(x_0, x_1, x_2\) which were normalized so that the limiting case \(\mu \rightarrow 0\) gives the classical external solid spherical harmonics. The nonorthogonality of the basic harmonics in the \(L_2\)-Hilbert space structure led to the discussion of the proper external spheroidal harmonics. Underlying our manipulations is a set of conversion formulas that relate the coefficients of the expansions among the spheroidal and spherical harmonic systems. The basic external spheroidal monogenic functions were calculated explicitly, and formulas for their nonscalar parts were obtained in terms of the proper harmonics. Ordinary methods based on decomposing a function space into subspaces of homogeneous functions fail to prove the completeness of a monogenic function system constructed through the basic external spheroidal monogenics because of the appearance of logarithmic functions. The technique we employed to build an orthogonal basis for the one-parameter family of monogenic \(L_2\)-Hilbert spaces in the space exterior of \(\Omega _{\mu }\) was based on the harmonic extension of a function defined on the boundary of the spheroid \(\Omega _{\mu }\) to the exterior domain \(\Omega ^{*}_{\mu }\).
We will extend our results in a future paper to external spheroidal monogenics taking values in the space of full quaternions (with a right \(\mathbb {R}\)-linear structure), using the theoretical basis presented in [21].
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The author acknowledges financial support by the Asociación Mexicana de Cultura, A. C. The author would like to express his sincere thanks to Raybel A. García A. for his help in creating the tables in Sect. 3.
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Morais, J. Orthogonal Harmonic and Quaternionic Monogenic Functions in the Exterior of a Spheroid. Complex Anal. Oper. Theory 17, 71 (2023). https://doi.org/10.1007/s11785-023-01369-1
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DOI: https://doi.org/10.1007/s11785-023-01369-1