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OF COMPUTATION Volume 84, Number 293, May 2015, Pages 1317–1337 S 0025-5718(2014)02888-3 Article electronically published on August 29, 2014

QUATERNION ZERNIKE SPHERICAL

J. MORAIS AND I. CAC¸ AO˜

Abstract. Over the past few years considerable attention has been given to the role played by the (ZPs) in many different fields of geometrical , optical engineering, and astronomy. The ZPs and their applications to corneal surface modeling played a key role in this develop- ment. These polynomials are a complete set of over the unit circle and are commonly used to describe balanced aberrations. In the present paper we introduce the Zernike spherical polynomials within quater- nionic analysis ((R)QZSPs), which refine and extend the Zernike moments (defined through their counterparts). In particular, the underlying functions are of three real variables and take on either values in the reduced and full (identified, respectively, with R3 and R4). (R)QZSPs are orthonormal in the unit . The representation of these functions in terms of spherical monogenics over the unit are explicitly given, from which several recurrence formulae for fast computer implementations can be derived. A summary of their fundamental properties and a further second or- der homogeneous differential equation are also discussed. As an application, we provide the reader with plot simulations that demonstrate the effectiveness of our approach. (R)QZSPs are new in literature and have some consequences that are now under investigation.

1. Introduction 1.1. The Zernike spherical polynomials. The complex Zernike polynomials (ZPs) have long been successfully used in many different fields of optics. These circle polynomials were introduced by F. Zernike’s in connection with his phase contrast and knife-edge tests; see [36]. ZPs are the product of a normal- ization constant by radial polynomials and a pair of (sine and cosine) [37], and are especially useful in numerical calculations. The ZPs also played a fundamental role in B.A. Nijboer’s theory of aberrations; see [30]. Besides being complete and orthogonal, they possess many remarkable properties such as rotation invariance, robustness to noise, expression efficiency, fast computation and multilevel representation for describing the shapes of patterns. As shown partic- ularly by A.B. Bhatia and E. Wolf (see [2]), ZPs arise more or less uniquely as

Received by the editor March 13, 2013 and, in revised form, August 14, 2013. 2010 Mathematics Subject Classification. Primary 26C05, 30G35; Secondary 33C45, 42C05. Key words and phrases. Quaternionic analysis, Ferrer’s associated Legendre functions, Cheby- shev polynomials, Zernike polynomials, spherical aberrations, spherical monogenics. This work was supported by Portuguese funds through the CIDMA – Center for Research and Development in Mathematics and Applications, and the Portuguese foundation for Sci- ence and Technology (“FCT–Funda¸c˜ao para a Ciˆencia e a tecnologia”), within project PEst- OE/MAT/UI4106/2014. Support from FCT via the post-doctoral grant SFRH/BPD/66342/2009 is also acknowledged by the first author.

c 2014 American Mathematical Society 1317

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orthogonal functions with polynomial radial dependence satisfying form invariance under rotations of the unit disk. For the reconstruction, recognition, and discrimination of 3D objects, the usage of a 2D system seems to be inappropriate and 3D systems are used instead. Gen- eralized Zernike functions were first studied by W.J. Tango in [35], which provided many of their analytical and algebraical properties, as well as properties that sup- port their appropriateness in optical diffraction theory. In particular, 3D Zernike spherical polynomials (ZSPs) orthonormal on the unit sphere were introduced by N. Canterakis in [10]. The 3D Zernike moments (through ZSPs) have the advan- tage of capturing global information about the 3D shape without requiring closed boundaries as in boundary based methods. Recently, there is renewed interest in the theoretical and computational aspects of the ZSPs as well as in the applications of these aspects to optical design, dynamic analysis and computational geometry. It is from these applications that methods for the detection and characterization of intracranial aneurysms also emerged, and these ideas are now being used in the area of bioinformatics and molecular biology. These studies motivated the implementa- tion of Zernike moments for protein tertiary structure retrieval and comparison of properties on protein surface. Also, Zernike moments are promising descriptors in the field of biological imaging; for instance, these descriptors can be used to build up 3D views of important biological entities such as proteins, nucleic acids, cells, tumors, tissues and whole organs or organisms.

1.2. Purpose and scope of the paper. During the past few years the theory of generalized holomorphic functions with values in the Hamiltonian algebra (the so-called monogenic functions) has proved its value in a variety of fields from theoretical to practical problems; it has been developed for decades as a generalization of the holomorphic theory in the complex plane to higher dimensions and also considered as a refinement of classical ; see e.g. [13, 14, 19, 20, 32–34]. The present article poses the question of whether the ZSPs can be generalized to the context of quaternionic analysis. We shall mostly confine ourselves to the examination of functions with values in the reduced and full quaternions (identified, respectively, with R3 and R4); in particular, we use mono- genic functions as null-solutions of the well-known Riesz and Moisil-Th´eodoresco systems; see e.g. [26]. One main reason for pursuing this direction is that a suffi- ciently well-developed theory already exists for the construction of spherical mono- genic systems by means of quaternionic analysis tools. For a general orientation the reader is suggested to check some of the existing pioneering works [3,5–7,9,12] and [8, 11, 16, 22, 26], which describe the R3 → R4 and R3 → R3 cases. The idea of the construction of the supported systems is a approach based on the factorization of the 3D Laplacian operator into first order differential operators. In this paper, we rely on the system of spherical monogenics presented in [9]; cf. also [26]. In particular, in [4] it is proved that the rate of convergence for a monogenic function approximated by this system has the same quality as in the similar case of a harmonic function approximated by (for comparison see [17]). This gives us an interesting way of motivating addition itself. With respect to the already mentioned applications it is of theoretical interest to check whether the ZSPs functions can be extended if the underlying structure is not commutative as in the case of the quaternion algebra. This understanding could be a for more generalizations. Also, it may be fruitful both to enrich

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the understanding of the properties of these polynomials in higher dimensions and as tools in such applications of quaternionic analysis in diffraction theory and re- lated fields. These connections will be revealed to mathematicians, who are now working in what has been largely separate areas, in forthcoming publications. We do honestly expect that the popularity of the ZSPs is still on the rise and will likely see even more growth in the future, due to their promising applications in many fields. The paper is laid out as follows. In order to make it self-contained, Section 2 gives an overview on some definitions and basic properties of quaternionic anal- ysis. It also recalls a complete orthogonal system of 3D spherical monogenics. The former were introduced in the papers [8, 9, 26] of which could be explicitly expressed in terms of products of Ferrer’s associated Legendre functions multiplied by Chebyshev polynomial factors. Section 3 introduces a set of Zernike spherical polynomials taking values in the reduced quaternions (RQZSPs) (see Definition 3.1 below). RQZSPs are orthonormal in the unit ball. The representations of these functions in terms of spherical monogenics on the surface of the unit ball are ex- plicitly given. Some important properties, and efficient recurrence formulae for the RQZSPs are discussed. This helps to reduce the amount of calculations in practice. Section 4 introduces the quaternion Zernike spherical polynomials (QZSPs) defined on spaces of square integrable quaternion functions (see Definition 4.1 below). Some important properties of the system are also mentioned; amongst its periodicity and , we show that they satisfy a second order homogeneous differential equation as well (see Theorem 4.8 below). Further to this, in Section 5 we illustrate our approach with plot examples. The final section shows the concluding remarks and discusses how the methods used can be extended within this context. We will not consider concrete applications of the RQZSPs and QZSPs in this paper.

2. Preliminaries 2.1. Basic notions. Throughout the paper, let

H := {z = z0 + z1i + z2j + z3k : zl ∈ R,l=0, 1, 2, 3} be the Hamiltonian skew field of quaternions, where the imaginary units i, j,and k obey the laws of multiplication: i2 = j2 = k2 = −1; ij = k = −ji, jk = i = −kj, and ki = j = −ik. As usual, the real R4 may be embedded in H by 4 identifying the element z := (z0,z1,z2,z3) ∈ R with z := z0 + z1i + z2j + z3k ∈ H. The scalar and vector parts of z,Sc(z)andVec(z), are defined as the z0 and z1i + z2j + z3k terms, respectively. Like in the complex case, the conjugate of z is the quaternion z = z0 − z1i − z2j − z3k,andthenorm|z| of z is defined | |2 2 2 2 2 by z = zz = zz = z0 + z1 + z2 + z3 . The quaternion conjugation is a linear anti-involution:

z = z, z1 + z2 = z1 + z2, z1z2 = z2 z1, λz = λz, ∀ λ ∈ R. In the sequel, let

A := spanR{1, i, j}⊂H

be the space of reduced quaternion elements of the form x := x0 + x1i + x2j, emphasizing that A is a real vectorial subspace, but not a subalgebra, of H.The real vector space R3 is embedded in A via the identification 3 x := (x0,x1,x2) ∈ R ↔ x := x0 + x1i + x2j ∈A.

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Now, let Ω be an open subset of R3 with a piecewise smooth boundary. We say that

(2.1) f :Ω→ H, f(x):=[f(x)]0 +[f(x)]1i +[f(x)]2j +[f(x)]3k

is a quaternion-valued function or, briefly, a function H-valued, where [f]l (l = 0, 1, 2, 3) are scalar-valued functions defined in Ω. By now, it is clear that if [f(x)]3 ≡ 0, then f is itself reduced quaternion-valued. Continuity, differentiability, integrability, and so on, which are ascribed to f are defined componentwise. The A-(resp.rightH-) linear Hilbert spaces L2(Ω; A; R)(resp. L2(Ω; H; H)) consist of all square integrable functions in Ω taking values in A (resp. H)thatsatisfy | |2 ∞ Ω f(x) dV < ; dV denotes the Lebesgue measure on Ω normalized so that V (Ω) = 1. In this assignment, the scalar and (right) quaternionic inner products are defined by

(2.2) f, g L2(Ω;A;R) = Sc(fg) dV Ω and

(2.3) f, g L2(Ω;H;H) = fgdV . Ω Depending on the case, these will be over the unit ball B or the unit sphere S, and we shall always mention this explicitly. A central notion in quaternionic analysis is that of monogenicity, which is intro- duced by means of the so-called generalized Cauchy-Riemann operator ∂ ∂ ∂ (2.4) ∂ = + i + j . ∂x0 ∂x1 ∂x2

It corresponds to the generalization of the classical Cauchy-Riemann operator ∂z = 1 ∂ ∂ ∈ C 2 ∂x + i ∂y , z = x + iy . A C1 function f with values in H is called (left) monogenic in Ω if ∂f =0inΩ. As the generalized Cauchy-Riemann operator (2.4) and its conjugate ∂ ∂ ∂ (2.5) ∂ = − i − j ∂x0 ∂x1 ∂x2 3 factorize the in R in the sense that Δ3 = ∂∂ = ∂∂, it follows that a monogenic function in Ω is also harmonic in Ω, and so are all of its compo- nents. This factorization establishes a special relationship between quaternionic and harmonic analyses wherein monogenic functions refine the properties of harmonic functions.

2.2. The ZSPs revisited. The ZSPs are most conveniently expressed in terms of the spherical coordinate system (ρ, θ, ϕ); we set

x0 = ρ cos θ, x1 = ρ sin θ cos ϕ, x2 = ρ sin θ sin ϕ, with 0 <ρ≤ 1, the radius of the sphere, θ is the polar so that 0 <θ≤ π,and ϕ is the azimuthal angle with −π<ϕ≤ π. The ZSPs form a complete orthonormal system of functions on the sphere (see [10]), and are 3D rotation invariant just like the 2D ZPs do in the unit circle. Let n and l be positive integers, with n − l even and nonnegative; these polynomials are defined as products of standard spherical

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{e (0) e (m) o (m)}1 harmonics Yl , Yl , Yl (considered e.g. in [31]) by radial polynomials R(l): n ⎧ (0) (0) (0) ⎪ eZ (ρ, θ, ϕ) ≡ N R(l)(ρ) eY (θ, ϕ), ⎨⎪ n,l l n l (2.6) eZ(m)(ρ, θ, ϕ) ≡ N (m)R(l)(ρ) eY (m)(θ, ϕ), ⎪ n,l l n l ⎩⎪ o (m) ≡ (m) (l) o (m) Zn,l (ρ, θ, ϕ) Nl Rn (ρ) Yl (θ, ϕ),m=1,...,l. (0) (m) Hereby Nl and Nl are, respectively, the normalization factors 2l +1 (2l +1) (l − m)! N (0) = and N (m) = (m ≥ 1). l 4π l 2π (l + m)! It is customary to refer to n as the radial degree and to m as the azimuthal order e (0) e (m) o (m) 2 of Zn,l , Zn,l and Zn,l ’s. The spherical harmonics are expressed as follows: ⎧ ⎪ eY (0)(θ, ϕ) ≡ P (cos θ), ⎨⎪ l l (2.7) eY (m)(θ, ϕ) ≡ P m(cos θ)cos(mϕ), ⎪ l l ⎩⎪ o (m) ≡ m Yl (θ, ϕ) Pl (cos θ) sin(mϕ),m=1,...,l (l =0, 1,...) where cos(mϕ) and sin(mϕ) are related to the Chebyshev polynomials of the first and second kinds, Tm and Um, in the following way: cos(mϕ)=:Tm(cos ϕ)and m sin(mϕ)=:Um−1(cos ϕ) sin(ϕ); Pl are Ferrer’s associated Legendre functions of the first kind of l-th degree and m-th order. In this assignment, the sign convention of including the Condon-Shortley phase is adopted: m m − m m d Pl (cos θ):=( 1) sin θ m [Pl(t)] . dt t=cos θ e (0) The polynomials Zn,0 independent of θ are the spherical aberrations, which play a fundamental role in the diffraction theory of aberrations; see [30]. The fundamental references for the ZSPs are [10, 21] (and the references cited therein). (l) The radial polynomials Rn are given explicitly as follows, see [21]: √ 0,l+ 1 (l) l ( 2 ) 2 − (2.8) Rn (ρ)= 2n +3ρ P n−l (2ρ 1), 2 and normalized so that 1 2 (l) (l) ρ Rn1 (ρ)Rn2 (ρ)dρ = δn1,n2 . 0 (α,β) In (2.8), Pλ (x) denotes the classical Jacobi polynomial of degree λ in x,and (α,β) with real parameters α and β (see [1, Ch. 22]); the Pλ (x) satisfy the second order linear homogeneous differential equation (2.9) (1 − x2)y +[β − α − (α + β +2)x] y + λ (λ + α + β +1)y =0.

1Although the notation is uncommon, we denote by eY and oY accordingtotheparityofthe Chebyshev polynomials; see (2.7) below. 2 (0) Although the Pl are actually a special case of Ferrer’s associated m e (0) e (m) Legendre functions Pl , we would prefer to treat Yl and Yl separately; this dissociation 2 e (0) e (m) becomes important when we calculate the L norms of Yl and Yl .

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The radial polynomials are either even or odd, depending on n or l: (l) − n (l) − − l (l) − (2.10) Rn (ρ)=( 1) Rn ( ρ)=( 1) Rn ( ρ). √ (l) A noteworthy special value is Rn (1) = 2n +3. Before we proceed further, we need to prove some preliminary results. Proposition 2.1. Let n and l be positive integers with n − l even and nonnegative. A recurrence is3 ⎧ √ (0) ⎪ R (ρ)= 3, ⎪ 0 ⎨⎪ 2n +3 (n + l +3)ρR(l)(ρ) − (2l +3) R(l+1)(ρ)=(n − l)ρR(l+2)(ρ), (n ≥ 0), ⎪ n 2n +5 n+1 n ⎪ ⎩⎪ (l) (l) (l) ≥ C1(n, l)Rn+2(ρ)+C2(n, l)Rn−2(ρ)=D(n, l, ρ)Rn (ρ)(n 2),

where the coefficients C1 and C2 are constants given by 1 (2.11) C1(n, l)= √ (2n +1)(n +3+l)(n +2− l), 2n +7 1 (2.12) C2(n, l)= √ (2n +5)(n +1+l)(n − l), 2n − 1 and D(n, l, ρ) is a polynomial of degree 2 as 1√ (2.13) D(n, l, ρ)= 2n +3 (2n + 5)(2n + 1)(2ρ2 − 1) − (2l +1)2 . 2 Proof. It follows straightforwardly from known recurrence formulae for the Jacobi polynomials.  Proposition 2.2. Let n and l be positive integers with n − l even and nonnegative. The polynomials R(l) satisfy the second order differential equation n (2.14) ρ2(1 − ρ2)y +2ρ(1 − 2ρ2)y − l(l +1)− n(n +3)ρ2 y =0.

(l) Proof. Since by definition the radial polynomials Rn areexpressedintermsof 1 (0,l+ 2 ) the Jacobi polynomials P n−l , using (2.9) the result follows in a straightforward 2 way.  2.3. The 3D spherical monogenics revisited. As we have seen the ZSPs are defined as products of radial polynomials and angular functions; in 3D the angular function set was chosen as the complete orthonormal system of spherical harmonics (2.7) defined on the sphere. In quaternionic analysis, a suitable set of angular functions would be an orthonormal and complete system of spherical monogenics. The (reduced and) quaternion Zernike polynomials under consideration depend fundamentally on the underlying angular spherical monogenics that are chosen in each case. More specifically, the construction of ZSPs with values in A requires the consideration of an orthogonal system of spherical monogenics that is complete in L2(S; A; R) ker ∂. In this case, for each n and l such that n − l is even and nonnegative, we choose a set of 2l+3 functions whose extensions in the ball form an orthogonal basis for the space of monogenic A-valued polynomials in the sense of

3 (l) For a more unified formulation we make the convention that Rn are the zero functions for l>n.

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the scalar product (2.2) (see Property 2 below). For the construction of ZSPs taking values in H the choice of the angular functions is made by the consideration of a set of l + 1 spherical monogenics, complete in L2(S; H; H) ker ∂ and orthogonal in the sense of the quaternionic product (2.3) (see [34] and [12, Remark 1]). We now recall an explicit complete orthogonal system of spherical monogenics in 3D required for subsequent derivations. The treatment is only introductory since we have not attempted to cover all the ongoing research. For a more detailed presentation, we refer the reader to [9, 26]; cf. also [16]. e (·) l+1 e (·) o (·) In all that follows, we denote by Xl (θ, ϕ):=∂ ρ Yl+1(θ, ϕ) , Xl (θ, ϕ) ρ=1 l+1 o (·) ≤ ≤ := ∂ ρ Yl+1(θ, ϕ) where the azimuthal order m is such that 1 m l +1 ρ=1 (l =0, 1,...)4. Having in mind the Laplacian factorization, it is easily seen that the e (0),† le (0) extensions in the ball of the spherical monogenics, respectively, Xl := ρ Xl , e (m),† le (m) o (m),† lo (m) Xl := ρ Xl and Xl := ρ Yl are monogenic (more specifically, there are a total of 2l + 3 homogeneous monogenic polynomials with values in A). The spherical monogenics are of the form; see [26]:

e (0) e (0) e (1) o (1) (2.15) Xl (θ, ϕ):=(l +1) Yl (θ, ϕ)+ Yl (θ, ϕ)i + Yl (θ, ϕ)j,

e (m) e (m) (2.16) Xl (θ, ϕ):=(l + m +1) Yl (θ, ϕ) 1 + eY (m+1) (θ, ϕ)i + oY (m+1) (θ, ϕ)j 2 l l 1 − − − (l + m +1)(l + m) eY (m 1) (θ, ϕ)i − oY (m 1) (θ, ϕ)j , 2 l l

o (m) o (m) (2.17) Xl (θ, ϕ):=(l + m +1) Yl (θ, ϕ) 1 + oY (m+1) (θ, ϕ)i − eY (m+1) (θ, ϕ)j 2 l l 1 − − − (l + m +1)(l + m) oY (m 1) (θ, ϕ)i + eY (m 1) (θ, ϕ)j , 2 l l for m =1,...,l+1(l =0, 1,...). (·) The spherical monogenics Xl can be seen, respectively, as a refinement of the aforementioned spherical harmonics, and correspondingly they constitute an ex- tension of the role played by the well-known Chebyshev and Ferrer’s associated Legendre functions; for more details see [16]. We shall conclude this survey by mentioning some properties of these functions. e (0) 1. For each l =0, 1,..., the scalar parts of the spherical monogenics Xl , e (m) o (m) 2 Xl and Xl (m =1,...,l+ 1) are complete and orthogonal in L (S) in the sense of the scalar product (2.2); e (0) e (m) o (m) 2. For each l =0, 1,..., the spherical monogenics Xl , Xl and Xl (m =1,...,l + 1) are complete and orthogonal in L2(S; A; R) ker ∂ in the sense of the scalar product (2.2), and their norms are explicitly given

4 (·) {e (0) e (m) o (m) For the sake of simplicity, we denote by Xl any polynomial of the set Xl , Xl , Xl : m =1,...,l+1},foreachl ∈ N0.

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by e (0) Xl L2(S;A;R) = 4π(l +1),

e (m) o (m) (l +1+m)!  X  2 A R =  X  2 A R = 2π(l +1) ; l L (S; ; ) l L (S; ; ) (l +1− m)! e (l+1) o (l+1) ∈ 2 A R 3. The sectorial monogenics Xl , Xl L (S; ; ) ker ∂ ker ∂. They are a combination of sectorial harmonics:

eX(l+1)i = oX(l+1)j l l e (l) − o (l) =(l + 1)(2l +1) Yl Yl k . These functions play the role of constants with respect to the hypercomplex derivability (known as hyperholomorphic constants); see [15].

3. Definition and Properties of RQZSPs This section introduces a set of generalized Zernike spherical polynomials and studies some of its important properties. The consideration of a standard system of ZSPs for its construction allows us to use some well-known results like its orthogo- nality and completeness on the unit sphere. As a consequence of the interrelation between ZSPs and Chebyshev and Ferrer’s associated Legendre functions, the con- structed Zernike A-valued polynomials are related to the Chebyshev and Ferrer’s functions as well. The solution procedure is presented as a generalization of the 3D treatment using the aforementioned spherical monogenics. To the best of our knowledge, this subject has not been studied before.

3.1. RQZSPs. The strategy adopted for the construction of reduced quaternion- valued Zernike polynomials (RQZSPs) is the following: we start by considering the aforementioned complete orthogonal system of spherical monogenics {e (0) e (m) o (m) } Xl , Xl , Xl : m =1,...,l+1;l =0, 1,... as the desired set of angular functions; next we multiply the resulting expressions by the radial polynomials and the underlying normalization factors. Carrying this further, we are thus led to the following definition. Definition 3.1 (RQZSPs). Let n and l be any integers such that n − l is even and nonnegative. For m =1,...,l+1(l =0, 1,...) the reduced quaternion Zernike spherical polynomials (RQZSPs) are defined by

e (0) ≡ (0) (l) e (0) Zn,l (ρ, θ, ϕ) Ml Rn (ρ) Xl (θ, ϕ), e (m) ≡ (m) (l) e (m) (3.1) Zn,l (ρ, θ, ϕ) Ml Rn (ρ) Xl (θ, ϕ), o (m) ≡ (m) (l) o (m) Zn,l (ρ, θ, ϕ) Ml Rn (ρ) Xl (θ, ϕ),

(l) (0) e (0)−1 where the radial polynomials Rn are given by (2.8), and Ml = Xl L2(S;A;R) (m) e (m)−1 o (m)−1 and Ml = Xl L2(S;A;R) = Xl L2(S;A;R) are the normalization factors.

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(·) For the sake of simplicity, we denote by Zn,l any polynomial of the set

{e (0) e (m) o (m) } Zn,l , Zn,l , Zn,l : m =1,...,l+1 ,

for each n, l ∈ N0 so that n − l is even and nonnegative.

Remark 3.2. We note that over the sphere the RQZSPs coincide, up to a real (·) constant depending on l, m and n,withtheXn ’s.

Remark 3.3. A direct observation shows that in case l = m = 0 the underlying e (0) e (0) polynomials Zn,0 coincide with the classical Zernike polynomials Zn,0.Conse- quently, they are scalar-valued for all n ≥ 0, and do not depend of the variables θ e (0) e (1) o (1) and ϕ.The Zn,0(ρ), Zn,0(ρ)and Zn,0(ρ) are the spherical aberrations.

e (0) e (m) o (m) Proposition 3.1. For each n =0, 1,...,theset{ Zn,n, Zn,n , Zn,n : m = 1,...,n+1} formsabasisforthespaceofA-valued homogeneous monogenic poly- nomials of degree n, and is orthogonal in the sense of the scalar product (2.2) with Ω=B.

(l) Proof. For the case l = n the√ radial polynomials Rn simplify to monomials of (n) n (·) degree n, such that Rn (ρ)= 2n +3ρ . Hence the polynomials Zn,n coincide, (·),† n (·) up to a real constant depending on n and m to Xn = ρ Xn , introduced above. The follows from the relation

(·),† (·),† 1 (·) (·) X X 2 A R X X 2 A R n1 , n2 L (B; ; ) = δn1,n2 n1 , n2 L (S; ; ), n1 + n2 +3

for n1,n2 =0, 1,.... 

(·) Remark 3.4. Except for n equal to l, the polynomials Zn,l are, in general, not ho- mogeneous. Nevertheless, they can be rewritten as a product of radial polynomials of order n − l by homogeneous monogenic polynomials:

(l) · · R (ρ) · † Z( ) (ρ, θ, ϕ)=M ( ) n X( ), (ρ, θ, ϕ). n,l l ρl l

e (l+1) In addition, notice that when m = l + 1 the resulting polynomials Zn,l and o (l+1) Zn,l have no scalar part. Based on the previous representations we formulate a first preliminary result.

Lemma 3.5. Let n and l be any integers such that n − l is even and nonnegative. e (m) o (m) ≥ The polynomials Zn,l and Zn,l are the zero functions for m l +2.

The RQZSPs are given in Cartesian coordinates for the cases n =0, 1, 2inthe following tables.

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Table 1. n =0, 1

Radial degree (n) l Azimuthal order (m) RQZSPs 0 0 0 eZ(0) = 1 3 0,0 2 π 1 eZ(1) = − 1 3 i 0,0 2 π oZ(1) = − 1 3 j 0,0 2 π 1 1 0 eZ(0) = 1 5 (2x + x i + x j) 1,1 2 2π 0 1 2 1 eZ(1) = 1 15 (x − x i) 1,1 2 2π 1 0 oZ(1) = 1 15 (x − x j) 1,1 2 2π 2 0 2 eZ(2) = − 1 15 (x i − x j) 1,1 2 2π 1 2 o (2) − 1 15 Z1,1 = 2 2π (x2i + x1j)

Table 2. n =2

n l m RQZSPs 2 0 0 eZ(0) = 1 7 (−3+5x2 +5x2 +5x2) 2,0 4 π 0 1 2 1 eZ(1) = − 1 7 (−3+5x2 +5x2 +5x2)i 2,0 4 π 0 1 2 o (1) − 1 7 − 2 2 2 Z2,0 = 4 π ( 3+5x0 +5x1 +5x2)j e (0) 1 21 2 − 2 − 2 2 0 Z2,2 = 4 π (2x0 x1 x2)+2x0x1i +2x0x2j e (1) 1 14 − 2 2 2 1 Z2,2 = 8 π 8x0x1 +( 4x0 +3x1 + x2)i +2x1x2j oZ(1) = 1 14 8x x +2x x i +(−4x2 + x2 +3x2)j 2,2 8 π 0 2 1 2 0 1 2 2 eZ(2) = 1 35 (x2 − x2 − 2x x i +2x x j) 2,2 4 π 1 2 0 1 0 2 o (2) 1 35 − − Z2,2 = 2 π (x1x2 x0x2i x0x1j) e (3) − 1 210 2 − 2 − 3 Z2,2 = 8 π (x1 x2)i 2x1x2j o (3) − 1 210 2 − 2 Z2,2 = 8 π 2x1x2i +(x1 x2)j

The next proposition shows the interesting relations between the RQZSPs and the classical ZSPs. Besides their obvious importance, these formulae also permit to determine orthogonal relations between the coordinates of the corresponding polynomials in a direct way but they are not needed in the present article and will not be discussed further here.

Proposition 3.2. Let n and l be any integers such that n − l is even and nonneg- e (0) e (m) o (m) ative. The polynomials Zn,l , Zn,l and Zn,l can be represented in the following

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way: l +1 l eZ(0) := eZ(0) + eZ(1)i + oZ(1)j , n,l 2l +1 n,l 2(2l +1) n,l n,l (l +1− m)(l +1+m) eZ(m) := eZ(m) n,l (l + 1)(2l +1) n,l (l +1− m)(l − m) + eZ(m+1)i + oZ(m+1)j 4(l + 1)(2l +1) n,l n,l (l +1+m)(l + m) e (m−1) o (m−1) − (1 + δ − ) Z i − Z j , 0,m 1 4(l + 1)(2l +1) n,l n,l (l +1− m)(l +1+m) oZ(m) := oZ(m) n,l (l + 1)(2l +1) n,l (l +1− m)(l − m) + oZ(m+1)i − eZ(m+1)j 4(l + 1)(2l +1) n,l n,l (l +1+m)(l + m) o (m−1) e (m−1) − (1 + δ − ) Z i + Z j , 0,m 1 4(l + 1)(2l +1) n,l n,l for m =1,...,l+1 (l =0, 1,...).

Proof. The proof follows from Definition 3.1 and expressions (2.15)–(2.17); also, (0) (0) 1 (m) (m) l+1−m  note that Ml = Nl (l+1)(2l+1) and Ml = Nl (l+1)(2l+1)(l+1+m) .

3.2. Recurrence relations. To motivate the relevance of the expressions stated beforehand next we illustrate explicit recurrence rules between the polynomials in use that are plain to be integrated in a computational framework. Proposition 3.3. Let n and l be any integers such that n − l is even and nonneg- e (0) e (m) o (m) ative. The polynomials Zn,l , Zn,l and Zn,l satisfy the recurrence formulae: l +1 (3.2) eZ(0) = eZ(1)i + oZ(1)j , n,l 2(l +2) n,l n,l 1 (1 + δ − )(l +1+m) − − eZ(m) = − 0,m 1 eZ(m 1)i − oZ(m 1)j n,l 2 l +2− m n,l n,l 1 l +1− m + eZ(m+1)i + oZ(m+1)j , 2 l + m +2 n,l n,l 1 (1 + δ − )(l +1+m) − − oZ(m) = − 0,m 1 oZ(m 1)i + eZ(m 1)j n,l 2 l +2− m n,l n,l 1 l +1− m + oZ(m+1)i − eZ(m+1)j , 2 l + m +2 n,l n,l for m =1,...,l+1 (l =0, 1,...).

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Remark 3.6. Under these conditions, the previous recurrence relations allow us to find all of the functions, since for each degree n it is easy to find all of the initial values eZ(1) and oZ(1) directly from the initial value 0,0 0,0 1 3 (3.3) eZ(0) = eZ(1)i = oZ(1)j = ≡ eZ(0). 0,0 0,0 0,0 2 π 0,0

e (m) Proof of Proposition 3.3. For simplicity we just present the calculations for Zn,l (m ≥ 1). To begin, we set 1 (1 + δ − )(l +1+m) 1 l +1− m A (m, l):=− 0,m 1 ,A(m, l):= . 1 2 l +2− m 2 2 l +2+m By direct inspection of previous expressions one has e (m−1) − o (m−1) e (m+1) o (m+1) A1(m, l) Zn,l i Zn,l j + A2(m, l) Zn,l i + Zn,l j (l +2− m)(l − m +1) = eZ(m) −A (m, l) n,l 1 (l + 1)(2l +1) (1 + δ )(l +2+m)(l +1+m) +A (m, l) 0,m 2 (l + 1)(2l +1) − − (l +2− m)(l + m) + eZ(m 1)i − oZ(m 1)j A (m, l) n,l n,l (l + 1)(2l +1) 1 − (l − m)(l +2+m) + eZ(m+1)i + oZ(m 1)j A (m, l) n,l n,l (l + 1)(2l +1) 2

e (m) := Zn,l ,m=1,...,l+1. e (0) o (m) ≥ The proofs for Zn,l and Zn,l (m 1) follow the same principle and are therefore straightforward.  As a direct consequence, we obtain the following relation. Corollary 3.1. Let n and l be any integers such that n−l is even and nonnegative. The polynomials eZ(0), eZ(m) and oZ(m) satisfy the following formula: n,l n,l n,l 2(l +1) eZ(0) − eZ(1)i − oZ(1)j =0, l +1 n,l n,l n,l (1 + δ − )(l +1+m) − − 0,m 1 eZ(m 1) − oZ(m 1)k − eZ(m)i − oZ(m)j =0, l +2− m n,l n,l n,l n,l for m =1,...,l+1 (l =0, 1,...), with the starting value (3.3). In view of the practical applicability of the aforementioned polynomials, it is also of interest to prove an explicit three-term type recurrence rule between them. Proposition 3.4. Let n and l be any integers such that n − l is even and nonneg- ≥ (·) ative. For each n 2, the polynomials Zn,l satisfy the three-term type recurrence formulae: (·) (·) (·) C1(n, l) Zn+2,l + C2(n, l) Zn−2,l = D(n, l, ρ) Zn,l,

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where C1 and C2 are given by (2.11)–(2.12), and the polynomial D(ρ) as (2.13). Proof. It follows straightforwardly from Proposition 2.1.  (·) Remark 3.7. In the case l = n, C2 vanishes and Zn+2,n can be obtained directly (·) from Zn,n for each n ≥ 0. 3.3. General properties. In the sequel, we summarize some fundamental prop- erties of the RQZSPs. Theorem 3.8. For any integers n and l such that n − l is even and nonnegative, (·) Zn,l satisfy the following properties: (·) (1) Zn,l are either even or odd, depending on n or l; (·) (2) Zn,l are 2π-periodic with respect to the variable ϕ; (·) (3) the scalar parts of the polynomials Zn,l are complete and orthogonal in the unit ball in the sense of the scalar product (2.2); (·) (4) the coordinates of the polynomials Zn,l are mutually orthogonal in the unit ball in the sense of the scalar product (2.2); (·) (5) Zn,l are orthonormal in the unit ball in the sense of the scalar product (2.2). (l) Proof. Statement (1) follows from the parity property (2.10) of Rn . The proof of Statement (2) follows from the properties of the Chebyshev polynomials. State- ments (3) and (4) are a consequence of Proposition 3.2. Now we prove Statement (5). By definition of the scalar product (2.2) a direct computation shows that e (m1) e (m2) e (m1) e (m2) Z , Z 2 A R =Sc Z Z dV l1 n2,l2 L (B; ; ) n1,l1 n2,l2 B 1 (m ) (m ) (m ) (m ) = M 1 M 2 R(l1)(ρ)R(l2)(ρ)ρ2dρ Sc eX 1 eX 2 dσ l1 l2 n1 n2 l1 l2 0 S

= δn1,n2 δl1,l2 δm1,m2 . Similarly, one may show that all of the remaining polynomials are mutually or- thonormal in the sense of the scalar product (2.2).  n+1 n Remark 3.9. There are precisely 2 2 +3 2 +1 linearly independent RQZSPs of degree n.

4. Definition and Properties of QZSPs 4.1. QZSPs. We shall now be able to extend our results to a quaternionic Hilbert subspace; in particular, we construct the quaternion Zernike spherical polynomials (QZSPs), whose study will require us to explore a new situation. In continuation of the previous results we define the new polynomials by Definition 4.1 (QZSPs). Let n and l be any integers such that n − l is even and nonnegative. For m =0,...,l (l =0, 1,...) the polynomials defined by l +1 (4.1) QZ(m)(ρ, θ, ϕ):= eZ(m+1)(ρ, θ, ϕ)i + oZ(m+1)(ρ, θ, ϕ)j n,l 2(l +2+m) n,l n,l are called the quaternion Zernike spherical polynomials (QZSPs).

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Remark 4.2. The chosen combination of the RQZSPs stated in Definition 4.1 is due to their underlying symmetry; other possible combinations of those polynomials could also be considered. Based on previous representations we formulate a first preliminary result. Lemma 4.3. Let n and l be any integers so that n − l is even and nonnegative. Q (m) ≥ The polynomials Zn,l are the zero functions for m l +1. Remark 4.4. Having in mind relation (3.2) a direct observation shows that if l = Q (0) m = 0 the underlying polynomials Zn,0 are spherical aberrations; they are scalar- valued for all n ≥ 0 and do not depend of θ and ϕ. Proposition 4.1. For each n =0, 1,...,theset {Q (m) } Zn,n : m =0,...,n formsabasisforthespaceofH-valued homogeneous monogenic polynomials of degree n, and is orthogonal in the sense of the quaternionic product (2.3) with Ω=B. Q (m) Proof. For the special case l = n the polynomials Zn,n coincide, up to a real con- stant depending on n and m, with the following system of homogeneous monogenic polynomials: {eX(m+1,†)i + oX(m+1,†)j : m =0,...,n; n =0, 1,...}. n n This system is complete in L2(B; H; H) ker ∂ and orthogonal in the sense of the quaternionic inner product (2.3); see [3].  Remark 4.5. Notice that, following the usual definition of ZSPs, we could also define the QZSPs by the help of the angular function set constituted by the restrictions to the sphere of the above basis of monogenic polynomials.

The next proposition shows interesting relations between the QZSPs and the classical ZSPs. Proposition 4.2. Let n and l be any integers such that n − l is even and nonneg- ative. The polynomials QZ(m) can be represented in the following way: n,l (1 + δ )(l +1+m) QZ(m) = 0,m eZ(m) − oZ(m)k n,l 2(2l +1) n,l n,l l − m + eZ(m+1)i + oZ(m+1)j , 2(2l +1) n,l n,l for m =0,...,l (l =0, 1,...). Proof. The proof follows from Definition 4.1 and Proposition 3.2.  The QZSPs are given in Cartesian coordinates for the cases n =0, 1, 2inthe following table. Proposition 4.3. Let n and l be any integers such that n − l is even and nonneg- ≥ Q (m) ative. For each n 2, the polynomials Zn,l satisfy the recurrence formula: Q (m) Q (m) Q (m) C1(n, l) Zn+2,l + C2(n, l) Zn−2,l = D(n, l, ρ) Zn,l

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n l m QZSPs 0 0 0 QZ(0) = 1 3 = eZ(0) 0,0 2 π 0,0 1 1 0 QZ(0) = 1 5 (2x + x i + x j)=eZ(0) 1,1 2 2π 0 1 2 1,1 1 QZ(1) = 1 15 (x − x k) 1,1 2 2π 1 2 Q (0) 1 7 − 2 2 2 e (0) 2 0 0 Z2,0 = 4 π ( 3+5x0 +5x1 +5x2)= Z2,0 Q (0) 1 21 2 − 2 − 2 e (0) 2 0 Z2,2 = 4 π (2x0 x1 x2)+2x0x1i +2x0x2j = Z2,2 Q (1) 1 21 2 − 2 − 1 Z2,2 = 4 2π 4x0x1 +(x1 x2)i +2x1x2j 2x0x2k Q (2) 1 210 2 − 2 − 2 Z2,2 = 8 π (x1 x2) 2x1x2k

for m =0,...,l (l =0, 1,...), where the constants C1 and C2 are given by (2.11)– (2.12), and the polynomial D(ρ) as (2.13).

Proof. It follows directly from Proposition 2.1. 

4.2. General properties. This subsection summarizes some fundamental proper- ties of the QZSPs and discusses a second order homogeneous differential equation.

Theorem 4.6. Let n and l be any integers such that n−l is even and nonnegative. Q (m) The Zn,l satisfy the following properties: Q (m) (1) Zn,l are either even or odd, depending on n or l; Q (m) (2) Zn,l are 2π-periodic with respect to the variable ϕ; Q (m) (3) Zn,n are homogeneous monogenic polynomials; Q (m) (4) Zn,l are orthonormal in the unit ball in the sense of the quaternionic product (2.3).

Proof. Statements (1) and (2) follow directly from the parity property (2.10) of (l) Rn and the properties of the Chebyshev polynomials. The proof of Statement (3) follows from Definition 4.1 and Remark 4.5. Now we prove Statement (4). By definition of the quaternionic inner product (2.3) a straightforward computation shows that Q (m1) Q (m2) Q (m1) Q (m2) Z , Z 2 H H = Z Z dV n1,l1 n2,l2 L (B; ; ) n1,l1 n2,l2 B (l +1)(l +1) = − 1 2 i QZ(m1+1) QZ(m2+1) dV i n1,l1 n2,l2 4(l1 + m1 +2)(l2 + m2 +2) B + i QZ(m1+1) QZ(m2+1) dV j + j QZ(m1+1) QZ(m2+1) dV i n1,l1 n2,l2 n1,l1 n2,l2 B B + j QZ(m1+1) QZ(m2+1) dV j . n1,l1 n2,l2 B

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In particular, each on the right-hand side is of the form QZ(m1+1) QZ(m2+1) dV =(I)+(II)i + (III)j +(IV)k n1,l1 n2,l2 B where in each integral (I), (II), (III) and (IV) we have a sum of functions scalar- valued. For simplicity of presentation, we compute (I). The remaining integrals follow the same procedure and are therefore straightforward. (I) = [QZ(m1+1)] [QZ(m2+1)] +[QZ(m1+1)] [QZ(m2+1)] n1,l1 0 n2,l2 0 n1,l1 1 n2,l2 1 B +[QZ(m1+1)] [QZ(m2+1)] +[QZ(m1+1)] [QZ(m2+1)] dV n1,l1 2 n2,l2 2 n1,l1 3 n2,l2 3

= δn1,n2 δl1,l2 δm1,m2 . The previous step follows from Proposition 4.2 and the orthogonality of the classical ZSPs.  Remark 4.7. The total number of linearly independent QZSPs of degree n is n+1 n 1+ 2 2 +1 . We now formulate the main result of this subsection. Theorem 4.8. Let n and l be any integers such that n−l is even and nonnegative. The polynomials QZ(m) are solutions of the homogeneous equation: n,l

(4.2) Δ3 − E(E +3)+n(n +3) f =0 2 ∂ where E := xi is the classical Euler operator. i=0 ∂xi (·) Proof. We begin by proving that Zn,l satisfy equation (4.2). We first compute the e (m) action of the Laplacian and Euler operators (in spherical coordinates) on Zn,l e (m) separately. Now, having in mind that Zn,l can be rewritten as (l) R (ρ) † eZ(m)(ρ, θ, ϕ)=M (m) n eX(m), (ρ, θ, ϕ), n,l l ρl l it follows that 2 (l) (l) 1 ∂ R (ρ) 2 ∂R (ρ) l(l +1) † Δ eZ(m) = M (m) n + n − R(l)(ρ) eX(m), , 3 n,l l ρl ∂ρ2 ρl+1 ∂ρ ρl+2 n l e (m) (l) ∂ Z 1 ∂R (ρ) † E eZ(m) := ρ n,l = M (m) n eX(m), , n,l ∂ρ l ρl−1 ∂ρ l 2 (l) (l) 1 ∂ R (ρ) 1 ∂R (ρ) † E2 eZ(m) = M (m) n + n eX(m), . n,l l ρl−2 ∂ρ2 ρl−1 ∂ρ l From the linearity of the operator Δ − E(E +3)weobtain 3 ∂2R(l)(ρ) l(l +1) Δ − E(E +3) eZ(m) = M (m) (1 − ρ2) n − R(l)(ρ) 3 n,l l ∂ρ2 ρ2 n 2 ∂R(l)(ρ) + (1 − 2ρ2) n eX(m)(ρ, θ, ϕ). ρ ∂ρ l

− e (m) = n(n +3) Zn,l ,

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where the last step follows from the differential equation (2.14). We can prove a o (m) similar result for the polynomials Zn,l . Since the operator is scalar its action on Q (m) Zn,l is given by l +1 Δ − E(E +3) QZ(m) = −n(n +3) eZ(m)i +o Z(m)j 3 n,l 2(l + m +2) n,l n,l

− Q (m)  = n(n +3) Zn,l .

5. Numerical Simulations 5.1. 3D plots of QZSPs. In this subsection we illustrate the QZSPs using plot Q (0) examples. Figures 1 and 2 visualize, respectively, the first two coordinates of Z2,2 in the spatial domain. Figures 3 and 4 visualize the two coordinates of 9 1 QZ(1) = (7x2 +7x2 +7x2 − 5)(x − x k). 3,1 4 6π 0 1 2 1 2

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5.2. Images of the unit ball under the RQZSPs and QZSPs. This subsection presents some numerical examples showing approximations up to degree 10 for the image of a unit ball under the RQZSPs and QZSPs. From the geometrical point of view, we want to map the unit ball to a domain in R3 (not necessarily a ball). The next figures visualize approximations up to degree e (m) o (m) n = 10 for the image of the unit ball under the mappings Zn,l and Zn,l . According to Proposition 4.2, a direct observation shows that for any integers − Q (0) n and l such that n l is even and nonnegative, the polynomials Zn,l are such Q (0) ≡ Q (0) →A∼ R3 that [ Zn,l ]3 0, i.e., Zn,l : B = . We use this insight to motivate the Q (0) numerical procedures for computing the image of the unit ball under Zn,l .We did not go further than n = 7, as our program becomes very time-consuming. The next figures visualize approximations up to degree 7 for the image of the Q (0) unit ball under the mapping Zn,l .

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6. Concluding Remarks and Perspectives The primary purpose of this study was to introduce an extension of the classical ZSPs within the quaternionic analysis setting. They are called the RQZSPs and QZSPs. They are orthonormal in the unit ball. In our point of view the systems can be fairly seen as generalizations of the 3D homogeneous monogenic polynomials recently exploited by several authors. The representations of these polynomials in terms of spherical monogenics on the surface of the unit ball are given explicitly, ready to be implemented on a computer. This is an important point for all the applications mentioned in the introduction (requiring the accurate and rapid calcu- lation of these functions) and it makes them accessible within the range of practical calculations. Amongst other properties of the polynomials, we show that they are of the second order differential operator Δ3 − E(E +3).Furtherre- search on this topic is now under investigation and will be reported in forthcoming papers. Recent studies have shown that these generalized ZSPs are closely related to the well-known prolate spheroidal wave functions (PSWFs) or Slepian functions on symmetric domains; see [18, 29]; the PSWFs have recently found useful practical applications in numerical analysis, nuclear modeling, signal processing and commu- nication theory, electromagnetic modeling and physics. The authors are currently attempting to explore this connection in detail, and they will report it in a forth- coming paper. Moreover, we aim to generalize the RQZSPs and QZSPs to 4D by choosing a suitable complete and orthogonal set of angular monogenic polynomials l (cf. [24]) while generalizing the radial polynomials Rn(ρ) via certain weights. At the same time, we attempt to extend them to different (and more general) domains such as prolate and oblate spheroids, and cylinders; see [23] and [25, 27, 28].

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Center for Research and Development in Mathematics and Applications, University of Aveiro, 3810-193 Aveiro, Portugal. E-mail address: [email protected] Department of Mathematics, Center for Research and Development in Mathematics and Applications, University of Aveiro, 3810-193 Aveiro, Portugal. E-mail address: [email protected]

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