# Fractal Spherical Harmonics

Authors: M. A. Navascués
Journal: International Journal of Analysis
 (2013)

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## Abstract

This paper tackles the construction of fractal maps on the unit sphere. The functions defined are a generalization of the classical spherical harmonics. The methodology used involves an iterated function system and a linear and bounded operator of functions on the sphere. For a suitable choice of the coefficients of the system, one obtains classical maps on the sphere. The different values of the system parameters provide Bessel sequences, frames, and Riesz fractal bases for the Lebesgue space of the square integrable functions on the sphere. The Laplace series expansion is generalized to a sum in terms of the new fractal mappings.


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## Body

## 1. Introduction

The spherical patterns have an increasing importance in many recent scientific and technical fields like brain mapping, meteorology, oceanography, environment, etc. The problem of interpolation and approximation on the sphere tackles the reconstruction of an unknown function from a finite set of data. The targeted variable may be an electric brain potential, temperature, pressure, etc. If the accessible information is localized in a small zone of the surface, one can use two-dimensional Cartesian methods. However, if the data are disseminated all over the sphere, the global methods are essential.The main objective of the present paper is the definition of new global functions on the sphere. The mappings are constructed here by means of a fractal methodology (an iterated function system and a linear and bounded operator), and they provide a perturbation of the spherical harmonics. For a suitable choice of the coefficients of the system, one obtains the classical functions. Different values of the parameters provide Bessel sequences, frames, and Riesz fractal bases for the Lebesgue space of square integrable functions on the sphere. The Laplace series expansion is generalized to a sum in terms of the new fractal maps.A very different approach for the problem of approximation on the sphere is used in [1].


## 2. A Special Type of Fractal Functions

Lett0<t1<⋯<tN be real numbers and I=[t0,tN]=[a,b] the closed interval that contains them. Let a set of data points {(tn,xn)∈I×R:n=0,1,2,…,N} be given. Set In=[tn-1,tn], and let Ln:I→In,n∈{1,2,…,N} be contractive homeomorphisms such that
(1)Ln(t0)=tn-1,Ln(tN)=tn,(2)|Ln(c1)-Ln(c2)|≤l|c1-c2|∀c1,c2∈I,
for some 0≤l<1.LetF=I×R and N continuous mappings, Fn:F→R, be given satisfying
(3)Fn(t0,x0)=xn-1,Fn(tN,xN)=xn,n=1,2,…,N(4)|Fn(t,x)-Fn(t,y)|≤r|x-y|,t∈I,x,y∈R,0≤r<1.
Now, define functions wn(t,x)=(Ln(t),Fn(t,x)), for all n=1,2,…,N. For any choices of Ln(t) and Fn(t,x) satisfying the conditions described and in particular (1), (2), (3), and (4), the next theorem holds.Theorem 1 (Barnsley [2]).
The iterated function system (IFS){F,wn:n=1,2,…,N} defined above admits a unique attractor G. G is the graph of a continuous function g:I→R which obeys g(tn)=xn  for n=0,1,2,…,N.The previous function is called a fractal interpolation function (FIF) corresponding to{(Ln(t),Fn(t,x))}n=1N. The map g is unique satisfying the functional equation [2]:
(5)g(t)=Fn(Ln-1(t),g∘Ln-1(t)),n=1,2,…,N,t∈In=[tn-1,tn].The most widely studied fractal interpolation functions so far are defined by the IFS(6)Ln(t)=ant+bn,Fn(t,x)=αnx+qn(t),
where -1<αn<1, for all n=1,2,…,N. αn is called a vertical scaling factor of the transformation wn, and α=(α1,α2,…,αN) is the scale vector of the IFS. Following the equalities (1),
(7)an=tn-tn-1tN-t0,bn=tNtn-1-t0tntN-t0.
Let f∈𝒞(I) be a continuous function. We consider the particular case
(8)qn(t)=f∘Ln(t)-αnb(t),
where b is continuous and such that b(t0)=x0, and b(tN)=xN.It is easy to check that condition (3) is fulfilled. The set of data points is here {(tn,xn=f(tn))∈I×R:n=0,1,2,…,N}. According to Theorem 1, the IFS provides in this case as an attractor the graph of a continuous function denoted here as fα:I→R such that
(9)fα(tn)=f(tn),
for n=0,1,…,N.The choice of IFS provides fractal analogues of any continuous function (see, e.g., [3–6]).In particular, we consider in this paper the case(10)b=L(f),
where L:𝒞[a,b]→𝒞[a,b] is an operator linear, bounded with respect to the least square norm
(11)∥f∥2=(∫ab‍|f|2dt)1/2
and such that Lf(t0)=f(t0),Lf(tN)=f(tN), and L≠Identity.An example of operatorL is defined by Lf(t)=v(t)f(t), where v(t) is continuous on I and v(t0)=v(tN)=1 (see page 958 in [3]).Definition 2.
LetΔ:a=t0<t1<⋯<tN=b, where N>1, be a partition of the interval I=[a,b]. A scale vector associated  with Δ is a vector α∈(-1,1)N.Definition 3.
Letfα be the continuous function defined by the IFS (6), (7), (8), and (10). fα is called α-fractal function associated with f with respect to L and the partition Δ.Figure1 represents the graph of the α-fractal function (P52)α(t) associated with  the Ferrers polynomial (23) P52(t), with respect to the elements described in the legend. The oscillations of the graph display the self-similarity of the mapping on every subinterval [tn-1,tn]. Their large amplitudes are due to the greatest scale factors (α3, α4, α6,  and α10.)Figure 1
Graph of theα-fractal function (P52)α(t) of the Ferrers polynomial P52(t) in the interval I=[-1,1], step h=0.1, operator Lf(t)=v(t)f(t), where v(t)=cos(2πt) and scale vector α={-0.3,0.4,-0.6,0.8,0.2,-0.7,0.4,0.5,0.2,-0.75}According to (5), fα satisfies the fixed point equation:
(12)fα(t)=f(t)+αn(fα-Lf)∘Ln-1(t)∀t∈In.fα interpolates to f at tn because of using (1), (12), and Barnsley’s theorem:
(13)fα(tn)=f(tn)+αn(fα-Lf)(tN)=f(tn)∀n=0,1,…,N.Let us callα-fractal operator ℱLα=ℱΔ,Lα with respect to Δ and L the transformation which assigns fα to the function f:
(14)ℱLα(f)=fα.The next results can be read in [4].Theorem 4.
(
 a 
)
ℱ
L
α: 𝒞[a,b]→𝒞[a,b] is linear and bounded with respect to the least square norm.
(
 b 
) If α=0, ℱLα=Identity.
(
 c

) The following inequalities hold:
(15)∥ℱLα∥≤1+|α|∞1-|α|∞∥I-L∥,∥I-ℱLα∥≤|α|∞1-|α|∞∥I-L∥,
where |α|∞=max{|αn|:n=1,2,…,N} and ∥T∥ is the operator norm defined as
(16)∥T∥=sup{∥T(f)∥2:∥f∥2=1,f∈𝒞[a,b]}.Consequence. According to Theorem 4(b), the collection of maps fα constitutes a family of continuous functions containing f as a particular case (for α=0).Theorem 5.
If|α|∞<1/(1+∥I-L∥), ℱLα has a bounded inverse and
(17)∥(ℱLα)-1∥≤1+|α|∞1-|α|∞∥L∥.


## 3. Fractal Spherical Harmonics

A homogeneous polynomialV of degree n in the variables x, y, and z satisfying the Laplace equation ΔV=0 is called a Laplace or harmonic polynomial of degree n. If we consider spherical coordinates (ρ,θ,φ) for P=(x,y,z)=ρ(ξ,η,ζ)∈R3, where
(18)ξ=sin(φ)cos(θ),η=sin(φ)sin(θ),ζ=cos(φ)
(θ is the longitude and φ is the colatitude), then
(19)V(x,y,z)=ρnV(ξ,η,ζ).
In this case, the function
(20)Y(θ,φ)=Yn(θ,φ)=V(sin(φ)cos(θ),sin(φ)sin(θ),cos(φ))
is called the Laplace function or spherical harmonic of order n on the unit sphere.Two spherical harmonics of different degree (or order) are orthogonal over the sphere:(21)∫S‍Yn(P)Ym(P)dS=0;n≠m,
where dS is the element of area of the sphere S. It is well known that the set of spherical harmonics of order n, ℋn, is a linear subspace of functions on the sphere with dimension 2n+1, and one of its orthogonal bases is
(22)Un0(Q)=Pn(cos(φ)),Unm(Q)=Pnm(cos(φ))cos(mθ),Vnm(Q)=Pnm(cos(φ))sin(mθ),
if Q=(θ,φ), m=1,2,…,n. Pn is the nth Legendre polynomial, and Pnm is the Ferrers or associated Legendre polynomial of degree n and order m defined as [7]
(23)Pnm(t)=(1-t2)m/2Pn(m)(t),
for m=1,2,…,n. The superindex m is only a notation on the left-hand side of (23). On the right, (m) represents the derivative of mth order.A historical survey of this kind of functions can be found in [8].The next equalities are satisfied by the basic spherical harmonics [7]:
(24)∫S‍(Un0(Q))2dS=4πn+1,∫S‍(Unm(Q))2dS=∫S‍(Vnm(Q))2dS=2π2n+1(n+m)!(n-m)!.
This basis is then unbounded. The family
(25){Un0,Unm,Vnm;n=0,1,2,…,m=1,2,…,n}
is an orthogonal and complete system of ℒ2(S). After normalization, one obtains an orthonormal basis (Xnj) of spherical harmonics, and every f∈ℒ2(S) can be expressed as
(26)f=∑n=0∞‍∑j=02n‍cnjXnj,
in ℒ2-sense. The expansion of a function f∈ℒ2(S) in terms of this system is called Laplace series of f.In the following, we extend the operatorℱLα to the functions on the sphere S⊂R3.Forf∈ℒ2(S), let us define the least square norm on the sphere:
(27)∥f∥2=(∫S‍f2(P)dS)1/2.Remark 6.
The notation∥·∥2 represents the least square norm of a square integrable function defined either on an interval I or on the sphere S.In [4], we defined a family of fractal functions close to the classical spherical harmonics and a finite sum corresponding to the new mappings. The fractal spherical harmonics are defined as in (22), substituting the functions Pn and Pnm by their fractal analogues Pnα and (Pnm)α. We look for an expansion of any continuous function on the sphere in terms of these new maps.The next result is proved in [4, Proposition 3.2].Proposition 7.
There exists an operator𝒮nα:ℋn→ℒ2(S), where ℋn is the space of spherical harmonics of order n, linear, and injective and such that
(28)∥𝒮nα∥≤∥ℱLα∥,
where ∥·∥ is the operator norm with respect to the least square norm and ℱLα is the operator of 𝒞[-1,1] studied in the previous section.The image of the elements of the basis is defined as(29)𝒮nα(Un0)(θ,φ)=(Un0)α(θ,φ)=Pnα(cos(φ)),𝒮nα(Unm)(θ,φ)=(Unm)α(θ,φ)=(Pnm)α(cos(φ))cos(mθ),𝒮nα(Vnm)(θ,φ)=(Vnm)α(θ,φ)=(Pnm)α(cos(φ))sin(mθ),
where (Pnm)α(cos(φ))=ℱLα(Pnm)(cos(φ)) and ℱLα is the operator defined in Section 2.We consider in this case the intervalI=[-1,1] and a uniform partition Δ of I in order to define the fractal analogues. By linearity, 𝒮nα is extended to the rest of ℋn. Let us denote
(30)ℋnα=𝒮nα(ℋn).ℋnα is spanned by {(Un0)α,(Unm)α,(Vnm)α;m=1,2,…,n}. These fractal elements are mutually orthogonal as well. For instance,
(31)((Unm)α,(Unj)α)2=∫02π‍∫0π‍(Pnm)α(cos(φ))(Pnj)α×(cos(φ))cos(mθ)cos(jθ)sinφdφdθ=0
if m≠j, due to the orthogonality of cos(mθ),  cos(jθ).Proposition 8.
For alln=0,1,… and for all m=1,2,…,n,
(32)∥(Un0)α∥2=2π∥Pnα∥2≤2π∥ℱLα∥∥Pn∥2,∥(Unm)α∥2=∥(Vnm)α∥2=π∥(Pnm)α∥2≤π∥ℱLα∥∥Pnm∥2.Proof.
For instance,(33)∥(Unm)α∥22=∫02π‍∫0π‍|(Pnm)α(cos(φ))|2cos2(mθ)sin(φ)dφdθ.
With the change of variable t=cos(φ) and the integration on the longitude θ, the following inequality holds:
(34)∥(Unm)α∥22=π∫-11‍|(Pnm)α(t)|2dt=π∥(Pnm)α∥22≤π∥ℱLα∥2∥Pnm∥22,
and the result is obtained.Let us denote by(Xnj) the orthonormal basis of spherical harmonics and by (Xnjα) the transformed elements by 𝒮nα. That is to say:
(35)𝒮nα(Xnj)=Xnjα.Theorem 9 (linear and bounded operator Theorem [9]).
If an operatorS:X→Y is linear and bounded, Y is Banach, and X is dense in X′, then S can be extended to X′ preserving the norm of S.Theorem 10.
For anyf∈ℒ2(S), let us consider the Laplace series (in terms of spherical harmonics)
(36)f=∑n=0∞‍∑j=02n‍cnjXnj.
The operator κα:ℒ2(S)→ℒ2(S) such that
(37)κα(f)=∑n=0∞‍∑j=02n‍cnjXnjα,
where Xnjα=𝒮nα(Xnj) is well defined, linear, and continuous.Proof.
Let us consider the linear mapping𝒢α:span(Xmj)⊂ℒ2(S)→ℒ2(S), defined by
(38)𝒢α(f)=𝒢α(∑m=m0M‍∑j=j0q‍cmjXmj)=∑m=m0M‍∑j=j0q‍cmjXmjα.
According to this definition, the operator 𝒢α is linear.
If one considers that for anyf∈span(Xmj), f∈ℋn for n sufficiently large, then
(39)𝒢α(f)=𝒮nα(f),
and thus (28)
(40)∥𝒢α(f)∥2=∥𝒮nα(f)∥2≤∥𝒮nα∥∥f∥2≤∥ℱLα∥∥f∥2.
Consequently,
(41)∥𝒢α∥≤∥ℱLα∥,
and 𝒢α is bounded and linear. Since
(42)span¯(Xnj)=ℒ2(S),
then by the bounded and linear operator theorem (Theorem 9), 𝒢α can be extended to ℒ2(S) preserving the norm. Let us denote this extension as
(43)κα:ℒ2(S)⟶ℒ2(S).
The linearity and continuity of κα imply that if
(44)f=∑n=0∞‍∑j=02n‍cnjXnj,
then
(45)κα(f)=∑n=0∞‍∑j=02n‍cnjXnjα.
Moreover,
(46)∥κα∥=∥𝒢α∥≤∥ℱLα∥.


## 4. Fractal Spanning Systems ofℒ2(S)

Let us consider now the process of expanding the operatorL:𝒞(I)→𝒞(I), linear and bounded with respect to the least square norm, quoted in Section 2, to the space ℒ2(S). The extension is similar to the method employed for ℱLα. Let LH be defined as
(47)LH(Unm)(θ,φ)=L(Pnm)(cos(φ))cos(mθ),LH(Vnm)(θ,φ)=L(Pnm)(cos(φ))sin(mθ)
and extended by linearity to the space of spherical harmonics of order n:
(48)LH:ℋn⟶ℒ2(S),
such that
(49)∥LH∥≤∥L∥.
The operator LH admits an extension to (see Theorem 10)
(50)LS:ℒ2(S)⟶ℒ2(S),
such that
(51)∥LS∥=∥LH∥≤∥L∥.The proof of the next result can be read in [4] (expression (17) of Theorem 2.4).Lemma 11.
For anyf∈𝒞(I),
(52)∥fα-f∥2≤|α|∞∥fα-L(f)∥2.Lemma 12.
Letκα be the operator defined in Theorem 10, then
(53)∥κα(Unm)-Unm∥2≤|α|∞∥κα(Unm)-LS(Unm)∥2,∥κα(Vnm)-Vnm∥2≤|α|∞∥κα(Vnm)-LS(Vnm)∥2.Proof.
κ
α is the extension of 𝒮nα. According to its definition,
(54)∥κα(Unm)-Unm∥22=∫02π‍∫0π‍|(Pnm)α(cos(φ))-Pnm(cos(φ))|2×cos2(mθ)sin(φ)dφdθ=π∫-11‍|(Pnm)α(t)-Pnm(t)|2dt=π∥(Pnm)α-Pnm∥22≤π|α|∞2∥(Pnm)α-LPnm∥22.
The last inequality agrees with (52) for f=Pnm. Furthermore,
(55)∥κα(Unm)-LS(Unm)∥22=∫02π‍∫0π‍|((Pnm)α-L(Pnm))(cos(φ))|2×cos2(mθ)sin(φ)dφdθ=π∫-11‍|(Pnm)α(t)-LPnm(t)|2dt=π∥(Pnm)α-LPnm∥22.
As a consequence,
(56)∥κα(Unm)-Unm∥2≤|α|∞∥κα(Unm)-LS(Unm)∥2.Proposition 13.
For anyf∈ℒ2(S),
(57)∥κα(f)-f∥2≤|α|∞∥κα(f)-LSf∥2,(58)∥κα(f)-f∥2≤|α|∞1-|α|∞∥I-LS∥∥f∥2.Proof.
Using arguments of density [3, Lemma 4.7], the inequalities of Lemma 12 can be extended to any f∈ℒ2(S):
(59)∥κα(f)-f∥2≤|α|∞∥κα(f)-LSf∥2.
Consequently,
(60)∥κα(f)-f∥2≤|α|∞(∥κα(f)-f∥2+∥f-LSf∥2),
and thus
(61)∥κα(f)-f∥2≤|α|∞1-|α|∞∥I-LS∥∥f∥2.Proposition 14.
If|α|∞<∥L∥-1, κα is injective and its range is closed.Proof.
Ifκα(f)=0, according to (57),
(62)∥f∥2≤|α|∞∥L∥∥f∥2.
Since |α|∞∥L∥<1, one obtains that f=0. The inequality (57)
(63)∥κα(f)-f∥2≤|α|∞∥κα(f)-LSf∥2,
implies that
(64)∥f∥2-∥κα(f)∥2≤|α|∞(∥κα(f)∥2+∥L∥∥f∥2),∥f∥2≤1+|α|∞1-|α|∞∥L∥∥κα(f)∥2.
By linearity,
(65)∥fn-fm∥2≤1+|α|∞1-|α|∞∥L∥∥κα(fn)-κα(fm)∥2.
The rest of the proof is similar to Theorem 3.5 of [3].Definition 15.
A sequence(xm)⊆H, where H is a Hilbert space, is a Bessel sequence if there exists a constant B>0 such that for all x∈H(66)∑m=0∞‍|〈x,xm〉|2≤B∥x∥2.
Let us consider the basis of spherical harmonics(Xnj) and its image by κα, (Xnjα).Proposition 16.
For any scale vector such that|α|∞<1, (Xnjα) is a Bessel sequence.Proof.
For anyf∈ℒ2(S), the definition of adjoint operator (κα)* of κα implies that
(67)∑n=0∞‍∑j=02n‍|〈f,Xnjα〉|2=∑n=0∞‍∑j=02n‍|〈f,κα(Xnj)〉|2=∑n=0∞‍∑j=02n‍|〈(κα)*(f),Xnj〉|2=∥(κα)*(f)∥22≤∥(κα)*∥2∥f∥22=B∥f∥22,
where B=∥κα∥2. The penultimate equality is due to Parseval’s formula applied to the basis (Xnj) and the element (κα)*(f). The norm of the adjoint operator (κα)* agrees with the norm of κα (see, e.g., Lemma 5.5.1 of [9]).Definition 17.
A sequence(xm)⊆H, where H is a Hilbert space, is a frame if there exist constants A,B>0 such that for all x∈H(68)A∥x∥2≤∑m=0∞‍|〈x,xm〉|2≤B∥x∥2.Definition 18.
A sequence(xm)⊆H, where H is a Hilbert space, is a frame sequence if it is a frame for its closed span [xm]=span¯(xm).Proposition 19.
If|α|∞<∥L∥-1, (Xnjα) is a frame sequence.Proof.
The right-hand side of (68) is proved in Proposition 16. For the left one, let us consider that κα is injective with closed range (Proposition 14). A closed space of a Hilbert space is Hilbert as well, and (κα)-1 is well defined, linear, and bounded as κα is (see, e.g., Theorem 3.5.3 of [10]).
Therefore,κα∘(κα)-1 is the identity on the range of κα (denoted by rg(κα)).
Ifg∈[Xnjα], then g∈rg(κα) since
(69)span(Xnjα)⊆rg(κα),
and thus
(70)[Xnjα]=span¯(Xnjα)⊆rg(κα),
as rg(κα) is closed. For any g∈[Xnjα],
(71)g=((κα)-1)*∘(κα)*(g),
and thus
(72)∥g∥22≤∥(κα)-1∥2∥(κα)*(g)∥22,
since
(73)∥(κα)-1∥=∥((κα)-1)*∥.
The definition of adjoint operator implies that (see the proof of Proposition 16)
(74)∥(κα)*(g)∥22=∑n=0∞‍∑j=02n‍|〈g,Xnjα〉|2.
Let us denote that
(75)A=∥(κα)-1∥-2.
The inequality (72) implies that,
(76)A∥g∥22≤∥(κα)*(g)∥22=∑n=0∞‍∑j=02n‍|〈g,Xnjα〉|2,
and the proof is complete.Definition 20.
A sequence(xm)⊆H, where H is a Hilbert space, is a Riesz sequence if there exist k1,k2>0 such that for any (cm)∈l2(77)k1∑m=0∞‍|cm|2≤∥∑m=0∞‍cmxm∥2≤k2∑m=0∞‍|cm|2.Proposition 21.
If|α|∞<∥L∥-1, (Xnjα) is a Riesz sequence.Proof.
If(cnj)∈l2,
(78)f=∑n=0∞‍∑j=02n‍cnjXnj∈ℒ2(S),∥κα(f)∥22=∥∑n=0∞‍∑j=02n‍cnjXnjα∥22≤∥κα∥2∥f∥22=k2∑n=0∞‍∑j=02n‍|cnj|2,
where
(79)k2=∥κα∥2.
Let us consider now the left inequality of (77). With the hypothesis on α, κα has closed range and is injective. (κα)-1 is defined on the range (rg(κα)), being continuous as well. Let l1 be defined by the expression
(80)l1=1-|α|∞∥L∥1+|α|∞.
Due to Parseval’s equality,
(81)∥f∥22=∑n=0∞‍∑j=02n‍|cnj|2.
Therefore, by (64),
(82)l12∥f∥22=l12∑n=0∞‍∑j=02n‍|cnj|2≤∥καf∥22=∥∑n=0∞‍∑j=02n‍cnjXnjα∥22,
and (Xnjα) is a Riesz sequence.Definition 22.
A sequence(xm)⊆H, where H is a Hilbert space, is a Riesz basis if it is equivalent to an orthonormal basis of H. That is to say, there exists a topological isomorphism T on H and an orthonormal basis (em), such that T(em)=(xm).Lemma 23 (see [9]).
IfL is a bounded and linear operator from a Banach space into itself such that ∥I-L∥<1, then L-1 exists and is bounded.Theorem 24.
If|α|∞<(1+∥I-L∥)-1, (Xnjα) is a Riesz basis.Proof.
The extension of the operatorI-L to ℒ2(S) is I-LS [3, Lemma 4.5], and thus
(83)∥I-L∥=∥I-LS∥.
If |α|∞<(1+∥I-LS∥)-1, then
(84)11-|α|∞<1+∥I-LS∥∥I-LS∥,|α|∞1-|α|∞∥I-LS∥<1,∥I-κα∥<1,
due to (58). The operator κα is an isomorphism (Lemma 23), and (Xnjα) is a Riesz basis.Using the identity,(85)f=κα∘(κα)-1(f),
one obtains the Laplace series of f∈ℒ2(S), with respect to the fractal spherical harmonics:
(86)f=κα((κα)-1(f))=∑n=0∞‍∑j=02n‍cnj((κα)-1(f))Xnjα.



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