Fractal Spherical Harmonics

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.

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

Let 𝑡₀<𝑡₁<⋯<𝑡_𝑁 be real numbers and 𝐼=[𝑡₀,𝑡_𝑁]=[𝑎,𝑏] the closed interval that contains them. Let a set of data points {(𝑡_𝑛,𝑥_𝑛)∈𝐼×𝑅∶𝑛=0,1,2,…,𝑁} be given. Set 𝐼_𝑛=[𝑡_𝑛−1,𝑡_𝑛], and let 𝐿_𝑛∶𝐼→𝐼_𝑛,𝑛∈{1,2,…,𝑁} be contractive homeomorphisms such that 𝐿_𝑛𝑡₀=𝑡_𝑛−1,𝐿_𝑛𝑡_𝑁=𝑡_𝑛,𝐿_𝑛𝑐₁−𝐿_𝑛𝑐₂≤𝑙𝑐₁−𝑐₂∀𝑐₁,𝑐₂∈𝐼, (1)(2) for some 0≤𝑙<1.

Let 𝐹=𝐼×𝑅 and 𝑁 continuous mappings, 𝐹_𝑛∶𝐹→𝑅, be given satisfying 𝐹_𝑛𝑡₀,𝑥₀=𝑥_𝑛−1,𝐹_𝑛𝑡_𝑁,𝑥_𝑁=𝑥_𝑛𝐹,𝑛=1,2,…,𝑁_𝑛(𝑡,𝑥)−𝐹_𝑛𝑡,𝑦≤𝑟𝑥−𝑦,𝑡∈𝐼,𝑥,𝑦∈𝑅,0≤𝑟<1. (3)(4) Now, define functions 𝑤_𝑛(𝑡,𝑥)=(𝐿_𝑛(𝑡),𝐹_𝑛(𝑡,𝑥)), for all 𝑛=1,2,…,𝑁. For any choices of 𝐿_𝑛(𝑡) and 𝐹_𝑛(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,𝑤_𝑛∶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(𝑡_𝑛)=𝑥_𝑛  for 𝑛=0,1,2,…,𝑁.

The previous function is called a fractal interpolation function (FIF) corresponding to {(𝐿_𝑛(𝑡),𝐹_𝑛(𝑡,𝑥))}^𝑁_𝑛=1. The map 𝑔 is unique satisfying the functional equation [2]: 𝑔(𝑡)=𝐹_𝑛𝐿_𝑛⁻¹(𝑡),𝑔∘𝐿_𝑛⁻¹(𝑡),𝑛=1,2,…,𝑁,𝑡∈𝐼_𝑛=𝑡_𝑛−1,𝑡_𝑛. (5)

The most widely studied fractal interpolation functions so far are defined by the IFS 𝐿_𝑛(𝑡)=𝑎_𝑛𝑡+𝑏_𝑛,𝐹_𝑛(𝑡,𝑥)=𝛼_𝑛𝑥+𝑞_𝑛(𝑡), (6) where −1<𝛼_𝑛<1, for all 𝑛=1,2,…,𝑁. 𝛼_𝑛 is called a vertical scaling factor of the transformation 𝑤_𝑛, and 𝛼=(𝛼₁,𝛼₂,…,𝛼_𝑁) is the scale vector of the IFS. Following the equalities (1), 𝑎_𝑛=𝑡_𝑛−𝑡_𝑛−1𝑡_𝑁−𝑡₀,𝑏_𝑛=𝑡_𝑁𝑡_𝑛−1−𝑡₀𝑡_𝑛𝑡_𝑁−𝑡₀. (7) Let 𝑓∈𝒞(𝐼) be a continuous function. We consider the particular case 𝑞_𝑛(𝑡)=𝑓∘𝐿_𝑛(𝑡)−𝛼_𝑛𝑏(𝑡), (8) where 𝑏 is continuous and such that 𝑏(𝑡₀)=𝑥₀, and 𝑏(𝑡_𝑁)=𝑥_𝑁.

It is easy to check that condition (3) is fulfilled. The set of data points is here {(𝑡_𝑛,𝑥_𝑛=𝑓(𝑡_𝑛))∈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 𝑓^𝛼∶𝐼→𝑅 such that 𝑓^𝛼𝑡_𝑛=𝑓𝑡_𝑛, (9) 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 𝑏=𝐿𝑓, (10) where 𝐿∶𝒞[𝑎,𝑏]→𝒞[𝑎,𝑏] is an operator linear, bounded with respect to the least square norm 𝑓₂=^𝑏_𝑎𝑓²𝑑𝑡^1/2 (11) and such that 𝐿𝑓(𝑡₀)=𝑓(𝑡₀),𝐿𝑓(𝑡_𝑁)=𝑓(𝑡_𝑁), and 𝐿≠Identity.

An example of operator 𝐿 is defined by 𝐿𝑓(𝑡)=𝑣(𝑡)𝑓(𝑡), where 𝑣(𝑡) is continuous on 𝐼 and 𝑣(𝑡₀)=𝑣(𝑡_𝑁)=1 (see page 958 in [3]).

Definition 2. Let Δ∶𝑎=𝑡₀<𝑡₁<⋯<𝑡_𝑁=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. Let 𝑓^𝛼 be the continuous function defined by the IFS (6), (7), (8), and (10). 𝑓^𝛼 is called 𝛼-fractal function associated with 𝑓 with respect to 𝐿 and the partition Δ.

Figure 1 represents the graph of the 𝛼-fractal function (𝑃²₅)^𝛼(𝑡) associated with the Ferrers polynomial (23) 𝑃²₅(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 [𝑡_𝑛−1,𝑡_𝑛]. Their large amplitudes are due to the greatest scale factors (𝛼₃, 𝛼₄, 𝛼₆, and 𝛼₁₀.)

Figure 1 

Graph of the 𝛼-fractal function (𝑃25)𝛼(𝑡) of the Ferrers polynomial 𝑃25(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), 𝑓^𝛼 satisfies the fixed point equation: 𝑓^𝛼(𝑡)=𝑓(𝑡)+𝛼_𝑛𝑓^𝛼−𝐿𝑓∘𝐿_𝑛⁻¹(𝑡)∀𝑡∈𝐼_𝑛. (12)𝑓^𝛼 interpolates to 𝑓 at 𝑡_𝑛 because of using (1), (12), and Barnsley’s theorem: 𝑓^𝛼𝑡_𝑛=𝑓𝑡_𝑛+𝛼_𝑛𝑓^𝛼−𝐿𝑓𝑡_𝑁=𝑓𝑡_𝑛∀𝑛=0,1,…,𝑁. (13)

Let us call 𝛼-fractal operator ℱ^𝛼_𝐿=ℱ^𝛼_Δ,𝐿 with respect to Δ and 𝐿 the transformation which assigns 𝑓^𝛼 to the function 𝑓: ℱ^𝛼_𝐿𝑓=𝑓^𝛼. (14)

The next results can be read in [4].

Theorem 4. (a)ℱ^𝛼_𝐿: 𝒞[𝑎,𝑏]→𝒞[𝑎,𝑏] is linear and bounded with respect to the least square norm.
(b) If 𝛼=0, ℱ^𝛼_𝐿=𝐼𝑑𝑒𝑛𝑡𝑖𝑡𝑦.
(c) The following inequalities hold: ℱ^𝛼_𝐿|𝛼|≤1+^∞1−|𝛼|_∞‖𝐼−𝐿‖,𝐼−ℱ^𝛼_𝐿≤|𝛼|_∞1−|𝛼|_∞‖𝐼−𝐿‖, (15) where |𝛼|_∞=max{|𝛼_𝑛|∶𝑛=1,2,…,𝑁} and ‖𝑇‖ is the operator norm defined as ‖𝑇‖=sup𝑇𝑓₂∶𝑓₂=1,𝑓∈𝒞[𝑎,𝑏]. (16)

Consequence. According to Theorem 4(b), the collection of maps 𝑓^𝛼 constitutes a family of continuous functions containing 𝑓 as a particular case (for 𝛼=0).

Theorem 5. If |𝛼|_∞<1/(1+‖𝐼−𝐿‖), ℱ^𝛼_𝐿 has a bounded inverse and ℱ^𝛼_𝐿⁻¹≤1+|𝛼|_∞1−|𝛼|_∞‖𝐿‖. (17)

3. Fractal Spherical Harmonics

A homogeneous polynomial 𝑉 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 𝜉=sin𝜑cos(𝜃),𝜂=sin𝜑sin(𝜃),𝜁=cos𝜑 (18) (𝜃 is the longitude and 𝜑 is the colatitude), then 𝑉𝑥,𝑦,𝑧=𝜌^𝑛𝑉𝜉,𝜂,𝜁. (19) In this case, the function 𝑌𝜃,𝜑=𝑌_𝑛𝜑(𝜃)𝜑(𝜃)𝜑𝜃,𝜑=𝑉sincos,sinsin,cos (20) 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: _𝑆𝑌_𝑛(𝑃)𝑌_𝑚(𝑃)𝑑𝑆=0;𝑛≠𝑚, (21) where 𝑑𝑆 is the element of area of the sphere 𝑆. It is well known that the set of spherical harmonics of order 𝑛, ℋ_𝑛, is a linear subspace of functions on the sphere with dimension 2𝑛+1, and one of its orthogonal bases is 𝑈⁰_𝑛(𝑄)=𝑃_𝑛𝜑,𝑈cos_𝑚𝑛(𝑄)=𝑃^𝑚_𝑛𝜑(),𝑉coscos𝑚𝜃_𝑚𝑛(𝑄)=𝑃^𝑚_𝑛cos𝜑sin(𝑚𝜃), (22) if 𝑄=(𝜃,𝜑), 𝑚=1,2,…,𝑛. 𝑃_𝑛 is the 𝑛th Legendre polynomial, and 𝑃^𝑚_𝑛 is the Ferrers or associated Legendre polynomial of degree 𝑛 and order 𝑚 defined as [7] 𝑃^𝑚_𝑛(𝑡)=1−𝑡^2𝑚/2𝑃_𝑛^(𝑚)(𝑡), (23) 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]: _𝑆𝑈⁰_𝑛(𝑄)²𝑑𝑆=4𝜋,𝑛+1_𝑆𝑈^𝑚_𝑛(𝑄)²𝑑𝑆=_𝑆𝑉^𝑚_𝑛(𝑄)²𝑑𝑆=2𝜋()!2𝑛+1𝑛+𝑚()!.𝑛−𝑚 (24) This basis is then unbounded. The family 𝑈⁰_𝑛,𝑈^𝑚_𝑛,𝑉^𝑚_𝑛;𝑛=0,1,2,…,𝑚=1,2,…,𝑛 (25) is an orthogonal and complete system of ℒ²(𝑆). After normalization, one obtains an orthonormal basis (𝑋_𝑛𝑗) of spherical harmonics, and every 𝑓∈ℒ²(𝑆) can be expressed as 𝑓=^∞_𝑛=0^2𝑛_𝑗=0𝑐_𝑛𝑗𝑋_𝑛𝑗, (26) 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 operator ℱ^𝛼_𝐿 to the functions on the sphere 𝑆⊂𝑅³.

For 𝑓∈ℒ²(𝑆), let us define the least square norm on the sphere: 𝑓₂=_𝑆𝑓²(𝑃)𝑑𝑆^1/2. (27)

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 𝑃_𝑛 and 𝑃^𝑚_𝑛 by their fractal analogues 𝑃^𝛼_𝑛 and (𝑃^𝑚_𝑛)^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 operator 𝒮^𝛼_𝑛∶ℋ_𝑛→ℒ²(𝑆), where ℋ_𝑛 is the space of spherical harmonics of order 𝑛, linear, and injective and such that 𝒮^𝛼_𝑛≤ℱ^𝛼_𝐿, (28) where ‖⋅‖ is the operator norm with respect to the least square norm and ℱ^𝛼_𝐿 is the operator of 𝒞[−1,1] studied in the previous section.

The image of the elements of the basis is defined as 𝒮^𝛼_𝑛𝑈⁰_𝑛𝜃,𝜑=𝑈⁰_𝑛^𝛼𝜃,𝜑=𝑃^𝛼_𝑛𝜑,𝒮cos_𝛼𝑛𝑈^𝑚_𝑛𝜃,𝜑=𝑈^𝑚_𝑛^𝛼𝜃,𝜑=𝑃^𝑚_𝑛^𝛼𝜑(),𝒮coscos𝑚𝜃_𝛼𝑛𝑉^𝑚_𝑛𝜃,𝜑=𝑉^𝑚_𝑛^𝛼𝜃,𝜑=𝑃^𝑚_𝑛^𝛼cos𝜑sin(𝑚𝜃), (29) where (𝑃^𝑚_𝑛)^𝛼(cos(𝜑))=ℱ^𝛼_𝐿(𝑃^𝑚_𝑛)(cos(𝜑)) and ℱ^𝛼_𝐿 is the operator defined in Section 2.

We consider in this case the interval 𝐼=[−1,1] and a uniform partition Δ of 𝐼 in order to define the fractal analogues. By linearity, 𝒮^𝛼_𝑛 is extended to the rest of ℋ_𝑛. Let us denote ℋ^𝛼_𝑛=𝒮^𝛼_𝑛ℋ_𝑛. (30)ℋ^𝛼_𝑛 is spanned by {(𝑈⁰_𝑛)^𝛼,(𝑈^𝑚_𝑛)^𝛼,(𝑉^𝑚_𝑛)^𝛼;𝑚=1,2,…,𝑛}. These fractal elements are mutually orthogonal as well. For instance, 𝑈^𝑚_𝑛^𝛼,𝑈^𝑗_𝑛^𝛼₂=₀^2𝜋𝜋₀𝑃^𝑚_𝑛^𝛼cos𝜑𝑃^𝑗_𝑛^𝛼×cos𝜑cos(𝑚𝜃)cos𝑗𝜃sin𝜑𝑑𝜑𝑑𝜃=0 (31) if 𝑚≠𝑗, due to the orthogonality of cos(𝑚𝜃),  cos(𝑗𝜃).

Proposition 8. For all 𝑛=0,1,… and for all 𝑚=1,2,…,𝑛, 𝑈⁰_𝑛^𝛼₂=√2𝜋𝑃^𝛼_𝑛2≤√2𝜋ℱ^𝛼_𝐿𝑃_𝑛2,𝑈^𝑚_𝑛^𝛼₂=𝑉^𝑚_𝑛^𝛼₂=√𝜋𝑃^𝑚_𝑛^𝛼₂≤√𝜋ℱ^𝛼_𝐿𝑃^𝑚_𝑛2. (32)

Proof. For instance, 𝑈^𝑚_𝑛^𝛼2₂=₀^2𝜋𝜋₀𝑃^𝑚_𝑛^𝛼cos𝜑²cos²(𝑚𝜃)sin𝜑𝑑𝜑𝑑𝜃. (33) With the change of variable 𝑡=cos(𝜑) and the integration on the longitude 𝜃, the following inequality holds: (𝑈^𝑚_𝑛)^𝛼2₂=𝜋¹₋₁𝑃^𝑚_𝑛^𝛼(𝑡)²𝑑𝑡=𝜋𝑃^𝑚_𝑛^𝛼2₂≤𝜋ℱ^𝛼_𝐿²𝑃^𝑚_𝑛²₂, (34) and the result is obtained.

Let us denote by (𝑋_𝑛𝑗) the orthonormal basis of spherical harmonics and by (𝑋^𝛼_𝑛𝑗) the transformed elements by 𝒮^𝛼_𝑛. That is to say: 𝒮^𝛼_𝑛𝑋_𝑛𝑗=𝑋^𝛼_𝑛𝑗. (35)

Theorem 9 (linear and bounded operator Theorem [9]). If an operator 𝑆∶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 any 𝑓∈ℒ²(𝑆), let us consider the Laplace series (in terms of spherical harmonics) 𝑓=^∞_𝑛=0^2𝑛_𝑗=0𝑐_𝑛𝑗𝑋_𝑛𝑗. (36) The operator 𝜅^𝛼∶ℒ²(𝑆)→ℒ²(𝑆) such that 𝜅^𝛼𝑓=^∞_𝑛=0^2𝑛_𝑗=0𝑐_𝑛𝑗𝑋^𝛼_𝑛𝑗, (37) where 𝑋^𝛼_𝑛𝑗=𝒮^𝛼_𝑛(𝑋_𝑛𝑗) is well defined, linear, and continuous.

Proof. Let us consider the linear mapping 𝒢^𝛼∶span(𝑋_𝑚𝑗)⊂ℒ²(𝑆)→ℒ²(𝑆), defined by 𝒢^𝛼𝑓=𝒢^𝛼𝑀_𝑚=𝑚0^𝑞_𝑗=𝑗0𝑐_𝑚𝑗𝑋_𝑚𝑗=^𝑀_𝑚=𝑚0^𝑞_𝑗=𝑗0𝑐_𝑚𝑗𝑋^𝛼_𝑚𝑗. (38) According to this definition, the operator 𝒢^𝛼 is linear.
If one considers that for any 𝑓∈span(𝑋_𝑚𝑗), 𝑓∈ℋ_𝑛 for 𝑛 sufficiently large, then 𝒢^𝛼𝑓=𝒮^𝛼_𝑛𝑓, (39) and thus (28) 𝒢^𝛼𝑓₂=𝒮^𝛼_𝑛𝑓₂≤𝒮^𝛼_𝑛𝑓₂≤ℱ^𝛼_𝐿𝑓₂. (40) Consequently, 𝒢^𝛼≤ℱ^𝛼_𝐿, (41) and 𝒢^𝛼 is bounded and linear. Since span𝑋_𝑛𝑗=ℒ²(𝑆), (42) then by the bounded and linear operator theorem (Theorem 9), 𝒢^𝛼 can be extended to ℒ²(𝑆) preserving the norm. Let us denote this extension as 𝜅^𝛼∶ℒ²(𝑆)⟶ℒ²(𝑆). (43) The linearity and continuity of 𝜅^𝛼 imply that if 𝑓=^∞_𝑛=0^2𝑛_𝑗=0𝑐_𝑛𝑗𝑋_𝑛𝑗, (44) then 𝜅^𝛼𝑓=^∞_𝑛=0^2𝑛_𝑗=0𝑐_𝑛𝑗𝑋^𝛼_𝑛𝑗. (45) Moreover, 𝜅^𝛼=𝒢^𝛼≤ℱ^𝛼_𝐿. (46)

4. Fractal Spanning Systems of ℒ²(𝑆)

Let us consider now the process of expanding the operator 𝐿∶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 ℱ^𝛼_𝐿. Let 𝐿_𝐻 be defined as 𝐿_𝐻𝑈^𝑚_𝑛𝜃,𝜑=𝐿𝑃^𝑚_𝑛𝜑(),𝐿coscos𝑚𝜃_𝐻𝑉^𝑚_𝑛𝜃,𝜑=𝐿𝑃^𝑚_𝑛cos𝜑sin(𝑚𝜃) (47) and extended by linearity to the space of spherical harmonics of order 𝑛: 𝐿_𝐻∶ℋ_𝑛⟶ℒ²(𝑆), (48) such that 𝐿_𝐻≤‖𝐿‖. (49) The operator 𝐿_𝐻 admits an extension to (see Theorem 10) 𝐿_𝑆∶ℒ²(𝑆)⟶ℒ²(𝑆), (50) such that 𝐿_𝑆=𝐿_𝐻≤‖𝐿‖. (51)

The proof of the next result can be read in [4] (expression (17) of Theorem 2.4).

Lemma 11. For any 𝑓∈𝒞(𝐼), 𝑓^𝛼−𝑓₂≤|𝛼|_∞𝑓^𝛼−𝐿𝑓₂. (52)

Lemma 12. Let 𝜅^𝛼 be the operator defined in Theorem 10, then 𝜅^𝛼𝑈^𝑚_𝑛−𝑈^𝑚_𝑛2≤|𝛼|_∞𝜅^𝛼𝑈^𝑚_𝑛−𝐿_𝑆𝑈^𝑚_𝑛2,𝜅^𝛼𝑉^𝑚_𝑛−𝑉^𝑚_𝑛2≤|𝛼|_∞𝜅^𝛼𝑉^𝑚_𝑛−𝐿_𝑆𝑉^𝑚_𝑛2. (53)

Proof. 𝜅^𝛼 is the extension of 𝒮^𝛼_𝑛. According to its definition, 𝜅^𝛼𝑈^𝑚_𝑛−𝑈^𝑚_𝑛²₂=₀^2𝜋𝜋₀𝑃^𝑚_𝑛^𝛼cos𝜑−𝑃^𝑚_𝑛cos𝜑²×cos²(𝑚𝜃)sin𝜑𝑑𝜑𝑑𝜃=𝜋¹₋₁𝑃^𝑚_𝑛^𝛼(𝑡)−𝑃^𝑚_𝑛(𝑡)²𝑃𝑑𝑡=𝜋^𝑚_𝑛^𝛼−𝑃^𝑚_𝑛²₂≤𝜋|𝛼|²_∞𝑃^𝑚_𝑛^𝛼−𝐿𝑃^𝑚_𝑛²₂. (54) The last inequality agrees with (52) for 𝑓=𝑃^𝑚_𝑛. Furthermore, 𝜅^𝛼𝑈^𝑚_𝑛−𝐿_𝑆𝑈^𝑚_𝑛²₂=₀^2𝜋𝜋₀𝑃^𝑚_𝑛^𝛼−𝐿𝑃^𝑚_𝑛cos𝜑²×cos²(𝑚𝜃)sin𝜑𝑑𝜑𝑑𝜃=𝜋¹₋₁𝑃^𝑚_𝑛^𝛼(𝑡)−𝐿𝑃^𝑚_𝑛(𝑡)²𝑃𝑑𝑡=𝜋^𝑚_𝑛^𝛼−𝐿𝑃^𝑚_𝑛²₂. (55) As a consequence, 𝜅^𝛼𝑈^𝑚_𝑛−𝑈^𝑚_𝑛2≤|𝛼|_∞𝜅^𝛼𝑈^𝑚_𝑛−𝐿_𝑆𝑈^𝑚_𝑛2. (56)

Proposition 13. For any 𝑓∈ℒ²(𝑆), 𝜅^𝛼𝑓−𝑓₂≤|𝛼|_∞𝜅^𝛼𝑓−𝐿_𝑆𝑓₂,𝜅^𝛼𝑓−𝑓₂≤|𝛼|_∞1−|𝛼|_∞𝐼−𝐿_𝑆𝑓₂. (57)(58)

Proof. Using arguments of density [3, Lemma 4.7], the inequalities of Lemma 12 can be extended to any 𝑓∈ℒ²(𝑆): 𝜅^𝛼𝑓−𝑓₂≤|𝛼|_∞𝜅^𝛼𝑓−𝐿_𝑆𝑓₂. (59) Consequently, 𝜅^𝛼𝑓−𝑓₂≤|𝛼|_∞𝜅^𝛼𝑓−𝑓₂+𝑓−𝐿_𝑆𝑓₂, (60) and thus 𝜅^𝛼𝑓−𝑓₂≤|𝛼|_∞1−|𝛼|_∞𝐼−𝐿_𝑆𝑓₂. (61)

Proposition 14. If |𝛼|_∞<‖𝐿‖⁻¹, 𝜅^𝛼 is injective and its range is closed.

Proof. If 𝜅^𝛼(𝑓)=0, according to (57), 𝑓₂≤|𝛼|_∞‖𝐿‖𝑓₂. (62) Since |𝛼|_∞‖𝐿‖<1, one obtains that 𝑓=0. The inequality (57) 𝜅^𝛼𝑓−𝑓₂≤|𝛼|_∞𝜅^𝛼𝑓−𝐿_𝑆𝑓₂, (63) implies that 𝑓₂−𝜅^𝛼𝑓₂≤|𝛼|_∞𝜅^𝛼𝑓₂+‖𝐿‖𝑓₂,𝑓₂≤1+|𝛼|_∞1−|𝛼|_∞‖𝐿‖𝜅^𝛼𝑓₂. (64) By linearity, 𝑓_𝑛−𝑓_𝑚2≤1+|𝛼|_∞1−|𝛼|_∞‖𝐿‖𝜅^𝛼𝑓_𝑛−𝜅^𝛼𝑓_𝑚2. (65) The rest of the proof is similar to Theorem 3.5 of [3].

Definition 15. A sequence (𝑥_𝑚)⊆𝐻, where 𝐻 is a Hilbert space, is a Bessel sequence if there exists a constant 𝐵>0 such that for all 𝑥∈𝐻^∞_𝑚=0𝑥,𝑥_𝑚²≤𝐵‖𝑥‖². (66)
Let us consider the basis of spherical harmonics (𝑋_𝑛𝑗) and its image by 𝜅^𝛼, (𝑋^𝛼_𝑛𝑗).

Proposition 16. For any scale vector such that |𝛼|_∞<1, (𝑋^𝛼_𝑛𝑗) is a Bessel sequence.

Proof. For any 𝑓∈ℒ²(𝑆), the definition of adjoint operator (𝜅^𝛼)^∗ of 𝜅^𝛼 implies that ^∞_𝑛=0^2𝑛_𝑗=0𝑓,𝑋^𝛼_𝑛𝑗²=^∞_𝑛=0^2𝑛_𝑗=0𝑓,𝜅^𝛼𝑋_𝑛𝑗²=^∞_𝑛=0^2𝑛_𝑗=0𝜅^𝛼∗𝑓,𝑋_𝑛𝑗²=𝜅^𝛼∗𝑓²₂≤𝜅^𝛼∗2𝑓²₂=𝐵𝑓²₂, (67) where 𝐵=‖𝜅^𝛼‖². The penultimate equality is due to Parseval’s formula applied to the basis (𝑋_𝑛𝑗) and the element (𝜅^𝛼)^∗(𝑓). The norm of the adjoint operator (𝜅^𝛼)^∗ agrees with the norm of 𝜅^𝛼 (see, e.g., Lemma 5.5.1 of [9]).

Definition 17. A sequence (𝑥_𝑚)⊆𝐻, where 𝐻 is a Hilbert space, is a frame if there exist constants 𝐴,𝐵>0 such that for all 𝑥∈𝐻𝐴‖𝑥‖²≤^∞_𝑚=0𝑥,𝑥_𝑚²≤𝐵‖𝑥‖². (68)

Definition 18. A sequence (𝑥_𝑚)⊆𝐻, where 𝐻 is a Hilbert space, is a frame sequence if it is a frame for its closed span [𝑥_𝑚]=span(𝑥_𝑚).

Proposition 19. If |𝛼|_∞<‖𝐿‖⁻¹, (𝑋^𝛼_𝑛𝑗) is a frame sequence.

Proof. The right-hand side of (68) is proved in Proposition 16. For the left one, let us consider that 𝜅^𝛼 is injective with closed range (Proposition 14). A closed space of a Hilbert space is Hilbert as well, and (𝜅^𝛼)⁻¹ is well defined, linear, and bounded as 𝜅^𝛼 is (see, e.g., Theorem 3.5.3 of [10]).
Therefore, 𝜅^𝛼∘(𝜅^𝛼)⁻¹ is the identity on the range of 𝜅^𝛼 (denoted by 𝑟𝑔(𝜅^𝛼)).
If 𝑔∈[𝑋^𝛼_𝑛𝑗], then 𝑔∈𝑟𝑔(𝜅^𝛼) since span𝑋^𝛼_𝑛𝑗⊆𝑟𝑔𝜅^𝛼, (69) and thus 𝑋^𝛼_𝑛𝑗=span𝑋^𝛼_𝑛𝑗⊆𝑟𝑔𝜅^𝛼, (70) as 𝑟𝑔(𝜅^𝛼) is closed. For any 𝑔∈[𝑋^𝛼_𝑛𝑗], 𝑔=𝜅^𝛼−1∗∘𝜅^𝛼∗𝑔, (71) and thus 𝑔²₂≤𝜅^𝛼−12𝜅^𝛼∗𝑔²₂, (72) since 𝜅^𝛼−1=𝜅^𝛼−1∗. (73) The definition of adjoint operator implies that (see the proof of Proposition 16) 𝜅^𝛼∗𝑔²₂=^∞_𝑛=0^2𝑛_𝑗=0𝑔,𝑋^𝛼_𝑛𝑗². (74) Let us denote that 𝐴=𝜅^𝛼−1−2. (75) The inequality (72) implies that, 𝐴𝑔²₂≤𝜅^𝛼∗𝑔²₂=^∞_𝑛=0^2𝑛_𝑗=0𝑔,𝑋^𝛼_𝑛𝑗², (76) and the proof is complete.

Definition 20. A sequence (𝑥_𝑚)⊆𝐻, where 𝐻 is a Hilbert space, is a Riesz sequence if there exist 𝑘₁,𝑘₂>0 such that for any (𝑐_𝑚)∈𝑙²𝑘₁^∞_𝑚=0𝑐_𝑚²≤^∞_𝑚=0𝑐_𝑚𝑥_𝑚²≤𝑘₂^∞_𝑚=0𝑐_𝑚². (77)

Proposition 21. If |𝛼|_∞<‖𝐿‖⁻¹, (𝑋^𝛼_𝑛𝑗) is a Riesz sequence.

Proof. If (𝑐_𝑛𝑗)∈𝑙², 𝑓=^∞_𝑛=0^2𝑛_𝑗=0𝑐_𝑛𝑗𝑋_𝑛𝑗∈ℒ²(𝑆),𝜅^𝛼𝑓²₂=^∞_𝑛=0^2𝑛_𝑗=0𝑐_𝑛𝑗𝑋^𝛼_𝑛𝑗²₂≤𝜅^𝛼2𝑓²₂=𝑘₂^∞_𝑛=0^2𝑛_𝑗=0𝑐_𝑛𝑗², (78) where 𝑘₂=𝜅^𝛼2. (79) Let us consider now the left inequality of (77). With the hypothesis on 𝛼, 𝜅^𝛼 has closed range and is injective. (𝜅^𝛼)⁻¹ is defined on the range (𝑟𝑔(𝜅^𝛼)), being continuous as well. Let 𝑙₁ be defined by the expression 𝑙₁=1−|𝛼|_∞‖𝐿‖1+|𝛼|_∞. (80) Due to Parseval’s equality, 𝑓²₂=^∞_𝑛=0^2𝑛_𝑗=0𝑐_𝑛𝑗². (81) Therefore, by (64), 𝑙²₁𝑓²₂=𝑙²₁^∞_𝑛=0^2𝑛_𝑗=0𝑐_𝑛𝑗²≤𝜅^𝛼𝑓²₂=^∞_𝑛=0^2𝑛_𝑗=0𝑐_𝑛𝑗𝑋^𝛼_𝑛𝑗²₂, (82) and (𝑋^𝛼_𝑛𝑗) is a Riesz sequence.

Definition 22. A sequence (𝑥_𝑚)⊆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 (𝑒_𝑚), such that 𝑇(𝑒_𝑚)=(𝑥_𝑚).

Lemma 23 (see [9]). If 𝐿 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, (𝑋^𝛼_𝑛𝑗) is a Riesz basis.

Proof. The extension of the operator 𝐼−𝐿 to ℒ²(𝑆) is 𝐼−𝐿_𝑆 [3, Lemma 4.5], and thus ‖𝐼−𝐿‖=𝐼−𝐿_𝑆. (83) If |𝛼|_∞<(1+‖𝐼−𝐿_𝑆‖)⁻¹, then 11−|𝛼|_∞<1+𝐼−𝐿_𝑆𝐼−𝐿_𝑆,|𝛼|_∞1−|𝛼|_∞𝐼−𝐿_𝑆<1,𝐼−𝜅^𝛼<1, (84) due to (58). The operator 𝜅^𝛼 is an isomorphism (Lemma 23), and (𝑋^𝛼_𝑛𝑗) is a Riesz basis.

Using the identity, 𝑓=𝜅^𝛼∘𝜅^𝛼−1𝑓, (85) one obtains the Laplace series of 𝑓∈ℒ²(𝑆), with respect to the fractal spherical harmonics: 𝑓=𝜅^𝛼𝜅^𝛼−1𝑓=^∞_𝑛=0^2𝑛_𝑗=0𝑐_𝑛𝑗𝜅^𝛼−1𝑓𝑋^𝛼_𝑛𝑗. (86)