Fractal Spherical Harmonics

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.


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 twodimensional 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].
The previous function is called a fractal interpolation function (FIF) corresponding to {(  (),   (, ))}  =1 .The map  is unique satisfying the functional equation [2]: The most widely studied fractal interpolation functions so far are defined by the IFS where −1 <   < 1, for all  = 1, 2, . . ., .   is called a vertical scaling factor of the transformation   , and  = ( 1 ,  2 , . . .,   ) is the scale vector of the IFS.Following the equalities (1), Let  ∈ C() be a continuous function.We consider the particular case where  is continuous and such that ( 0 ) =  0 , and (  ) =   .
In this case, the function is called the Laplace function or spherical harmonic of order  on the unit sphere.Two spherical harmonics of different degree (or order) are orthogonal over the sphere: where  is the element of area of the sphere .It is well known that the set of spherical harmonics of order , H  , is a linear subspace of functions on the sphere with dimension 2 + 1, and one of its orthogonal bases is if  = (, ),  = 1, 2, . . ., .   is the th Legendre polynomial, and    is the Ferrers or associated Legendre polynomial of degree  and order  defined as [7] for  = 1, 2, . . ., .The superindex  is only a notation on the left-hand side of (23).On the right, () represents the derivative of th 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]: This basis is then unbounded.The family is an orthogonal and complete system of L 2 ().After normalization, one obtains an orthonormal basis (  ) of spherical harmonics, and every  ∈ L 2 () can be expressed as in L 2 -sense.The expansion of a function  ∈ L 2 () in terms of this system is called Laplace series of .
In the following, we extend the operator F   to the functions on the sphere  ⊂  3 .
For  ∈ L 2 (), let us define the least square norm on the sphere: Remark 6.The notation ‖ ⋅ ‖ 2 represents the least square norm of a square integrable function defined either on an interval  or on the sphere .
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   and    by their fractal analogues    and (   )  .We look for an expansion of any continuous function on the sphere in terms of these new maps.
The  With the change of variable  = cos() and the integration on the longitude , the following inequality holds: and the result is obtained.
Let us denote by (  ) the orthonormal basis of spherical harmonics and by (   ) the transformed elements by S   .That is to say: Theorem 9 (linear and bounded operator Theorem [9]).If an operator  :  →  is linear and bounded,  is Banach, and  is dense in   , then  can be extended to   preserving the norm of .
Theorem 10.For any  ∈ L 2 (), let us consider the Laplace series (in terms of spherical harmonics) The operator where    = S   (  ) is well defined, linear, and continuous.
Proof.Let us consider the linear mapping According to this definition, the operator G  is linear.If one considers that for any  ∈ span(  ),  ∈ H  for  sufficiently large, then and thus (28) Consequently, and G  is bounded and linear.Since then by the bounded and linear operator theorem (Theorem 9), G  can be extended to L 2 () preserving the norm.Let us denote this extension as The linearity and continuity of   imply that if then and extended by linearity to the space of spherical harmonics of order : such that           ≤ ‖‖ .
The operator   admits an extension to (see Theorem 10) The rest of the proof is similar to Theorem 3.5 of [3].

Figure 1
Figure 1 represents the graph of the -fractal function ( 2 5 )  () associated with the Ferrers polynomial (23)  2 5 (), with respect to the elements described in the legend.The oscillations of the graph display the self-similarity of the mapping on every subinterval [ −1 ,   ].Their large amplitudes are due to the greatest scale factors ( 3 ,  4 ,  6 , and  10 .)Accordingto(5),   satisfies the fixed point equation:
where ‖ ⋅ ‖ is the operator norm with respect to the least square norm and F   is the operator of C[−1, 1] studied in the previous section. (cos ()) (   )  × (cos ()) cos () cos () sin Let us consider now the process of expanding the operator  : C() → C(), linear and bounded with respect to the least square norm, quoted in Section 2, to the space L 2 ().