Research Article | Open Access
M. A. Navascués, "Fractal Spherical Harmonics", International Journal of Analysis, vol. 2013, Article ID 927368, 7 pages, 2013. https://doi.org/10.1155/2013/927368
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.
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 .
2. A Special Type of Fractal Functions
Let be real numbers and the closed interval that contains them. Let a set of data points be given. Set , and let be contractive homeomorphisms such that for some .
Let and continuous mappings, , be given satisfying Now, define functions , for all . For any choices of and satisfying the conditions described and in particular (1), (2), (3), and (4), the next theorem holds.
Theorem 1 (Barnsley ). The iterated function system (IFS) defined above admits a unique attractor . is the graph of a continuous function which obeys for .
The previous function is called a fractal interpolation function (FIF) corresponding to . The map is unique satisfying the functional equation :
The most widely studied fractal interpolation functions so far are defined by the IFS where , for all . is called a vertical scaling factor of the transformation , and is the scale vector of the IFS. Following the equalities (1), Let be a continuous function. We consider the particular case where is continuous and such that , and .
It is easy to check that condition (3) is fulfilled. The set of data points is here . According to Theorem 1, the IFS provides in this case as an attractor the graph of a continuous function denoted here as such that for .
In particular, we consider in this paper the case where is an operator linear, bounded with respect to the least square norm and such that ,, and .
An example of operator is defined by , where is continuous on and (see page 958 in ).
Definition 2. Let , where , be a partition of the interval . A scale vector associated with is a vector .
Figure 1 represents the graph of the -fractal function associated with the Ferrers polynomial (23) , with respect to the elements described in the legend. The oscillations of the graph display the self-similarity of the mapping on every subinterval . Their large amplitudes are due to the greatest scale factors (, , , and .)
Let us call -fractal operator with respect to and the transformation which assigns to the function :
The next results can be read in .
Theorem 4. : is linear and bounded with respect to the least square norm.
If , .
The following inequalities hold: where and is the operator norm defined as
Consequence. According to Theorem 4(b), the collection of maps constitutes a family of continuous functions containing as a particular case (for ).
Theorem 5. If , has a bounded inverse and
3. Fractal Spherical Harmonics
A homogeneous polynomial of degree in the variables , , and satisfying the Laplace equation is called a Laplace or harmonic polynomial of degree . If we consider spherical coordinates for , where ( is the longitude and is the colatitude), then 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 , , is a linear subspace of functions on the sphere with dimension , and one of its orthogonal bases is if , . is the th Legendre polynomial, and is the Ferrers or associated Legendre polynomial of degree and order defined as  for . 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 .
The next equalities are satisfied by the basic spherical harmonics : This basis is then unbounded. The family is an orthogonal and complete system of . After normalization, one obtains an orthonormal basis of spherical harmonics, and every can be expressed as in -sense. The expansion of a function in terms of this system is called Laplace series of .
In the following, we extend the operator to the functions on the sphere .
For , let us define the least square norm on the sphere:
Remark 6. The notation represents the least square norm of a square integrable function defined either on an interval or on the sphere .
In , 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 next result is proved in [4, Proposition 3.2].
Proposition 7. There exists an operator , where is the space of spherical harmonics of order , linear, and injective and such that where is the operator norm with respect to the least square norm and is the operator of studied in the previous section.
The image of the elements of the basis is defined as where and is the operator defined in Section 2.
We consider in this case the interval and a uniform partition of in order to define the fractal analogues. By linearity, is extended to the rest of . Let us denote is spanned by . These fractal elements are mutually orthogonal as well. For instance, if , due to the orthogonality of , .
Proposition 8. For all and for all ,
Proof. For instance, With the change of variable 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 . That is to say:
Theorem 9 (linear and bounded operator Theorem ). 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 , let us consider the Laplace series (in terms of spherical harmonics) The operator such that where is well defined, linear, and continuous.
Proof. Let us consider the linear mapping , defined by
According to this definition, the operator is linear.
If one considers that for any , for sufficiently large, then and thus (28) Consequently, and is bounded and linear. Since then by the bounded and linear operator theorem (Theorem 9), can be extended to preserving the norm. Let us denote this extension as The linearity and continuity of imply that if then Moreover,
4. Fractal Spanning Systems of
Let us consider now the process of expanding the operator , linear and bounded with respect to the least square norm, quoted in Section 2, to the space . The extension is similar to the method employed for . Let be defined as and extended by linearity to the space of spherical harmonics of order : such that The operator admits an extension to (see Theorem 10) such that
The proof of the next result can be read in  (expression of Theorem 2.4).
Lemma 11. For any ,
Lemma 12. Let be the operator defined in Theorem 10, then
Proof. is the extension of . According to its definition, The last inequality agrees with (52) for . Furthermore, As a consequence,
Proposition 13. For any ,
Proposition 14. If , is injective and its range is closed.
Definition 15. A sequence , where is a Hilbert space, is a Bessel sequence if there exists a constant such that for all
Let us consider the basis of spherical harmonics and its image by , .
Proposition 16. For any scale vector such that , is a Bessel sequence.
Proof. For any , the definition of adjoint operator of implies that where . The penultimate equality is due to Parseval’s formula applied to the basis and the element . The norm of the adjoint operator agrees with the norm of (see, e.g., Lemma 5.5.1 of ).
Definition 17. A sequence , where is a Hilbert space, is a frame if there exist constants such that for all
Definition 18. A sequence , where is a Hilbert space, is a frame sequence if it is a frame for its closed span .
Proposition 19. If , is a frame sequence.
Proof. The right-hand side of (68) is proved in Proposition 16. For the left one, let us consider that is injective with closed range (Proposition 14). A closed space of a Hilbert space is Hilbert as well, and is well defined, linear, and bounded as is (see, e.g., Theorem 3.5.3 of ).
Therefore, is the identity on the range of (denoted by ).
If , then since and thus as is closed. For any , and thus since The definition of adjoint operator implies that (see the proof of Proposition 16) Let us denote that The inequality (72) implies that, and the proof is complete.
Definition 20. A sequence , where is a Hilbert space, is a Riesz sequence if there exist such that for any
Proposition 21. If , is a Riesz sequence.
Proof. If , where Let us consider now the left inequality of (77). With the hypothesis on , has closed range and is injective. is defined on the range (), being continuous as well. Let be defined by the expression Due to Parseval’s equality, Therefore, by (64), and is a Riesz sequence.
Definition 22. A sequence , where is a Hilbert space, is a Riesz basis if it is equivalent to an orthonormal basis of . That is to say, there exists a topological isomorphism on and an orthonormal basis , such that .
Lemma 23 (see ). If is a bounded and linear operator from a Banach space into itself such that , then exists and is bounded.
Theorem 24. If , is a Riesz basis.
Using the identity, one obtains the Laplace series of , with respect to the fractal spherical harmonics:
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