Function Spaces, Compact Operators, and Their ApplicationsView this Special Issue
On Analog of Fourier Transform in Interior of the Light Cone
We introduce an analog of Fourier transform in interior of light cone that commutes with the action of the Lorentz group. We describe some properties of , namely, its action on pseudoradial functions and functions being products of pseudoradial function and space hyperbolic harmonics. We prove that -transform gives a one-to-one correspondence on each of the irreducible components of quasiregular representation. We calculate the inverse transform too.
One of the most valuable integral transforms used in many-dimensional analysis is the classical Fourier transform. It is caused by the fact that this transform has a very simple transformation law at tensions and commutes with action of Lie group in . As a consequence of these properties, the decomposes in a direct sum of irreducible subspaces that are invariant under rotations (e.g., [1, chapter 4] and [2, chapter 9]). This decomposition is an analog of decomposition in a direct integral of irreducible representations: which act in one-dimensional subspaces invariant under translations. There exists a general theorem that guarantees the existence of a direct integral decomposition into irreducible subrepresentations: it suffices that the topological group have a countable dense subset.
The goal of this paper is to introduce an analog of Fourier transform in the interior of the light cone on which Lie group acts. Suppose is a space of locally integrable functions on pseudosphere of radius , so this space allows a direct integral decomposition into irreducible subspaces invariant under action of Lie group that is similar to decompositions of and in classical case. Actually, this decomposition was obtained by Gel’fand et al. in  in the sixties of the last century.
Our analog of Fourier transform is an intertwining operator of quasiregular representation of Lie group , so it maps each of the irreducible components of decomposition in itself. Following Stein and Weiss in Euclidean space , we describe action of on pseudoradial functions and functions that represent a product of pseudoradial function and space hyperbolic harmonics. The obtained formulas allow us to write the inverse transform with ease. These results may be applicable to constructing an equivariant extension of wave operator in interior of the light cone. For Laplace operator it was completed in .
2. Spherical Harmonics and Classical Funk-Hecke Theorem
Let be the space of homogeneous harmonic polynomials of degree in variables. If belongs to then its restriction to sphere is called the surface spherical harmonics of degree and is denoted by , . The relation between and follows from homogeneity condition: Surface spherical harmonics of degree form a linear space over too, and we denote it by . It is quite evident that, for any , the inclusion is valid. But in the same space acts the so-called quasiregular representation of Lie group , defined by the next equality: This representation is unitary with respect to the standard inner product in . The next theorem is widely known (see, e.g., [2, 5, 6]).
Theorem 1. (a) The next decomposition is valid (decomposition of Hilbert space into orthogonal direct sum).
(b) The subspaces consisting of the space harmonics of degree are invariant under Fourier transform .
(c) Each of the subspaces is invariant with respect to the quasiregular representation of Lie group and is isomorphic to the irreducible representation with a highest weight .
(d) The quasiregular representation of in has a simple spectrum.
(e) The space has a dimension and an orthogonal basis consisting of the next surface spherical harmonics : where are Euler angles on sphere ; ; and is multi-index such that , .
The known Funk-Hecke theorem states that for integral operators whose kernels depend only on the distance (in spherical geometry) between points and where every surface spherical harmonics is an eigenvector. We give a contemporary formulation of the Funk-Hecke theorem following the monograph  of Erdélyi.
Theorem 2 (Funk, Hecke). Let be a function of a real variable which is absolutely Lebesgue integrable on together with its square. Then, for any unit vector , where
The simple consequences of the Funk-Hecke theorem are the following two propositions.
Proposition 3 (see [1, chapter IV, Theorems 3.3, 3.10]). (a) Let function be a product of radial function and space spherical harmonics of degree : where is such that . Then its Fourier transform has a form: where (b) In particular, Fourier transform for radial function is also radial: (one sets in the above formula).
As Proposition 3 implies, the infinite-dimensional subspaces where runs over the set of radial functions satisfying conditions of Proposition 3 and runs over the set of space spherical harmonics of degree , are invariant under Fourier transform in .
On the other hand, if we fix the function we get a subspace in , which is invariant with respect to quasiregular representation of group in . It can be easily verified that it is isomorphic to the irreducible representation with a highest weight . Since spaces of irreducible nonisomorphic unitary representations of compact group are mutually orthogonal (H. Weyl's theorem), we have one more important consequence of the classical Funk-Hecke theorem.
Corollary 4. The next decomposition into orthogonal direct sum is valid:
We will try to extend the classical theorem of Funk and Hecke and its corollaries on the hyperbolic space with indefinite inner product.
3. Hyperbolic Harmonics and Generalized Funk-Hecke Theorem
Let be the pseudo-Euclidean space with the indefinite inner product This inner product may be used for definition of a distance between two points that do not belong to the light cone . We assume, for such two points, Such distance may take either real nonnegative or pure imaginary values. However, if we restrict ourselves by the interior of the light cone’s upper sheet then, for all , we have .
Let us call the set of all points of , for which holds, pseudosphere of radius . We will use the designation for pseudosphere of radius and for pseudosphere of radius 1 in . Recall that is a manifold of a constant negative curvature in on the one hand and a homogeneous symmetric space with respect to the action of Lie group on the other hand, because It follows from here that possesses the unique up to constant multiplier left-invariant with respect to measure :
We denote by the space of complex-valued functions on locally integrable in measure . In acts the quasiregular representation of Lie group , defined by We need the notion of space and surface hyperbolic harmonics to decompose the representation into irreducible ones. We will consider in functions where are adjoined Legendre functions of genus one, , , with , and all parameters are integers.
It is easy to see that if we extend functions from pseudosphere to the interior of the light cone's upper sheet “by homogeneity” with the degree then obtained functions on are solutions of the wave equation ; that is, they are space hyperbolic harmonics. This means that we may call surface hyperbolic harmonics and consider them analogs of surface spherical harmonics .
The relation between and follows from homogeneous condition:
Suppose is the minimal closed subspace in containing all surface hyperbolic harmonics . Similarly to Euclidean case, denote by the minimal closed subspace in containing all space hyperbolic harmonics. It is obvious, from what is stated above, that the are linearly independent for different and all of them are subspaces in the space of wave equation solutions.
Basic properties of are proved in . We formulate them in a compact form now.
Theorem 5 (analog of Theorem 1). (a) The next decomposition is valid (the decomposition into continuous direct sum).
(b) Each of the subspaces is invariant with respect to the quasiregular representation of Lie group .
(c) The representations of in are irreducible and mutually nonisomorphic.
(d) The quasiregular representation of in has a simple spectrum.
(e) The space is infinite-dimensional and has a basis generated by functions of the form .
Theorem 6. Suppose is a function of a real variable such that (a); (b) can be continued analytically to a function of the complex variable that is bounded and analytic in the lower half-plane ;(c) has Fourier preimage .Let be an arbitrary surface hyperbolic harmonic of homogeneity degree . Then, for any vector , the following equality holds: where the eigenvalue does not depend on index of harmonics and equals where is the McDonald function and
The idea of the proof lies in using of intertwining operators theory. Namely, let us define an operator in by the equality It can be easily seen that is an intertwining operator of quasiregular representation . Because the spectrum of is simple, can map the invariant subspace only to itself. Schur's lemma implies that , where does not depend on the multi-index and is identity operator.
Thus one can assume that ; that is, examine the zonal hyperbolic harmonics instead of arbitrary spherical harmonics .
The nontrivial part of the proof lies in calculation of eigenvalue rather than in verifying if surface hyperbolic harmonics are eigenvectors for .
4. The Hyperbolic Fourier Transform and Some of Its Properties
To obtain analogs of Proposition 3 we need some integral transform in similar to the Fourier transform in .
Definition 7. The hyperbolic Fourier transform in space is a transform , defined by (this integral should be understood in a regularized value sense) where and is an invariant measure on hyperboloid .
Note that hyperbolic Fourier transform is dependent on ; the reason is that this transform acts on its “own” component (i.e., for ) simply as a scalar operator (see Corollary 11 from Proposition 10). Perhaps, uniform integral operator, acting on all subspaces as a scalar and invariant under , does not exist.
Proposition 8. The hyperbolic Fourier transform is an intertwining operator of quasiregular representation in .
Proof. By definition of quasiregular representation and hyperbolic Fourier transform, we have On the other hand, Change variable . Then, and since the measure on is invariant under action of Lie group . We have So, ; that is, is an intertwining operator.
Proposition 9. Let be a pseudoradial function belonging to the space ; that is, for almost all . Then its integral transform is pseudoradial for all : where
Proof. Let . Then, taking use of Proposition 8,
We take into account that because is pseudoradial.
This proves the first part of the proposition.
We fix now and . Introduce the Euler coordinates on hyperboloids and : By definition of hyperbolic Fourier transform, where . Consider the inner integral on hyperboloid in more detail: where and belong to spheres, which are intersections of hyperboloids and by hyperplanes and correspondingly: We calculate the inner integral on sphere in a standard way: first we integrate on a parallel , orthogonal to vector ; then we integrate by the obtained function in variable , : where is area of surface of -dimensional sphere with radius .
After changing variables , , we get Putting the found value of integral on sphere into the expression of hyperbolic Fourier transform, we get: Change variables , , and , so we have Now we apply integral from , Section , formula , and set , , , , and . Finally we have Hence, Proposition 9 is proved.
Proposition 10. Suppose function is a product of pseudoradial function and space hyperbolic harmonic of homogeneity degree : and then its -transform is where
Proof. We have, by definition of hyperbolic Fourier transform, Let , , where . Because is a homogeneous function of homogeneity degree , Hence, We make use of formula (32) from  to calculate the integral on . Namely, for each , the equality takes place: Now we have where Proposition 10 is proved.
Corollary 11. If , then hyperbolic Fourier transform acts on the space of space hyperbolic harmonics as a scalar operator: where
Proof. Consider . Let us use Proposition 10. Because , we have We apply the integral from , Section , formula , and set and . Finally we have (this integral should be understood in a regularized value sense too) the following: Corollary 11 is proved.
Corollary 12. The inverse hyperbolic Fourier transform on each of the spaces has the next form: The proof evidently follows from Corollary 11.
Remark 13. A well-known theorem asserts that any intertwining operator of the quasiregular representation of a compact group is a convolution [5, chapter V, Section 2, Theorem 2.3]. However, the question whether this theorem is true for representation of Lie group in is still open. Due to the exact sequence any function defined on a Lobachevsky space could be raised to function on that is constant on the left cosets under subgroup . An analog of this theorem in is valid as it was shown in the author’s paper . Our proof method uses Fourier transform and an ordinary convolution of functions on : We hope that the technique developed in this work (including the hyperbolic Fourier transform) will be able to prove that intertwining operators of quasiregular representation of Lorentz group are also involutions in interior of the light cone.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean Spaces, Princeton University Press, 1971.View at: MathSciNet
N. Y. Vilenkin, Special Functions and Representations of Groups, Nauka, Moscow, Russia, 2nd edition, 1991.View at: MathSciNet
I. M. Gel'fand, M. I. Graev, and N. Y. Vilenkin, Integral Geometry and Representation Theory, Generalized Functions, vol. 5, Academic Press, New York, NY, USA, 1966.
S. Helgason, Groups and Geometric Aanalysis, vol. 113, Academic Press, 1984.View at: MathSciNet
A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series: Special Functions, vol. 2, Gordon & Breach, New York, NY, USA, 1990.View at: MathSciNet
T. V. Shtepina, “About representation as convolution of the operator, permutable with the operator quasiregular representations of group of Lorentz,” Trudy Instituta Prikladnoj Matematiki i Mekhaniki, vol. 7, pp. 225–228, 2002.View at: Google Scholar