# On Analog of Fourier Transform in Interior of the Light Cone

Authors: Tatyana Shtepina
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We introduce an analog of Fourier transformFhρ in interior of light cone that commutes with the action of the Lorentz group. We describe some properties of Fhρ, namely, its action on pseudoradial functions and functions being products of pseudoradial function and space hyperbolic harmonics. We prove that Fhρ-transform gives a one-to-one correspondence on each of the irreducible components of quasiregular representation. We calculate the inverse transform too.


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## Body

## 1. Introduction

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 groupSO(n) in Rn. As a consequence of these properties, the L2(Rn) 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 L2(R) in a direct integral of irreducible representations:
(1)L2(R)=∫-∞∞C·eixξdξ,
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 transformFhρ in the interior of the light cone on which Lie group G=SO0(n-1,1) acts. Suppose Lloc⁡1(SH(R)) is a space of locally integrable functions on pseudosphere SH(R) of radius R, so this space allows a direct integral decomposition into irreducible subspaces invariant under action of Lie group G that is similar to decompositions of L2(Rn) and L2(R) in classical case. Actually, this decomposition was obtained by Gel’fand et al. in [3] in the sixties of the last century.Our analog of Fourier transform is an intertwining operator of quasiregular representation of Lie groupG, so it maps each of the irreducible components of decomposition in itself. Following Stein and Weiss in Euclidean space [1], we describe action of Fhρ 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 (Fhρ)-1 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 [4].


## 2. Spherical Harmonics and Classical  Funk-Hecke Theorem

LetRn,l be the space of homogeneous harmonic polynomials of degree l in n variables. If f(x) belongs to Rn,l then its restriction to sphere Sn-1 is called the surface spherical harmonics of degree l and is denoted by f(ξ), ξ∈Sn-1. The relation between f(x) and f(ξ) follows from homogeneity condition:
(2)f(x)=f(r·ξ)=rlf(ξ),r=|x|.
Surface spherical harmonics of degree l form a linear space over C too, and we denote it by Rn,l. It is quite evident that, for any l=0,1,2,…, the inclusion Rn,l⊂L2(Sn-1) is valid. But in the same space acts the so-called quasiregular representation of Lie group SO(n), defined by the next equality:
(3)[T(g)f](ξ)=f(g-1ξ),ξ∈Sn-1,g∈SO(n).
This representation is unitary with respect to the standard inner product in L2(Sn-1). The next theorem is widely known (see, e.g., [2, 5, 6]).Theorem 1.
(a)    The next decomposition is validL2(Sn-1)=⨁l=0∞Rn,l (decomposition of Hilbert space into orthogonal direct sum).
(b) The subspacesRn,l consisting of the space harmonics of degree l are invariant under Fourier transform F.
(c)   Each of the subspacesRn,l is invariant with respect to the quasiregular representation T of Lie group SO(n) and is isomorphic to the irreducible representation T(l,0,…,0) with a highest weight (l,0,…,0).
(d) The quasiregular representation ofSO(n) in L2(Sn-1) has a simple spectrum.
(e) The spaceRn,l has a dimension (n+l-3)!(n+2l-2)/l!(n-2)! and an orthogonal basis consisting of the next surface spherical harmonics SKl(θ):
(4)SKl(θ)=(∏j=0n-3‍Cmj-mj+1n/2-j/2-1+mj+1(cos⁡θn-j-1)×sinmj+1(θn-j-1)∏j=0n-3‍)×e±imn-2θ1,
where θ1,θ2,…,θn-1 are Euler angles on sphere Sn-1; l=m0; and K is multi-index K=(m1,…,mn-3;±mn-2) such that m0≥m1≥⋯≥mn-3≥mn-2≥0, mi∈Z.The known Funk-Hecke theorem states that for integral operators whose kernels depend only on the distanceρ (in spherical geometry) between points ξ and η where ξ,η∈Sn-1 every surface spherical harmonics is an eigenvector. We give a contemporary formulation of the Funk-Hecke theorem following the monograph [7] of Erdélyi.Theorem 2 (Funk, Hecke).
LetF(x) be a function of a real variable x which is absolutely Lebesgue integrable on [-1,1] together with its square. Then, for any unit vector η,
(5)∫Sn-1F[(ξ,η)]SKl(ξ)dξ=λn,lSKl(η),
where
(6)λn,l=il(2π)n/2∫-∞∞t(2-n)/2Jl+n/2-1(t)f(t)dt,f(t)=12π∫-11e-ixtF(x)dx.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 functionf(x) be a product of radial function and space spherical harmonics of degree l:
(7)f(x)=f0(|x|)SKl(x),
where f0(r) is such that f(x)∈L1(Rn)∩L2(Rn). Then its Fourier transform has a form:
(8)[Ff](x)=Fl(|x|)SKl(x),
where
(9)Fl(r)=2πilr(n-2)/2+l×∫0+∞f0(s)sl+n/2J(n+2l-2)/2(2πrs)ds.
(b) In particular, Fourier transform for radial function f(x)=f0(|x|) is also radial:
(10)[Ff](x)=F0(|x|)
(one sets l=0 in the above formula).As Proposition3 implies, the infinite-dimensional subspaces
(11)Hl=span⁡{f(|x|)SKl(x)},
where f(r) runs over the set of radial functions satisfying conditions of Proposition 3 and SKl(x) runs over the set of space spherical harmonics of degree l, are invariant under Fourier transform in L2(Rn).On the other hand, if we fix the functionf(|x|) we get a subspace in Hl, which is invariant with respect to quasiregular representation of group SO(n) in L2(Rn). It can be easily verified that it is isomorphic to the irreducible representation T(l,0,…,0) with a highest weight (l,0,…,0). 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:(12)L2(Rn)=⨁l=0∞Hl.We will try to extend the classical theorem of Funk and Hecke and its corollaries on the hyperbolic spaceRn-1,1 with indefinite inner product.


## 3. Hyperbolic Harmonics and Generalized Funk-Hecke Theorem

LetRn-1,1 be the pseudo-Euclidean space with the indefinite inner product
(13)[x,y]=-x1y1-⋯-xn-1yn-1+xnyn.
This inner product may be used for definition of a distance r(x,y) between two points x,y∈Rn-1,1 that do not belong to the light cone [z,z]=0. We assume, for such two points,
(14)cosh⁡r(x,y)=[x,y][x,x]·[y,y].
Such distance may take either real nonnegative or pure imaginary values. However, if we restrict ourselves by the interior U of the light cone’s upper sheet
(15)U={x∈Rn-1,1∣[x,x]>0,xn>0},
then, for all x,y∈U, we have r(x,y)⩾0.Let us call the set of all pointsx of U, for which [x,x]=R2 holds, pseudosphere of radius R. We will use the designation SH(R) for pseudosphere of radius R and SH for pseudosphere of radius 1 in Rn-1,1. Recall that SH is a manifold of a constant negative curvature in Rn-1,1 on the one hand and a homogeneous symmetric space with respect to the action of Lie group G=SO0(n-1,1) on the other hand, because
(16)SH≅SO0(n-1,1)/SO(n-1).
It follows from here that SH possesses the unique up to constant multiplier left-invariant with respect to G measure dξ:
(17)dξ=sinh⁡n-2θn-1sinn-3θn-2·…·sinθ2dθ1dθ2⋯dθn-1.We denote byLloc⁡1(SH,dξ) the space of complex-valued functions on SH locally integrable in measure dξ. In Lloc⁡1(SH,dξ) acts the quasiregular representation R of Lie group G, defined by
(18)(R(g)f)(x)=f(g-1x),x∈SH,g∈SO0(n-1,1).
We need the notion of space and surface hyperbolic harmonics to decompose the representation R into irreducible ones. We will consider in Lloc⁡1(SH,dξ) functions
(19)HLn,σ(θ)=sinh⁡(3-n)/2θn-1Pσ+(n-3)/2(3-n)/2-m1(cosh⁡θn-1)×SKm1(θ1,θ2,…,θn-2),
where Pνμ(x) are adjoined Legendre functions of genus one, L=(k0,K), K=(k1,…,kn-4,±kn-3), with k0⩾k1⩾⋯⩾kn-2⩾0, and all parameters ki are integers.It is easy to see that if we extend functionsHLn,σ(θ) from pseudosphere SH to the interior U of the light cone's upper sheet “by homogeneity” with the degree
(20)σ=-n-22+iρ,ρ∈[0,+∞),
then obtained functions H~Ln,σ on U are solutions of the wave equation □H~Ln,σ=0; that is, they arespace hyperbolic harmonics. This means that we may call HLn,σ(θ)surface hyperbolic harmonics and consider them analogs of surface spherical harmonics SKl(θ).The relation betweenHLnσ and H~Ln,σ follows from homogeneous condition:
(21)H~Ln,σ(u)=H~Ln,σ(r·x)=rσH~Ln,σ(x),r=|x|.SupposeGn,σ is the minimal closed subspace in Lloc⁡1(SH,dξ) containing all surface hyperbolic harmonics HLn,σ(θ). Similarly to Euclidean case, denote by Gn,σ the minimal closed subspace in Lloc⁡1(U,du) containing all space hyperbolic harmonics. It is obvious, from what is stated above, that the HLn,σ(θ) are linearly independent for different σ and all of them are subspaces in the space of wave equation solutions.Basic properties ofGn,σ are proved in [6]. We formulate them in a compact form now.Theorem 5 (analog of Theorem1).
(a) The next decomposition is validLloc1(SH,dξ)=∫0∞Gn,σdρ (the decomposition into continuous direct sum).
(b) Each of the subspacesGn,σ is invariant with respect to the quasiregular representation R of Lie group SO0(n-1,1).
(c) The representations ofSO0(n-1,1) in Gn,σ are irreducible and mutually nonisomorphic.
(d) The quasiregular representationR of G in Lloc1(SH,dξ) has a simple spectrum.
(e) The spaceGn,σ is infinite-dimensional and has a basis generated by functions of the form HLn,σ(θ).The following theorem generalizes the classical Funk-Hecke theorem to the case of hyperbolic space. This theorem, for casesn=2 and n=4, was proved in [8]. But the general case was published in [6].Theorem 6.
SupposeF(x) is a function of a real variable x such that (a)
F(x)∈L1(-∞,+∞)∩L2(-∞,+∞);(b)
F(x) can be continued analytically to a function F(α) of the complex variable α=x+iy that is bounded and analytic in the lower half-plane y⩽0;(c)
F(x) has Fourier preimage f(t)∈L1(0,+∞).Let HLn,σ be an arbitrary surface hyperbolic harmonic of homogeneity degree σ. Then, for any vector η∈SH, the following equality holds:
(22)∫SHF([ξ,η])HLn,σ(ξ)dξ=λn,σHLn,σ(η),
where the eigenvalue λn,σ does not depend on index L of harmonics HLn,σ and equals
(23)λn,σ=2n/2(-πi)(n-2)/2×∫0+∞α2-n/2Kσ+n/2-1(iα)f(α)dα,
where Kν(z) is the McDonald function and
(24)f(α)=12π∫-∞+∞eiαtF(t)dt.The idea of the proof lies in using of intertwining operators theory. Namely, let us define an operatorA in Lloc⁡1(SH) by the equality
(25)[Af](η)=∫SHF[(ξ,η)]f(ξ)dξ.
It can be easily seen that A is an intertwining operator of quasiregular representation R. Because the spectrum of R is simple, A can map the invariant subspace Gn,σ only to itself. Schur's lemma implies that A|Gn,σ=λn,σ·E, where λn,σ does not depend on the multi-index L and E is identity operator.Thus one can assume thatL=(0,0,…,0); that is, examine the zonal hyperbolic harmonics HOnσ(ξ) instead of arbitrary spherical harmonics HLnσ(ξ).The nontrivial part of the proof lies in calculation of eigenvalueλn,σ rather than in verifying if surface hyperbolic harmonics are eigenvectors for A.


## 4. The Hyperbolic Fourier Transform and Some of Its Properties

To obtain analogs of Proposition3 we need some integral transform Fhρ in Lloc⁡1(U) similar to the Fourier transform in L2(Rn).Definition 7.
The hyperbolic Fourier transform in space Lloc⁡1(U) is a transform Fhρ, defined by (this integral should be understood in a regularized value sense)(26)[Fhρf](ξ)=|ξ|1+iρ∫Ue-i[ξ,x]|x|-1-iρf(x)dx,
where dx=drds and ds is an invariant measure on hyperboloid SH(r).Note that hyperbolic Fourier transformFhρ is dependent on ρ; the reason is that this transform acts on its “own” component Gnσ (i.e., for σ=-(n-2)/2+iρ) simply as a scalar operator (see Corollary 11 from Proposition 10). Perhaps, uniform integral operator, acting on all subspaces Gnσ as a scalar and invariant under R(g), does not exist.Proposition 8.
The hyperbolic Fourier transformFhρ is an intertwining operator of quasiregular representation R in Lloc1(U).Proof.
By definition of quasiregular representation and hyperbolic Fourier transform, we have(27)[Fhρ∘R(g)f](ξ)=Fhρ(R(g)f)(ξ)=|ξ|1+iρ∫Ue-i[x,ξ]|x|-1-iρ(R(g)f)(x)dx=|ξ|1+iρ∫Ue-i[x,ξ]|x|-1-iρf(g-1x)dx.On the other hand, (28)R[(g)∘Fhρf](ξ)=R(g)(Fhρf)(ξ)=R(g)(|ξ|1+iρ∫Ue-i[x,ξ]|x|-1-iρf(x)dx)=|ξ|1+iρ∫Ue-i[x,g-1ξ]|x|-1-iρf(x)dx=|ξ|1+iρ∫Ue-i[gx,ξ]|x|-1-iρf(x)dx.
Change variable gx=t. Then, x=g-1t and dx=dt since the measure on U is invariant under action of Lie group SO0(n-1,1). We have
(29)[R(g)∘Fhρf](ξ)=|ξ|1+iρ∫Ue-i[t,ξ]|g-1t|-1-iρf(g-1t)dt=|ξ|1+iρ∫Ue-i[t,ξ]|t|-1-iρf(g-1t)dt.
So, Fhρ∘R(g)=R(g)∘Fhρ; that is, Fhρ is an intertwining operator.Proposition 9.
Letf(x) be a pseudoradial function belonging to the space Lloc1(U); that is, f(x)=f0([x,x]1/2) for almost all x∈U. Then its integral transform Fhρ is pseudoradial for all ξ∈U:
(30)[Fhρf](ξ)=F0([ξ,ξ]1/2),
where
(31)F0(s)=(-πi)(n-2)/2·2n/2·s-(n-4)/2+iρ×∫0+∞f0(r)r(n-2)/2-iρK(n-2)/2(irs)dr.Proof.
Letf^(ξ)=(Fhρf)(ξ). Then, taking use of Proposition 8,
(32)[R(g)f^](ξ)=(R(g)∘Fhρf)(ξ)=(Fhρ∘R(g)f)(ξ)=(Fhρf)(ξ)=f^(ξ).
We take into account that R(g)f=f because f is pseudoradial.
This proves the first part of the proposition.
We fix now[x,x]=r and [ξ,ξ]=s. Introduce the Euler coordinates on hyperboloids SH(r) and SH(s):
(33)x1=rsinh⁡θn-1sinθn-2·…·cos⁡θ1,x2=rsinh⁡θn-1sinθn-2·…·sinθ1,⋮xn-1=rsinh⁡θn-1cos⁡θn-2,xn=rcosh⁡θn-1,ξ1=ssinh⁡φn-1sinφn-2·…·cos⁡φ1,ξ2=ssinh⁡φn-1sinφn-2·…·sinφ1,⋮ξn-1=ssinh⁡φn-1cos⁡φn-2,ξn=scosh⁡φn-1.
By definition of hyperbolic Fourier transform,
(34)[Fhρf](ξ)=|ξ|1+iρ∫0+∞∫SH(r)e-i[x,ξ]|x|-1-iρf0([x,x])drds=|ξ|1+iρ∫0+∞f0(r)rn-1r-1-iρdr·∫SH(r)e-i[rx′,sξ′]sinh⁡n-2θn-1·sinn-3θn-2·…·sinθ2dθn-1dθn-2⋯dθ1,
where x′,ξ′∈SH. Consider the inner integral I on hyperboloid in more detail:
(35)I=∫0∞∫Sn-2e-irscosh⁡θn-1cosh⁡φn-1×eirssinh⁡θn-1sinh⁡φn-1〈x′′,ξ′′〉·sinh⁡n-2θn-1sinn-3θn-2·…·sinθ2dθn-1⋯dθ1,
where x′′ and ξ′′ belong to spheres, which are intersections of hyperboloids [x,x]=r2 and [ξ,ξ]=s2 by hyperplanes xn=rcosh⁡θn-1 and ξn=scosh⁡φn-1 correspondingly:
(36)I=∫0∞e-irscosh⁡θn-1cosh⁡φn-1sinh⁡n-2θn-1dθn-1·∫Sn-2eirssinh⁡θn-1sinh⁡φn-1〈x′′,ξ′′〉sinn-3θn-2·…·sinθ2dθn-2⋯dθ1.
We calculate the inner integral on sphere in a standard way: first we integrate on a parallel 〈x′′,ξ′′〉=cos⁡a, orthogonal to vector ξ′′; then we integrate by a the obtained function in variable a, 0⩽a⩽π:
(37)∫Sn-2eirssinh⁡θn-1sinh⁡φn-1cos⁡adS=∫0πeirssinh⁡θn-1sinh⁡φn-1cos⁡a(2π(n-2)/2Γ((n-2)/2))(sina)n-3da,
where (2π(n-2)/2/Γ((n-2)/2))(sina)n-3 is area of surface of (n-3)-dimensional sphere with radius sina.
After changing variablescos⁡a=t, sina=1-t2, we get
(38)∫Sn-2eirssinh⁡θn-1sinh⁡φn-1cos⁡adS=2π(n-2)/2Γ((n-2)/2)∫-11eirssinh⁡θn-1sinh⁡φn-1·t×(1-t2)(n-4)/2dt=2π(n-2)/2Γ((n-2)/2)·Γ((n-2)/2)Γ(1/2)(-rssinh⁡θn-1sinh⁡φn-1/2)(n-3)/2·(-rssinh⁡θn-1sinh⁡φn-1/2)(n-3)/2Γ((n-2)/2)Γ(1/2)·∫-11ei(rssinh⁡θn-1sinh⁡φn-1)t(1-t2)(2((n-3)/2)-1)/2dt=(2π)(n-2)/2Γ((n-2)/2)·Γ((n-2)/2)Γ(1/2)(-rssinh⁡θn-1sinh⁡φn-1)(n-3)/2·J(n-3)/2(-rssinh⁡θn-1sinh⁡φn-1)=(2π)(n-1)/2(-rssinh⁡θn-1sinh⁡φn-1)(n-3)/2×J(n-3)/2(-rssinh⁡θn-1sinh⁡φn-1).
Putting the found value of integral on sphere into the expression of hyperbolic Fourier transform, we get:
(39)[Fhρf](ξ)=(2π)(n-1)/2s1+iρ(-ssinh⁡φn-1)(n-3)/2∫0+∞f0(r)rn-1r(n-3)/2r-1-iρ·∫0+∞e-irscosh⁡θn-1cosh⁡φn-1×sinh⁡(n-1)/2θn-1×J(n-3)/2(-rssinh⁡θn-1sinh⁡φn-1)dθn-1dr.
Change variables  z=sinh⁡θn-1, cosh⁡θn-1=1+z2, and dz=1+z2dθn-1, so we have
(40)∫0+∞e-irscosh⁡θn-1cosh⁡φn-1sinh⁡(n-1)/2θn-1×J(n-3)/2(-rssinh⁡θn-1sinh⁡φn-1)dθn-1=∫0+∞e-irscosh⁡φn-11+z2z(n-1)/21+z2·J(n-3)/2(-rssinh⁡φn-1z)dz.
Now we apply integral from [9], Section 2.12.10, formula (10), and set ρ=irscosh⁡φn-1, z=1, x=z, c=-rssinh⁡φn-1, and ν=(n-3)/2. Finally we have
(41)∫0+∞e-irscosh⁡φn-11+z2z(n-1)/21+z2·J(n-3)/2(-rssinh⁡φn-1z)dz=2π(-rssinh⁡φn-1)(n-3)/2(irs)-(n-2)/2K(n-2)/2(irs).
Hence,
(42)[Fhρf](ξ)=2n/2π(n-2)/2×s1+iρ∫0+∞f0(r)rn-1r-1-iρ·(irs)-(n-2)/2K(n-2)/2(irs)dr=(-πi)(n-2)/22n/2s-(n-4)/2+iρ×∫0∞f0(r)r(n-2)/2-iρK(n-2)/2(irs)dr.
Proposition 9 is proved.Proposition 10.
Suppose functionf(x)∈Lloc1(U) is a product of pseudoradial function and space hyperbolic harmonic of homogeneity degree σ:
(43)f(x)=f0([x,x]1/2)H~Ln,σ(x),
and then its Fhρ-transform is
(44)[Fhρf](ξ)=Fn,σ([ξ,ξ]1/2)H~Ln,σ(ξ),
where
(45)Fn,σ(s)=2n/2(-πi)(n-2)/2·s∫0+∞f0(r)Kσ+(n-2)/2(irs)dr.Proof.
We have, by definition of hyperbolic Fourier transform,(46)[Fhρf](ξ)=|ξ|1+iρ∫Ue-i[x,ξ]|x|-1-iρf0([x,x])H~Lnσ(x)dx=|ξ|1+iρ∫0+∞f0(r)r-1-iρ×(∫SH(r)e-i[x,ξ]H~Lnσ(x)dx)dr.
Let x=rx′, ξ=sξ′, where  x′,ξ′∈SH. Because H~Lnσ is a homogeneous function of homogeneity degree σ,
(47)H~Lnσ(x)=H~Lnσ(rx′)=rσHLnσ(x′).
Hence,
(48)[Fhρf](ξ)=|ξ|1+iρ∫Ue-i[x,ξ]|x|-1-iρf0([x,x])H~Lnσ(x)dx(wechangevariablex=rx′,dx=rn-1dx′)=|ξ|1+iρ∫0+∞f0(r)r-1-iρ×(∫SHe-irs[x′,ξ′]rσHLnσ(x′)rn-1dx′)dr=|ξ|1+iρ∫0+∞f0(r)rσ+n-1r-1-iρ×(∫SHe-irs[x′,ξ′]HLnσ(x′)dx′)dr.
We make use of formula (32) from [6] to calculate the integral on SH. Namely, for each ξ′∈SH, the equality takes place:
(49)∫SHe-irs[x′,ξ′]HLnσ(x′)dx′=2n/2(-πirs)(n-2)/2Kσ+(n-2)/2(irs)HLnσ(ξ′).
Now we have
(50)[Fhρf](ξ)=2n/2s1+iρHLnσ(ξ′)·∫0+∞f0(r)rσ+n-1r-1-iρ×(-πirs)(n-2)/2Kσ+(n-2)/2(irs)dr=2n/2H~Lnσ(ξ)s·(-πi)(n-2)/2×∫0+∞f0(r)Kσ+(n-2)/2(irs)dr=FnσH~Lnσ(ξ),
where
(51)Fnσ(s)=2n/2H~Lnσ(ξ)s·(-πi)(n-2)/2×∫0+∞f0(r)Kσ+(n-2)/2(irs)dr.
Proposition 10 is proved.Corollary 11.
Ifσ=-(n-2)/2+iρ, then hyperbolic Fourier transform Fhρ acts on the space Gnσ of space hyperbolic harmonics as a scalar operator:
(52)[FhρH~Lnσ](ξ)=λnσH~Lnσ(ξ),
where
(53)λnσ=(-2πi)n/22cosh⁡(πρ/2).Proof.
Consider[FhρH~nσ](ξ). Let us use Proposition 10. Because f0([ξ,ξ]1/2)≡1, we have
(54)[FhρH~nσ](ξ)=2n/2(-πi)(n-2)/2sH~nσ(ξ)×∫0+∞Kσ+(n-2)/2(irs)dr.
We apply the integral from [9], Section 2.16.2, formula (1), and set ν=iρ and c=is. Finally we have (this integral should be understood in a regularized value sense too) the following:
(55)[FhρH~nσ](ξ)=2n/2(-πi)(n-2)/2·s·π2is·1cos⁡(iπρ/2)H~nσ(ξ)=(-2πi)n/22cosh⁡(πρ/2)H~nσ(ξ).
Corollary 11 is proved.Corollary 12.
The inverse hyperbolic Fourier transform(Fhρ)-1 on each of the spaces Gnσ has the next form:
(56)[(Fhρ)-1f](ξ)=2cosh⁡(πρ/2)(-2πi)n/2|ξ|1+iρ∫Uei[ξ,x]|x|-1-iρf(x)dx.
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 R of Lie group SO0(n-1,1) in Lloc⁡1(U) is still open. Due to the exact sequence
(57)1⟶SO(n-1)⟶SO0(n-1,1)⟶SH⟶1,
any function φ(x) defined on a Lobachevsky space SH could be raised to function φ~ on G=SO0(n-1,1) that is constant on the left cosets under subgroup SO(n-1). An analog of this theorem in L1∩L2(SH) is valid as it was shown in the author’s paper [10]. Our proof method uses Fourier transform and an ordinary convolution of functions on G:
(58)[φ~*ψ~](x)=∫Gφ~(g)ψ~(g-1x)dg.
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.



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*Source: 685794-2014-08-06.xml*