BESSEL-TYPE SIGNED HYPERGROUPS ON IR

BESSEL-TYPE SIGNED HYPERGROUPS ON IR ¨ MARGIT ROSLER Mathematisches Institut, Technische Universit¨at M¨ unchen Arcisstr. 21, 80290 M¨ unchen, Germany

ABSTRACT In this paper we introduce a family of convolution structures on the real line which are derived from the Bessel-Kingman hypergroups by a certain complexification of spherical Bessel functions. Though their convolution is not positivity-preserving, these “Bessel-type signed hypergroups” provide a natural extension of the usual group structure on the real line.

Introduction While on the half-line IR+ = {x ∈ IR : x ≥ 0} a variety of non-isomorphic hypergroup structures is known, there do not exist any nontrivial hypergroup structures on IR (Zeuner[13]). As Zeuner’s proof shows, this is mainly due to the hypergroup axiom on support continuity of the convolution with respect to the Michael topology. So one may ask whether the situation becomes different if the hypergroup axioms are weakened. One possibility of doing so is provided by the framework of so-called signed hypergroups as introduced by the author in [8]. The concept of signed hypergroups generalizes the hypergroup axiomatics in several points, mainly in abandoning positivity and support-continuity of the convolution. It is aimed to allow the development of a commutative harmonic analysis in close analogy to the hypergroup case; see R¨ osler [8, 9], and for a survey, Ross [10]. In this contribution we construct a class of commutative signed hypergroups Xα , α > − 12 on IR, which are intimately connected with the Bessel-Kingman hypergroups on IR+ . The characters of Xα are given as complex combinations of the normalized spherical Bessel functions jα and jα+1 , namely, Ψα λ (x) = jα (λx) + iCα λx jα+1 (λx),

λ, x ∈ IR,

with Cα = 21 (α + 1)−1 . This is a natural extension of the formula eiλx = cos λx + i sin λx = j−1/2 (λx) + iλx j1/2 (λx) for the usual group characters on IR.

1. Preliminaries If not stated otherwise, our basic notation is the same as in the monograph of Bloom and Heyer [2], to which we also refer for an introduction to hypergroups. Here we recapitulate some facts on Bessel-Kingman hypergroups which will be of interest later on, and give a short summary on commutative signed hypergroups. For a background on Bessel hypergroups we refer to Kingman [7]; for details concerning signed hypergroups see R¨ osler [8, 9]. 1.1. Bessel-Kingman hypergroups For α > −1 , let

∞ X Jα (z) (−1)k z
2k jα (z) := 2 Γ(α + 1) α = Γ(α + 1) , z k! Γ(k + α + 1) 2 α

k=0

z ∈ C,

denote the normalized spherical Bessel functions of order α. In case α > − 21 , they satisfy the product formula Z π p jα (x)jα (y) = Mα jα ( x2 + y 2 − 2xy cos θ) sin2α θ dθ (1.1) 0

for x, y > 0 , with Mα =

Γ(α + 1) ; Γ(α + 21 )Γ( 12 )

see e.g. Watson [12], 11.4. This implies the following product formula for the 1 functions ϕα λ (x) := jα (λx), with α > − 2 and parameter λ ∈ C : Z ∞ α α 2α+1 ϕλ (x)ϕλ (y) = dz for x, y > 0, ϕα λ (z) kα (x, y, z)z 0

where ∆(x, y, z)2α−1 · 1[|x−y|,x+y] (z). (xyz)2α is the indicator function of A and 1p ∆(x, y, z) := (x + y + z)(x + y − z)(x − y + z)(y + z − x) 4 kα (x, y, z) = 22α−1 Mα ·

Here 1A

(1.2)

denotes the area of the triangle with sides x, y, z > 0. The ϕα λ , λ ∈ C are exactly the multiplicative functions of the Bessel-Kingman hypergroup Hα on IR+ , with the convolution on M b (IR+ ) being defined by d(εx ∗ εy )(z) = kα (x, y, z)z 2α+1 dz. Hα is a commutative hypergroup with 0 as neutral element and the identity mapping as involution. Its dual space, that is the space of bounded multiplicative and symmetric functions on Hα , is given

b α = {ϕα : λ ∈ IR+ }. Hα is self-dual with Haar- and Plancherel measure by H λ α −1 2α+1 (2 Γ(α + 1)) x dx. For details, we refer to Bloom and Heyer [2]. 1.2. Commutative signed hypergroups b For a locally compact Hausdorff space X let MIR (X) denote the subspace of real b b measures from M (X), and τ∗ the σ(M (X), C0 (X)) -topology on M b (X). A commutative signed hypergroup is a triple (X, m, ω) consisting of a locally compact, σ -compact Hausdorff space X, a distinguished positive Radon measure m ∈ M+ (X) with supp m = X and a commutative τ∗ -continuous mapping b ω : X × X → MIR (X), (x, y) 7→ εx ∗ εy , satisfying the following axioms:

(A1) For each x ∈ X and f ∈ Cb (X), the translate T x f : y 7→ εx ∗ εy (f ) again belongs to Cb (X) . Furthermore, for f ∈ Cc (X) and any compact subset S K ⊂ X, the set x∈K supp (T x f ) is relatively compact in X. (A2) kεx ∗ εy k ≤ C for all x, y ∈ X, where C > 0 is a constant. (A3) The canonical continuation ∗ of ω to M b (X) , which is given by Z µ ∗ ν(f ) := εx ∗ εy (f ) d(µ ⊗ ν)(x, y) for f ∈ C0 (X), X×X

is associative. (A4) There exists a neutral element e ∈ X with εe ∗ εx = εx for all x ∈ X. (A5) There exists an involutive homeomorphism − on X such that (εx ∗ εy )− = εy ∗ εx

for all x, y ∈ X,

where µ− (A) := µ(A− ) for Borel measures µ on X and Borel sets A ⊂ X. (A6) For all f, g ∈ Cc (X) and x ∈ X the following adjoint relation holds: Z Z x (T f )g dm = f (T x g) dm. X

X

The algebra (M b (X), ∗) becomes a commutative Banach-*-algebra with unit εe , the involution µ 7→ µ− , and with the norm kµk′ := kLµ k , where Lµ (ν) := µ ∗ ν for µ, ν ∈ M b (X) . L1 (X, m) with the same multiplication and norm is a closed ∗ -ideal in (M b (X), ∗, k.k′ ) . Equipped with the topology of uniform convergence on compact subsets of X, the character spaces Xb (X) := {ϕ ∈ Cb (X) : ϕ 6≡ 0, εx ∗ εy (ϕ) = ϕ(x)ϕ(y) for all x, y ∈ X} b := {ϕ ∈ Xb (X) : ϕ(x) = ϕ(x) for all x ∈ X} X

and

are canonically homeomorphic to the spectrum ∆(L1 (X, m)) and its symmetric part ∆S (L1 (X, m)) respectively. Fourier transforms are defined in the usual way,

and there hold generalized Bochner and Plancherel theorems. In particular, there b the Plancherel measure of (X, m, ω), with exists a unique measure π ∈ M+ (X), Z Z 2 |fb|2 dπ for all f ∈ Cc (X). |f | dm = b X X 2. Modified Bessel functions on IR and their product formula For α > −1 and λ ∈ C we define the modified Bessel functions Ψα λ on C by α+1 α Ψα (z), λ (z) := ϕλ (z) + iCα λz ϕλ

with Cα =

1 . 2(α + 1)

The Ψα λ are holomorphic on C ; for λz ∈ C \ {x ∈ IR : x ≤ 0} they can also be written as
2α Γ(α + 1) J (λz) + iJ (λz) . Ψα (z) = α α+1 λ (λz)α −1/2

Note that in the special case α = − 12 we just have Ψλ

(z) = eiλz .

1 2.1. Lemma. (Laplace-representation of Ψα λ ). Let α > − 2 . Then Z 1 1 1 α Ψλ (z) = Mα eiλzt (1 − t)α− 2 (1 + t)α+ 2 dt for all λ, z ∈ C . −1

o [11], Proof. Mehler’s formula for Bessel functions with index α > − 12 (cf. Szeg¨ (1.71.6)) can be written as Z 1 1 α ϕλ (z) = Mα cos(λzt)(1 − t2 )α− 2 dt. −1

With this representation for ϕα+1 , partial integration yields λ Z 1 1 α+1 λzϕλ (z) = (2α + 1)Mα+1 sin(λzt)(1 − t2 )α− 2 t dt, −1

and from these two the assertion follows. |λ·Im(z)| 2.2. Corollary. For α ≥ − 21 and λ ∈ IR the estimate |Ψα λ (z)| ≤ e holds. In particular, α |Ψα λ (x)| ≤ 1 = Ψλ (0)

for all λ, x ∈ IR.

In case α = − 12 this is obvious; for α > − 12 , we have Z 1 1 1 α |λ·Im(z)| |Ψλ (z)| ≤ Mα e−λt·Im(z) (1−t)α− 2 (1+t)α+ 2 dt ≤ e|λ·Im(z)| ·Ψα . λ (0) = e −1

We will now derive a product formula with a quasi-positive kernel for the modified Bessel functions Ψα λ on IR. We shall use the following abbreviations: 2.3. Notation. For x, y, z ∈ IR put  1 (x2 + y 2 − z 2 ) σx,y,z := 2xy 0 as well as ̺(x, y, z) :=

if x, y 6= 0, otherwise,

1 (1 − σx,y,z + σz,x,y + σz,y,x ). 2

2.4. Theorem. (1) For α > − 12 the modified Bessel functions Ψα λ, λ ∈ C , satisfy the following product formula: Z α α α Ψλ (x)Ψλ (y) = Ψα for x, y ∈ IR. (2.1) λ (z) dµx,y (z) IR

b The µα x,y ∈ M (IR) are given by   Kα (x, y, z)|z|2α+1 dz dµα dε (z) x,y (z) :=  x dεy (z)

if x, y 6= 0, if y = 0, if x = 0,

with kernel

Kα (x, y, z) = kα (|x|, |y|, |z|)̺(x, y, z) where kα is the Bessel kernel (1.2). (2) The measures µα x,y have the following properties:     (i) supp µα = −|x| − |y|, − |x| − |y| ∪ |x| − |y| , |x| + |y| for x, y 6= 0 . x,y b α α (ii) µα x,y ∈ M (IR) with µx,y (IR) = 1 and kµx,y k ≤ 4 for all x, y ∈ IR.

Note that in general the measures µα 6 y, then x,y are not positive: if x, y 6= 0, x = ̺(x, y, y − x) = −1, and hence there exists a neighbourhood of y − x in supp µα x,y where z 7→ Kα (x, y, z) is strictly negative. Proof of Theorem 2.4. (1) We need two modified product formulas for spherd ical Bessel functions. First note that xjα+1 (x) = − C1α dx jα (x) (Szeg¨ o [11],

(1.71.5)). Therefore derivation of the usual product formula (1.1) yields Z π p xjα+1 (x) jα (y) = Mα jα+1 ( x2 + y 2 − 2xy cos θ)(x − y cos θ) sin2α θ dθ. 0

p Substitute z := x2 − y 2 − 2xy cos θ and note that x − y cos θ = z · σz,x,y . Thus Z ∞ xjα+1 (x) jα (y) = zjα+1 (z) σz,x,y kα (x, y, z) z 2α+1 dz (2.2) 0

for x, y > 0. Further we claim that for x, y > 0, Z ∞ 2 Cα xy jα+1 (x) jα+1 (y) = jα (z) σx,y,z kα (x, y, z) z 2α+1 dz.

(2.3)

0

For the proof of (2.3) recall the following expansions of Gegenbauer (see e.g. Askey [1], (4.36), (4.37)): For x, y > 0 and θ ∈ IR, p ∞ X Jα ( x2 + y 2 − 2xy cos θ) Jn+α (x) Jn+α (y) α α p Cn (cos θ) = 2 Γ(α) (n + α) α α x y x2 + y 2 − 2xy cos θ n=0

1 if α > − , α 6= 0, 2

∞ X p 2 2 J0 ( x + y − 2xy cos θ) = J0 (x)J0 (y) + 2 Jn (x)Jn (y) cos nθ , n=0

where the Cnα are the Gegenbauer polynomials Cnα (x) =

(2α)n P (α−1/2,α−1/2) (x). (α + 12 )n n

Both series converge uniformly in θ ∈ IR. Multiply by C1α (cos θ) and cos nθ respectively. By orthogonality of the Cnα , integration then gives Z π p 2 Cα x jα+1 (x) y jα+1 (y) = Mα jα ( x2 + y 2 − 2xy cos θ) cos θ sin2α θ dθ 0

for α > − 21 . Now the same substitution as before yields (2.3). By the homogeneity of kα (x, y, z)z 2α+1 and σx,y,z it is also clear that for λ, x, y > 0 the following identities hold: Z ∞ λx jα+1 (λx) jα (λy) = λzjα+1 (λz) kα (x, y, z) σz,x,y z 2α+1 dz, 0 Z ∞ 2 λ xy jα+1 (λx) jα+1 (λy) = jα (λz) kα (x, y, z) σx,y,z z 2α+1 dz. 0

Now suppose that λ > 0 and x, y ∈ IR \ {0}. Then of course, Z 1 jα (λz) kα (|x|, |y|, |z|) |z|2α+1 dz. jα (λx) jα (λy) = 2 IR

Further, z 7→ σx,y,z is even and z 7→ σz,x,y as well as z 7→ σz,y,x are odd. Thus Z ∞ λx jα+1 (λx) jα (λy) = sgn(x) λz jα+1 (λz) kα (|x|, |y|, z) σz,|x|,|y| z 2α+1 dz 0 Z 1 = λz jα+1 (λz) kα (|x|, |y|, |z|) σz,x,y |z|2α+1 dz, 2 IR as well as Cα2

Z

2

∞

λ xy jα+1 (λx) jα+1 (λy) = sgn(xy) jα (λz) kα (|x|, |y|, z) σ|x|,|y|,z z 2α+1 dz 0 Z 1 jα (λz) kα (|x|, |y|, |z|) σx,y,z |z|2α+1 dz. = 2 IR

Combining these gives Z 1 2α+1 α α Ψα dz. Ψλ (x)Ψλ (y) = λ (z) kα (|x|, |y|, |z|)(1 − σx,y,z + σz,x,y + σz,y,x )|z| 2 IR As Ψα λ (0) = 1 for all λ, formula (2.1) is thus proved in case λ > 0. For arbitrary λ ∈ C , it is obtained by analytic continuation: The mapping (λ, z) 7→ Ψα λ (z) is α continuous on C × IR and bounded on U × supp µx,y for each bounded U ⊂ C and x, y ∈ IR. For fixed z ∈ IR, the function λ 7→ Ψα λ (z) is holomorphic on C , and so both sides in (2.1) are holomorphic on C as functions of λ. (2) The statement on the support of µα x,y is clear, because z 7→ ̺(x, y, z) does not vanish on any open subset of IR. Furthermore, if x, y 6= 0 then z ∈ supp µα x,y holds if and only if |x|, |y|, |z| are the sides of a (perhaps degenerated) triangle. So if z ∈ supp µα x,y , then |σx,y,z | ≤ 1, |σz,x,y | ≤ 1 and |σz,y,x | ≤ 1. This yields the assertion about kµα x,y k. 3. Bessel-type signed hypergroups on IR The product formula of the modified Bessel functions gives rise to signed hypergroup structures on IR in a canonical way; this is stated in the following theorem. Furthermore, we shall derive some duality results. Throughout this section we assume that α > − 21 . b 3.1. Theorem. Define ωα : IR × IR → MIR (IR) by ωα (x, y) := µα x,y and put

dmα (x) :=

1 |x|2α+1 dx, 2α+1 Γ(α + 1)

x ∈ IR.

Then Xα := (IR, mα , ωα ) is a commutative signed hypergroup with neutral element 0 and involution x 7→ −x.

For the proof we need two preparatory lemmata: 3.2. Lemma. For f ∈ Cb (IR) write f = fe + fo with fe ∈ Cb (IR) even, fo ∈ Cb (IR) odd. Then Z Z π p α f dµx,y = Mα fe ( x2 + y 2 − 2|xy| cos θ) he (x, y, θ) sin2α θ dθ IR 0 (3.1) Z π p 2α o 2 2 + Mα fo ( x + y − 2|xy| cos θ) h (x, y, θ) sin θ dθ 0

for all x, y ∈ IR, where he (x, y, θ) = 1 − sgn(xy) cos θ,  + y)(1 − sgn(xy) cos θ)  (xp o h (x, y, θ) = x2 + y 2 − 2|xy| cos θ  0

if (x, y) 6= (0, 0), otherwise.

Proof. In case xy = 0 formula (3.1) is obvious. (Note that fe (|x|) = fe (x), Rπ fo (|x|)sgn(x) = fo (x), fo (0) = 0, and that 0 sin2α θ dθ = Mα−1 .) Now suppose xy 6= 0. First note that Z Z ∞ α f dµx,y = fe (z) kα (|x|, |y|, z)(1 − σx,y,z ) z 2α+1 dz IR 0 Z ∞ + fo (z) kα (|x|, |y|, z)(σz,x,y + σz,y,x ) z 2α+1 dz. 0

For z ∈ supp µα x,y we may substitute cos θ :=

x2 + y 2 − z 2 with θ ∈ [0, π] . A 2|xy|

short calculation then yields that 1 − σx,y,z = he (x, y, θ), σz,x,y + σz,y,x = ho (x, y, θ), from which the assertion follows.

3.3. Lemma. Then µ = 0.

Suppose µ ∈ M b (IR) with

R

IR

Ψα λ (x) dµ(x) = 0 for all λ ∈ IR.

Proof. As the subspace M0 := {µ ∈ M b (IR) : µ({0}) = 0} is σ(M b (IR), Cb (IR)) dense in M b (IR) we may suppose that µ ∈ M0 . Further, we have the identities α α Ψα λ (x) + Ψ−λ (x) = 2 ϕλ (x),

α+1 α Ψα (x). λ (x) − Ψ−λ (x) = 2iCα λx ϕλ

By assumption of the lemma, they lead to the following two relations: Z ∞ Z α − ϕλ (x) d(µ + µ )(x) = ϕα λ (x) dµ(x) = 0 for all λ ∈ IR, 0 IR Z ∞ Z α+1 − ϕλ (x) x d(µ − µ )(x) = ϕα+1 (x) x dµ(x) = 0 for all λ ∈ IR \ {0}, λ 0

IR

and a continuity argument shows that the second relation also holds for λ = 0. So by injectivity of the Fourier transform on Bessel-Kingman hypergroups it follows that µ + µ− = 0 and x(µ − µ− ) = 0 , which implies µ = 0. α α Proof of Theorem 3.1. It is clear that the µα x,y are real with µx,y = µy,x . So the first thing to show is that for f ∈ Cb (IR) the mapping (x, y) 7→ µα x,y (f ) 2 α is continuous on IR . This yields τ∗ -continuity of (x, y) 7→ µx,y , and also that T x f ∈ Cb (IR) for f ∈ Cb (IR), by norm-boundedness of the µα x,y . The basis of our proof is (3.1), and we may restrict to even and odd test functions.

(a) If f ∈ Cb (IR) is even, then (3.1) becomes Z Z π p α f dµx,y = Mα f ( x2 + y 2 − 2|xy| cos θ) sin2α θ dθ IR 0 Z π p − sgn(xy) · Mα f ( x2 + y 2 − 2|xy| cos θ) cos θ sin2α θ dθ. 0

Both integrals are continuous functions in (x, y) ∈ IR2 , and for xy = 0, where sgn(xy) is discontinuous, the latter takes the value 0. But as sgn(xy) is bounded, 2 it follows that (x, y) 7→ µα x,y (f ) is continuous on IR . (b) For odd f the statement is clear other than along the lines |x| = |y| and xy = 0. If |x| = |y|, then the weight ho defined in Lemma 3.2 is continuous p in (x, y, θ) unless x2 + y 2 − 2|xy| cos θ = 0. But as f (0) = 0, and as |ho | is bounded by 2 according to the boundedness properties of σ, the product p f ( x2 + y 2 − 2|xy| cos θ) · ho (x, y, θ) becomes continuous in each (x, y, θ) with |x| = |y|. Hence (x, y) 7→ µα x,y (f ) is also continuous on the diagonals. To handle the remaining case xy = 0, |x| 6= |y|, consider the function p f ( x2 + y 2 − 2|xy| cos θ) . (x, y, θ) 7→ (x + y) p x2 + y 2 − 2|xy| cos θ

It is continuous in every (x, y, θ) with |x| 6= |y|. So by the same argument as used in (a), it is now easily seen that (x, y) 7→ µα x,y (f ) is continuous in all (x, y) with xy = 0, |x| 6= |y|. 2 Summing up, we have shown that (x, y) 7→ µα x,y (f ) is continuous on IR for each

f ∈ Cb (IR). Concerning axiom (A1), it remains to note that for f ∈ Cc (IR) with supp f ⊆ [−M, M ] the support of T x f is contained in [−|x| − M, M + |x|]. To prove associativity of the canonical continuation ∗ of ωα it suffices to consider point measures. But by definition of ωα ,

α α α α εx ∗ (εy ∗ εz ) (Ψα λ ) = Ψλ (x)Ψλ (y)Ψλ (z) = (εx ∗ εy ) ∗ εz (Ψλ )

for all x, y, z ∈ IR and λ ∈ IR. Lemma 3.3 now yields the assertion. (A4) is obvious; finally, (A5) and (A6) are immediate consequences of the symmetry relations Kα (−x, −y, −z) = Kα (x, y, z) and Kα (x, y, z) = Kα (−x, z, y) respectively. 3.4. Remark. The mapping (x, y) 7→ supp µα x,y is not continuous with respect to the Michael topology on the space C(IR) of compact subsets of IR. This is α because supp µα x,y ∩ {ξ ∈ IR : ξ < 0} 6= ∅ for all x, y > 0, while supp µx,0 = {x}. The characterization of multiplicative functions on our Bessel-type signed hypergroups can be accomplished by a reduction to the well-known corresponding results for Bessel-Kingman hypergroups. It leads to results which extend the group case in a natural way. cα , let X (Xα ) and Xα∗ denote In addition to the character spaces Xb (Xα ) and X the spaces of continuous multiplicative functions and semi-characters respectively: X (Xα ) := {ϕ ∈ C(X) : ϕ 6≡ 0, µα x,y (ϕ) = ϕ(x)ϕ(y) for all x, y ∈ IR}, Xα∗ := {ϕ ∈ X (Xα ) : ϕ(−x) = ϕ(x) for all x ∈ IR}.

3.5. Theorem. (a) X (Xα ) = {Ψα λ : λ ∈ C }. cα = {Ψα : λ ∈ IR}, (b) Xα∗ = Xb (Xα ) = X λ cα , λ 7→ Ψα , is a homeomorphism. and the mapping IR → X λ

Proof. By definition of the convolution on Xα it is clear that each Ψα λ, λ ∈ C 1 belongs to X (Xα ) . Now take ϕ ∈ X (Xα ) and define F (x) := 2 (ϕ(x) + ϕ(−x)). A short computation shows that for x, y > 0, Z Z ∞ 1 2α+1 F (x)F (y) = ϕ(z) kα (x, y, |z|)|z| dz = F (z) kα (x, y, z) z 2α+1 dz. 2 IR 0 Thus F is a multiplicative function on the Bessel-Kingman hypergroup Hα . According to Zeuner [13], it follows that F = ϕα λ for some λ ∈ C .

Now set G(x) := 12 (ϕ(x) − ϕ(−x)). Then for x, y > 0, Z 1 G(x)G(y) = − ϕ(z) kα (x, y, |z|)σx,y,z |z|2α+1 dz 2 IR Z ∞ 2α+1 = − ϕα dz. λ (z) kα (x, y, z)σx,y,z z 0

α But at the same time, this last expression is equal to Φα λ (x)Φλ (y) where
1 α α Φα λ (x) := 2 Ψλ (x) − Ψλ (−x) . Hence we have α G(x)G(y) = Φα λ (x)Φλ (y)

for all x, y > 0,

α α from which it follows that G = Φα λ or G = −Φλ = Φ−λ . We have shown that α ϕ = Ψα λ or ϕ = Ψ−λ , and thus the proof of (a) is complete.

For the proof of (b), note that by Corollary 2.2 it is clear that {Ψα λ : λ ∈ IR} ⊆ cα . On the other hand, if Ψα ∈ Xα∗ then X λ α+1 α+1 ϕα (x) = ϕα (x) λ (x) − iCα λx ϕλ λ (x) − iCα x λϕλ

for all x ∈ IR, and by comparison of even and odd parts in x we obtain that α+1 both ϕα (x) must be real for all x ∈ IR. But this is only possible λ (x) and λϕλ if λ is real. The final statement is standard; we omit its proof. cα and IR as above the Plancherel 3.6. Proposition. With the identification of X measure of the signed hypergroup Xα is given by dπα (λ) = dmα (λ) =

1 |λ|2α+1 dλ. α+1 2 Γ(α + 1)

Proof. Again we reduce our discussion to the associated Bessel-Kingman hypergroup Hα . For this, denote the Fourier transform on Hα by F, and let de πα (λ) = dm e α (λ) =

1

2α Γ(α

+ 1)

λ2α+1 dλ

(λ ≥ 0)

denote the Plancherel- and Haar meaure on Hα . Then, for all even functions f ∈ Cc (IR) and λ ∈ IR, Z Z ∞ α fb(λ) = Ψλ (x) f (x)dmα (x) = 2 ϕα (3.2) λ (x)f (x)dmα (x) = F(f )(|λ|). IR

0

Thus the Plancherel formula for Hα yields that for even f ∈ Cc (IR), Z Z ∞ Z ∞ Z 2 2 2 |fb| dmα = |F(f )| de πα = |f | dm eα = |f |2 dmα . IR

0

0

IR

Hence, by the Plancherel formula for the Bessel-type signed hypergroup Xα , we find that Z Z 2 |fb| dπα = |fb|2 dmα for all even f ∈ Cc (IR). (3.3) IR

IR

To complete the proof we have to show that (3.3) implies πα = mα . For this first notice that just as for usual hypergroups, the Plancherel measure of Xα must be symmetric, i.e. dπα (−λ) = dπα (λ). Therefore it is sufficient to prove R R g dmα for all g ∈ C0e,+ (IR) := {h ∈ C0 (IR) : h ≥ 0 and even }. g dπ = α IR IR In view of (3.3) this follows if we have shown that {|fb|2 : f ∈ Cc (IR) even } is

k.k∞ -dense in C0e,+ (IR). So take g ∈ C0e,+ (IR) and ε > 0. Then there exists √ f ∈ Cc (IR) with kfb − gk∞ < ε. Now write f = fe + fo with fe even and fo odd. Then by (3.2), fbe is even, and a similar calculation shows that fbo is odd. It follows that √ √ √ kfbe − gk∞ ≤ k(fbe − g) + fbo k∞ = kfb − gk∞ < ε, which immediately takes care of the claim.

3.7. Remarks. b and Plan1. Let (X, m, ω) be a commutative signed hypergroup with dual X b there exists a measure ̺α,β ∈ cherel measure π. Assume that for all α, β ∈ X, b b MIR (X) such that Z γ(x) d̺α,β (γ) for all x ∈ X. α(x)β(x) = b X b carries a dual signed hypergroup structure if (X, b π, Ω), with Then we say that X Ω(α, β) := ̺α,β , is a commutative signed hypergroup. In our case, α Ψα λ (x) = Ψx (λ)

for all λ, x ∈ IR.

bα carries a commutative signed Together with Proposition 3.6 this shows that X hypergroup structure which is isomorphic to Xα in the obvious way. 2. The Fourier transform on the Bessel-type signed hypergroup Xα is exactly the Dunkl transform with parameter α + 21 , associated with the reflection group ZZ2 on IR. This is a special case of a generalized Hankel transform introduced by Dunkl [3, 4] in connection with finite reflection groups on IRN , N ≥ 1. The Dunkl transform involves an integral kernel, which in the special case of ZZ2 and IR is just given by Kα (x, −iy) = Γ α+


α− 1 1
 |xy| −α+ 21 Jα− 1 (|xy|)−i sgn(xy)Jα+ 1 (|xy|) = Ψ−y 2 (x), 2 2 2 2

with parameter α > 0 (see Dunkl [4], Section 4). For a further study of the Dunkl transform we also refer to de Jeu [5].

References [1] [2] [3] [4]

[5] [6] [7] [8]

[9] [10]

[11] [12] [13]

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