PSEUDOMEASURE CHARACTER OF THE ULTRASPHERICAL SEMIGROUPS

H. EMAMIRAD^{†} AND G.S. HESHMATI^{††}

†Laboratoire de Modélisation Mécanique et de Mathématiques Appliquées. Université de Poitiers, Teleport 2 - Boulevard Marie et Pierre Curie BP 30179 86962 FUTUROSCOPE CHASSENEUIL CEDEX / France. emamirad@l3ma.univ-poitiers.fr ††Faculty of Engineering, Tehran University, P.O.Box 11365-4563 Tehran, IRAN heshmati@chamran.ut.ac.ir

Abstract. We prove that the diffusion semigroups generated by the second order differential ultraspherical (Gegenbauer) operator are pseudomeasure operators.

1. Introduction

The diffusion semigroups have been broadly studied from the functional analysis point of view (analyticity, positivity, Markov property, etc...) but much less so from the perspective of harmonic analysis. In this paper we confine our attention to the pseudomeasure character of diffusion semigroups, especially those which are naturally associated with a system of orthogonal polynomials. It is well-known that some orthogonal polynomials satisfy a second order differential equation of the form Lu + λ_{n}u = 0. In other words, the orthogonal polynomials are the eigenfunctions of the operator L for the eigenvalues λ_{n} in the appopriate Hilbert space. In particular for the Jacobi polynomials P_{n}^{α,β}(x), L := (1 − x^{2})\frac{d^{2}}{dx^{2}} + [(α − β) − (α + β + 2)x]\frac{d}{dx} and λ_{n} = n(n + α + β + 1) in the Hilbert space L^{2}([−1, 1], (1 − x)^{α}(1 + x)^{β}dx). Throughout this paper, for the sake of simplicity, we take α = β, which corresponds to the ultraspherical ( or Gegenbauer ) polynomials Q_{n}^{α}, but the same analysis can be readily generalized to the Jacobi polynomials. In this case, for any f ∈ L^{2}(I, μ_{α}) (I := [−1, 1] and dμ_{α} := (1 − x^{2})^{α}dx), the diffusion semigroup T_{α}(t) generated by L := (1 − x^{2})\frac{d^{2}}{dx^{2}} − 2(α + 1)x\frac{d}{dx} can be expressed by

01991 Mathematics Subject Classification. Primary 47D07, 33A45; Secondary 43Axx

0Keywords: Ultraspherical semigroups and polynomials, Gegenbauer kernel, Generalized convolution, Pseudomeasures.

0The authors research was supported by Tehran University under contract 611/1/474.

T_{α}(t)f = ∑_{n⩾0}e^{−λ_{n}t}f̂(n)Q_{n}^{α},  (1.1)

where

f = ∑_{n⩾0} f̂(n)Q_{n}^{α}  (1.2)

and f̂(n) is the ultraspherical Fourier coefficient of f given by

f̂(n) = ω_{n}^{α} ∫_{I} fQ_{n}^{α}dμ_{α}  (1.3)

( the constant ω_{n}^{α} is defined in (2.1)).

Now let us introduce the pseudomeasure character of a bounded linear operator. We denote by FL^{p} := FL^{p}(ℝ^{n}) the space of distributions on ℝ^{n} which are Fourier transforms of functions in L^{p}(ℝ^{n}). It is well-known that if f ∈ L^{1}(ℝ^{n}) then its Fourier transform f̂ is a continuous function vanishing at infinity. Hence FL^{1} ⊂ C_{0}(ℝ^{n}). Since the dual of C_{0}(ℝ^{n}) is M(ℝ^{n}), the space of the Radon measures on ℝ^{n}. The dual of FL^{1} (which can be identified with FL^{∞}) is referred to as the space of pseudomeasures, ( See [11] for more informations on this space). It is clear that M(ℝ^{n}) is a relatively small part of FL^{∞}; for instance, if f ∈ L^{∞} is not uniformly continuous, f̂ cannot be a measure.

The same formalism is carried out for Fourier series or orthogonal systems (see [10] or [3] for details). Let 𝔞_{α} be the space of the continuous functions on I := [−1, 1] of the form (1.2) such that ∥f∥_{α} = ∑_{n⩾0} |f̂(n)| < ∞. By a pseudomeasure is meant a distribution Φ ∈ C^{∞}(I)′ such that Φ̂ is a bounded function on ℕ. As in the Fourier transform case, we can identify the space of pseudomeasures with 𝔞′_{α}, the dual of 𝔞_{α}. We will see in Lemma 2.3 that 𝔞_{α} equipped with the norm ∥ · ∥_{α} is a commutative, semisimple Banach algebra for the simple product of functions. This topological property labels a part of the bounded linear operators Π_{α} ⊂ L(L^{2}(I, μ_{α})). In order to define Π_{α} we need to introduce the generalized convolution product and the corresponding hypergroup. In an appendix section to this paper we will introduce an intuitive representation of the hypergroups through the ultraspherical harmonics which affects particularly our framework. This convolution product is defined by

(f ∗_{α} g)(x) = ∫_{I} ∫_{I} G_{α}(x, y, z)f(y)g(z)dμ_{α}(y)dμ_{α}(z),  (1.4)

where the Gegenbauer kernel G_{α}(x, y, z) will be defined in (2.4).

The resulting convolution will have sufficient resemblance with the classical convolution properties which will be indicated in the next section; by means of this kernel one can construct the integral operator

[P_{α,x}f](y) = ∫_{I} G_{α}(x, y, z)f(z)dμ_{α}(z),  for all f ∈ L^{1}(I, μ_{α}).  (1.5)

This operator possesses many relevant properties. We have gathered some of these properties in an appendix section to this paper.

Let Π_{α} be the subspace of the Banach algebra L(L^{2}(I, μ_{α})) such that for any T ∈ Π_{α} we have

T (g ∗_{α} h) = (T g) ∗_{α} h ∀ g, h ∈ L^{2}(I, μ_{α}).  (1.6)

The elements of Π_{α} are called pseudomeasure operators; the following Theorem justifies this designation.

Theorem 1.1. For any T ∈ Π_{α} there exists a unique pseudomeasure Φ_{T} ∈ 𝔞′_{α} such that

Φ_{T} (g ∗_{α} h) = ⟨T g, h⟩,  ∀g, h ∈ L^{2}(I, μ_{α}),  (1.7)

and this correspondence is an isometric isomorphism from Π_{α} onto 𝔞′_{α}.

What is surprising in this result is that a purely algebraic relation like (1.6) on T implies its topological character. In Section 3 we prove the above Theorem and show that the ultraspherical semigroup T_{α}(t) satisfies the relation (1.6).

2. Ultraspherical convolution

Let

Q_{n}^{α}(x) = \frac{(−1)^{n}}{2^{n}(α + 1)...(α + n)}(1 − x^{2})^{−α} \frac{d^{n}}{dx^{n}} (1 − x^{2})^{α+n}

be the n^{th} order ultraspherical polynomial of index α⟩ − 1. The set {Q_{n}^{α}(x)} forms an orthogonal system on I for the measure dμ_{α}(x) = (1 − x^{2})^{α}dx ( cf. [4]), with the relation

∫_{I} Q_{m}^{α}(x)Q_{n}^{α}(x)dμ_{α}(x) = \begin{cases}0 & if n ≠ m \\ (ω_{n}^{α})^{−1} & if n = m, \end{cases}

where

ω_{n}^{α} = \frac{(2n + 2α + 1)Γ(n + 2α + 1)}{2^{2α+1}Γ(α + 1)^{2}Γ(n + 1)}.  (2.1)

The well-known properties of these polynomials are:

Q_{n}^{α}(1) = 1; (the normalization condition)

Q_{0}^{α}(x) = 1,  Q_{1}^{α}(x) = x;  (2.2)

Q_{n}^{α}(−x) = (−1)^{n}Q_{n}^{α}(x);
|Q_{n}^{α}(x)| < 1 if α > −\frac{1}{2}, |x| < 1 and n ⩾ 1;

Q_{1}^{α}(x)Q_{n}^{α}(x) = \frac{n}{2n + 2α + 1}Q_{n−1}^{α}(x) + \frac{n + 2α + 1}{2n + 2α + 1}Q_{n+1}^{α}(x). (n ⩾ 1) (2.3)

For α > −\frac{1}{2}, the kernel

G_{α}(x, y, z) = \frac{Γ(α + 1)}{Γ(\frac{1}{2})Γ(α + \frac{1}{2})} \frac{(1 − x^{2} − y^{2} − z^{2} + 2xyz)_{+}^{α− \frac{1}{2}}}{(1 − z^{2})^{α}(1 − x^{2})^{α}(1 − y^{2})^{α}} ,  (2.4)

where by f_{+} we designate the positive part of the function f, defines the Gegenbauer multiplication formula (cf. [5, 6] ):

Q_{n}^{α}(x)Q_{n}^{α}(y) = ∫_{I} G_{α}(x, y, z)Q_{n}^{α}(z)dμ_{α}(z),  (−1 < x, y < 1)  (2.5)

For any f, g ∈ L^{1}(I, μ_{α}), we can define the generalized convolution product of f by g by the formula (1.4)

In the sequel, all the functions f ∈ L^{p}(I, μ_{α}) are real valued and the norm of f is defined by

∥f∥_{p} := (∫_{I} |f(x)|^{p}dμ_{α}(x))^{1/p} .

For p = 2 the inner product of L^{2}(I, μ_{α}) is designated by

⟨f, g⟩ := ∫_{I} f(x)g(x)dμ_{α}(x).

In [5] and [6], G. Gasper discusses this construction not only for the ultraspherical polynomials P_{n}, but also for the Jacobi polynomials, where he shows the following Lemma.

Lemma 2.1. (see [5, 6]) (a) L^{1}(I, μ_{α}) endowed with the convolution product ∗_{α} is a semisimple commutative Banach algebra.

(b) For 1 ⩽ p, q, r ⩽ ∞ and \frac{1}{r} = \frac{1}{p} + \frac{1}{q} − 1, if f ∈ L^{p}(I, μ_{α}) and g ∈ L^{q}(I, μ_{α}), then f ∗_{α} g ∈ L^{r}(I, μ_{α}), and one has ∥f ∗_{α} g∥_{r} ⩽ ∥f∥_{p}∥g∥_{q}.

It is important to note ∗_{α} is a distinct convolution for each α ⩾ −\frac{1}{2}. For example, for α = 0 we deal with spherical harmonics Q_{n}^{0} which are exactly the Legendre polynomials.

Lemma 2.2. Suppose f, g ∈ L^{1}(I, μ_{α}), then

{ \atop f ∗_{α} g}(n) = (ω_{n}^{α})^{−1}f̂(n)ĝ(n).

Proof. This follows from

{ \atop f ∗_{α} g}(n) = ω_{n}^{α} ∫_{I} ∫_{I} f(x)g(y)dμ_{α}(x)dμ_{α}(y) ∫_{I} G_{α}(x, y, z)Q_{n}^{α}(z)dμ_{α}(z),

and the Gegenbauer multiplication formula (2.5). □ Let 𝔞_{α} be the algebra of the continuous functions on I of the form

f = ∑_{n⩾0} f̂(n)Q_{n}^{α}

such that ∥f∥_{α} = ∑_{n⩾0} |f̂(n)| < ∞.

Lemma 2.3. 𝔞_{α} equipped with the norm ∥ · ∥_{α} is a commutative, semisimple Banach algebra for the simple product of functions.

Proof. It suffices to prove that

∥fg∥_{α} ⩽ ∥f∥_{α}∥g∥_{α}.  (2.6)

From the relation (2.3) we can infer that

Q_{m}^{α}Q_{n}^{α} = ∑_{r} h_{m,n,r}^{α}Q_{r}^{α},  (2.7)

where r runs over a set of integers such that h_{m,n,r}^{α} ≠ 0. That is, |n − m| ⩽ r ⩽ n + m and m + n + r is even. Computation of (2.7) at the point x = 1 yields that

∑_{r} h_{n,m,r}^{α} = 1.  (2.8)

This implies that for f = ∑_{n⩾0} f̂(n)Q_{n}^{α}, g = ∑_{n⩾0} ĝ(n)Q_{n}^{α} and from orthogonality between Q_{r}^{α} and Q_{k}^{α} we get

f̂g(k) = ω_{k}^{α} ∫_{I} ∑_{n⩾0} ∑_{m⩾0} f̂(n)Q_{n}^{α}(x)ĝ(m)Q_{m}^{α}(x)Q_{k}^{α}(x)dμ_{α}(x)

= ω_{k}^{α} ∑_{n⩾0} ∑_{m⩾0} f̂(n)ĝ(m) ∫_{I} ∑_{r} h_{n,m,r}^{α}Q_{r}^{α}(x)Q_{k}^{α}(x)dμ_{α}(x)

= ∑_{n⩾0} ∑_{m⩾0}h_{n,m,k}^{α}f̂(n)ĝ(m).

Finally from (2.8) it follows that

∥fg∥_{α} = ∑_{k⩾0} |f̂g(k)| = | ∑_{n⩾0} ∑_{m⩾0} f̂(n)ĝ(m) (∑_{k⩾0}h_{n,m,k}^{α}) |

⩽ (∑_{n⩾0} |f̂(n)|) (∑_{m⩾0} |ĝ(n)|) ,

which implies (2.6). □

3. The proof of Theorem 1.1

For the proof of Theorem 1.1 we need the following Lemmas.

Lemma 3.1. For any f ∈ 𝔞_{α} there exist g, h ∈ L^{2}(I, μ_{α}) such that f = g ∗_{α} h and such that

∥f∥_{α} = ∥g∥_{2}^{2} = ∥h∥_{2}^{2}.  (3.1)

Conversely if g, h ∈ L^{2}(I, μ_{α}) then g ∗_{α} h ∈ 𝔞_{α} and

∥g ∗_{α} h∥_{α} ⩽ ∥g∥_{2}∥h∥_{2}.  (3.2)

Proof. For any f ∈ 𝔞_{α} we define ĝ(n) = \sqrt{ω_{n}^{α}|f̂(n)|},

ĥ(n) = \begin{cases}0 & if f̂(n) = 0 \\ \sqrt{ω_{n}^{α}} \frac{f̂(n)}{\sqrt{|f̂(n)|}} & if f̂(n) ≠ 0, \end{cases}

g = ∑_{n⩾0} ĝ(n)Q_{n}^{α} and h = ∑_{n⩾0} ĥ(n)Q_{n}^{α}.

From Lemma 2.2, { \atop g ∗_{α} h}(n) = (ω_{n}^{α})^{−1}ĝ(n)ĥ(n) = f̂(n). Since \{\sqrt{ω_{n}^{α}}Q_{n}^{α}\}_{n⩾0} is a complete orthonormal system in L^{2}(I, μ_{α}) and ∑_{n⩾0} |f̂(n)| < ∞, one finds that f = g ∗_{α} h, and (3.1) follows from Parseval’s identity.

Conversely for g, h ∈ L^{2}(I, μ_{α}), by the Cauchy–Schwarz inequality we have

∑_{n⩾0} |{ \atop g ∗_{α} h}(n)| = ∑_{n⩾0} |\frac{1}{ω_{n}^{α}} ĝ(n)ĥ(n)|

⩽ (∑_{n⩾0} \frac{1}{ω_{n}^{α}} |ĝ(n)|^{2})^{1/2} (∑_{n⩾0} \frac{1}{ω_{n}^{α}} |ĥ(n)|^{2})^{1/2} ,

which implies (3.2). □

Lemma 3.2. The injection 𝔞_{α} into L^{1}(I, μ_{α}) is continuous for any p (1 ⩽ p < +∞), and 𝔞_{α} is dense in L^{p}(I, μ_{α}).

Proof. Suppose f = ∑_{n⩾0} a_{n}Q_{n}^{α} with ∑_{n⩾0} |a_{n}| < +∞; since for |x| ⩽ 1, |Q_{n}^{α}(x)| ⩽ 1,

∥f∥_{1} ⩽ ∑_{n⩾0}|a_{n}| ∫_{I} |Q_{n}^{α}(x)|dμ_{α}(x) ⩽ \frac{1}{ω_{0}^{α}} ∑_{n⩾0}|a_{n}| = \frac{1}{ω_{0}^{α}} ∥f∥_{α}.  (3.3)

The density of 𝔞_{α} in L^{p}(I, μ_{α}) follows from the fact that C^{∞}(I) ⊂ 𝔞_{α}. □

Lemma 3.3. For any g, h ∈ L^{2}(I, μ_{α}),

[g ∗_{α} h](1) = ⟨g, h⟩.  (3.4)

Proof. Since

[g ∗_{α} h](1) = ∫_{I} ( ∫_{I} G_{α}(1, y, z)h(z)dμ_{α}(z))g(y)dμ_{α}(y),

it suffices to prove that G_{α}(1, y, z) is a reproducing kernel on L^{2}(I, μ_{α}), i.e.

∫_{I} G_{α}(1, y, z)h(z)dμ_{α} = h(y), for all h ∈ L^{2}(I, μ_{α}).

This identity will be proved in Section 5(Appendix). □

Proof of Theorem 1.1. Let T ∈ Π_{α} and f ∈ 𝔞_{α}. From Lemma 3.1 it follows that there exist g, h ∈ L^{2}(I, μ_{α}) such that f = g ∗_{α} h and so T f = T g ∗_{α} h ∈ 𝔞_{α}. Consequently T f is a continuous function on [−1, 1]. Thus, one can define the linear form Φ_{T} by Φ_{T} (f) = [T f](1). The continuity of Φ_{T} on 𝔞_{α} follows from Lemma 3.1 and Lemma 3.3. Actually, we note that Φ_{T} satisfies (1.7) and

|Φ_{T} (f)| = |Φ_{T} (g ∗_{α} h)| = |[(T g) ∗_{α} h](1)| = |⟨h, T g⟩|
⩽ ∥T g∥_{2}∥h∥_{2} ⩽ ∥T ∥_{Π_{α}}∥g∥_{2}∥h∥_{2} = ∥T ∥_{Π_{α}}∥f∥_{α}.

Hence Φ_{T} ∈ 𝔞′_{α} and we have

∥Φ_{T} ∥𝔞′_{α} ⩽ ∥T ∥_{Π_{α}}.  (3.5)

Conversely, if Φ ∈ 𝔞′_{α}, then for any g, h ∈ L^{2}(I, μ_{α}), one has

|Φ(g ∗_{α} h)| ⩽ ∥Φ∥𝔞′_{α}∥g ∗_{α} h∥_{α} ⩽ ∥Φ∥𝔞′_{α}∥g∥_{2}∥h∥_{2}.

Thus, (g, h) ↦ Φ(g ∗_{α} h) is a bilinear form on L^{2}(I, μ_{α}), which implies the existence of T_{Φ} ∈ L(L^{2}(I, μ_{α}) such that Φ(g ∗_{α} h) = ⟨h, T_{Φ}g⟩ and

∥T_{Φ}∥ ⩽ ∥Φ∥𝔞′_{α}.  (3.6)

Furthermore T_{Φ} ∈ Π_{α}. In fact, it is not hard to see that, for any f, g and h ∈ L^{2}(I, μ_{α}) we have ⟨f, g ∗_{α} h⟩ = ⟨f ∗_{α} g, h⟩ and consequently

⟨h, T_{Φ}(f ∗_{α} g)⟩ = Φ(h ∗_{α} (f ∗_{α} g)) = Φ((h ∗_{α} f) ∗_{α} g)
= ⟨h ∗_{α} g, T_{Φ}f⟩ = ⟨h, (T_{Φ}f) ∗_{α} g⟩,

which implies that T_{Φ} commutes with the generalized convolution products from the left.

The fact that

Φ_{T_{Φ}}(g ∗_{α} h) = [(T_{Φ}g) ∗_{α} h](1) = ⟨T_{Φ}g, h⟩ = Φ(g ∗_{α} h)

for any g, h ∈ L^{2}(I, μ_{α}) implies that Φ_{T_{Φ}} = Φ. Hence by (3.5) and (3.6) the mapping T ↦ Φ_{T} is an isometric isomorphism.

The following Corollary will be useful in the sequel.

Corollary 3.1. Suppose T ∈ Π_{α}. For any f ∈ 𝔞_{α}, T f ∈ 𝔞_{α} and

T̂ f(n) = Φ_{T} (Q_{n}^{α})f̂(n).

Proof. In the proof of Theorem 1.1, we have shown that if f ∈ 𝔞_{α}, then T f ∈ 𝔞_{α}. From (1.4) and (2.5) it follows that

(Q_{n}^{α} ∗_{α} Q_{m}^{α})(x) = ∫_{I} ∫_{I} G_{α}(x, y, z)Q_{n}^{α}(y)Q_{m}^{α}(z)dμ_{α}(y)dμ_{α}(z)

= ∫_{I} Q_{m}^{α}(x)Q_{m}^{α}(y)Q_{n}^{α}(y)dμ_{α}(y)

= \begin{cases}0 & if n ≠ m \\ (ω_{n}^{α})^{−1}Q_{n}^{α}(x) & if n = m. \end{cases}

Hence, from (1.7)

⟨T (Q_{n}^{α}), Q_{m}^{α}⟩ = Φ_{T} (Q_{n}^{α} ∗_{α} Q_{m}^{α}) = \begin{cases}0 & if n ≠ m \\ (ω_{n}^{α})^{−1}Φ_{T} (Q_{n}^{α}) & if n = m, \end{cases}

and consequently T (Q_{n}^{α}) = Φ_{T} (Q_{n}^{α})Q_{n}^{α}. This shows that

T̂ f(n) = ω_{n}^{α} ∫_{I}(T f)Q_{n}^{α}dμ_{α} = ω_{n}^{α} ∫_{I} ∑_{m⩾0} f̂(m)Φ_{T} (Q_{m}^{α})Q_{m}^{α}Q_{n}^{α}dμ_{α} = Φ_{T} (Q_{n}^{α})f̂(n).

□

4. Ultraspherical semigroups

The ultracontractivity and the Sobolev inequality for ultraspherical semigroups are already discussed in [13] and [1]. It is proved in [13] that T_{α}(t) is a positive semigroup and furthermore we have

Proposition 4.1. (see [13, Proposition 1.17]) (a) {T_{α}(t)}_{t⩾0} is a C_{0}–semigroup on L^{p}(I, μ_{α}) (1 ⩽ p < ∞) with norm 1;

(b) If f ⩾ 0 in L^{1}(I, μ_{α}), then T_{α}(t)f ⩾ 0;

(c) If f ∈ L^{1}(I, μ_{α}), then |T_{α}(t)f| ⩾ T_{α}(t)|f|;

(d) If H_{α} is the generator of T_{α}(t) with domain D(H_{α}) on L^{p}(I, μ_{α}) (1 ⩽ p < ∞), then C^{2}(I) ⊂ D(H_{α}).

From this Proposition it follows that the family {T_{α}(t)}_{t⩾0} belongs to L(L^{2}(I, μ_{α})). In the following theorem we prove that in fact it belongs to the subset Π_{α} of L(L^{2}(I, μ_{α})).

Theorem 4.1. Any ultraspherical semigroup T_{α}(t) forms a family of pseudomeasure operators of L(L^{2}(I, μ_{α})).

Proof. Since all the eigenvalues of the operator L := (1 − x^{2})\frac{d^{2}}{dx^{2}} − 2(α + 1)x\frac{d}{dx} are positive, the expression (1.1) implies that ∥T_{α}(t)f∥_{α} ⩽ ∥f∥_{α}. Hence for the pseudomeasure character of T_{α}(t) it suffices to prove that

T_{α}(t)[f ∗_{α} g] = [T_{α}(t)f] ∗_{α} g for all f, g ∈ L^{2}(I, μ_{α}).  (4.1)

For this purpose we shall prove in the following lemma a more general result which implies readily (4.1). □

Lemma 4.1. For any {β_{n}} ∈ ℓ^{∞}(ℕ), the operator T defined by

T f = ∑_{n⩾0}β_{n}f̂(n)Q_{n}^{α},

where f = ∑_{n⩾0} f̂(n)Q_{n}^{α}, belongs to Π_{α}. Conversely, if T ∈ Π_{α}, then there exists a unique sequence {β_{n}} ∈ ℓ^{∞}(ℕ) such that T f = ∑_{n⩾0} β_{n}f̂(n)Q_{n}^{α}, for any f = ∑_{n⩾0} f̂(n)Q_{n}^{α}, and we have ∥T ∥_{Π_{α}} = ∥{β_{n}}∥_{∞}.

Proof. It suffices to prove that T satisfies

T [f ∗_{α} g] = [T f] ∗_{α} g for all f, g ∈ L^{2}(I, μ_{α}).

From the definition of ∗_{α} it follows that

T [f ∗_{α} g](x) = ∑_{n⩾0}β_{n}{ \atop f ∗_{α} g}(n)Q_{n}^{α}(x)

= ∑_{n⩾0}β_{n}(ω_{n}^{α})^{−1}f̂(n)ĝ(n)Q_{n}^{α}(x)

= ∑_{n⩾0}β_{n}f̂(n)Q_{n}^{α}(x) ∫_{I} Q_{n}^{α}(z)g(z)dμ_{α}(z)

= ∑_{n⩾0}β_{n}f̂(n) ∫∫ _{I×I} G_{α}(x, y, z)Q_{n}^{α}(y)g(z)dμ_{α}(z)dμ_{α}(y)

(by (2.5))

= [∑_{n⩾0}β_{n}f̂(n)Q_{n}^{α}] ∗_{α} g(x)

= [T f] ∗_{α} g(x).

Conversely, let T ∈ Π_{α} and Φ_{T} ∈ 𝔞′_{α} be the corresponding pseudomeasure given by Theorem 1.1. By designating β_{n} := Φ_{T} (Q_{n}^{α}) for any f = ∑_{n⩾0} f̂(n)Q_{n}^{α} ∈ 𝔞_{α}, we have Φ_{T} (f) = ∑_{n⩾0} f̂(n)Φ_{T} (Q_{n}^{α}) = ∑_{n⩾0} β_{n}f̂(n) and consequently

∥Φ_{T} ∥𝔞′_{α} = sup_{∥f∥_{𝔞_{α}}⩽1}|Φ_{T} (f)| = sup ; ∑ |f̂(n)|⩽1| ∑_{n⩾0}β_{n}f̂(n)| = sup_{n⩾0}|β_{n}|.

Furthermore it follows from Corollary 3.1 that for any f ∈ 𝔞_{α}, T f ∈ 𝔞_{α} and

T f = ∑_{n⩾0} T̂ f(n)Q_{n}^{α} = ∑_{n⩾0}β_{n}f̂(n)Q_{n}^{α}.

□

5. Appendix

In this section we introduce an intuitive representation of the hypergroups through ultraspherical harmonics and we prove some results which have been used previously.

Let G be a locally compact topological group and M(G) the space of finite Borel measures on G. Then (M(G), ⋆) is a measure algebra endowed with a generalized convolution ⋆. This convolution, or convos as it is called by Jewett [9], is defined by (μ ⋆ ν)(E) = ∫_{G} μ(Ex^{−1})dν(x) and satisfies a set of axioms called hypergroup axioms. Now, if H is not a group but can be endowed with a generalized convolution ⋆ which satisfies the hypergroup axioms then (H, ⋆) is called a hypergroup. Here we are not going to give the definitive set of the hypergroup axioms. The interested reader is directed to [2] or [9] for a precise statement of these axioms. A typical example of such a convolution is given by I. I. Hirschmann, Jr. in [7] and [8]. He describes two measure algebras (M(G), ∗_{α}) and (M(ℕ_{0}), ⋆_{α}) by means of the ultraspherical polynomials. In Section 2 we gave the precise definition of (I, ∗_{α}) and (ℕ_{0}, ⋆_{α}). This convolution ∗_{α} is easily extended to the point masses by defining:

(δ_{x}^{(α)} ∗_{α} δ_{y}^{(α)})(z) = G_{α}(x, y, z)dμ_{α}(z),
δ_{x}^{(α)} ∗_{α} δ_{1}^{(α)} = δ_{x}^{(α)}, and δ_{x}^{(α)} ∗_{α} δ_{−1}^{(α)} = δ_{−x}^{(α)},

where the Gegenbauer kernel G_{α}(x, y, z) was be defined in (2.4). By this measure we can define an integral operator

[P_{α,x}f](y) := δ_{x}^{(α)} ∗_{α} f = ∫_{I} G_{α}(x, y, z)f(z)dμ_{α}(z)

on L^{p}(I, μ_{α}). Here we cite a list of relevant properties of this operator.

A1. P_{α,x} is a contraction operator on L^{p}(I, μ_{α}), 1 ⩽ p ⩽ ∞. This follows from the fact that

{ \atop P_{α,x}f}(n) = ω_{n}^{α} ∫_{I}(P_{α,x}f)Q_{n}^{α}dμ_{α}
= ω_{n}^{α} ∫_{I} ∫_{I} G_{α}(x, y, z)f(z)Q_{n}^{α}(y)dμ_{α}(z)dμ_{α}(y)
= ω_{n}^{α} ∫_{I} Q_{n}^{α}(x)Q_{n}^{α}(z)f(z)dμ_{α}(z)
= Q_{n}^{α}(x)f̂(n).

Since |Q_{n}^{α}(x)| ⩽ 1 for any x ∈ I, for any f ∈ 𝔞_{α} we have

∥P_{α,x}f∥_{p} = ∫_{I} |∑_{n⩾0}Q_{n}^{α}(x)Q_{n}^{α}(z)f̂(n)|^{p} dμ_{α}(y) ⩽ ∥f∥_{p},

and the density of 𝔞_{α} in L^{2}(I, μ_{α}) (see Lemma 3.2) yields the conclusion.

A2. P_{α,x} is a selfadjoint operator on L^{2}(I, μ_{α}). From the symmetry of G_{α} we get

⟨P_{α,x}f, g⟩ = ⟨f, P_{α,x}g⟩,

which in turn proves the self-adjointness of P_{α,x} in L^{2}(I, μ_{α}). Furthermore the symmetry of G_{α} with respect to x, y and z, implies that

[P_{α,x}f](y) = [P_{α,y}f](x).

A3. P_{α,x} is a Markov operator on L^{1}(I, μ_{α}). In a measure space (ω, μ) an operator P ∈ L(L^{1}(μ)) is called Markov if P f ⩾ 0 and ∥P f∥_{1} = ∥f∥_{1} for all f ∈ L^{1}(μ)_{+}. From (2.4) it follows that G_{α}(x, y, z) ⩾ 0. Hence P_{α,x} is positive, and by putting n = 0 in (2.5), (2.2) and the symmetry of G_{α} with respect to x, y and z, yield that

∫_{I} G_{α}(x, y, z)dμ_{α}(y) = 1,

which implies that P_{α,x} is a Markov operator. The space L^{1}(I, μ_{α}) being generating (i.e. L^{1}(I, μ_{α}) = L^{1}(I, μ_{α})_{+} − L^{1}(I, μ_{α})_{−}) we get that ∥P_{α,x}∥ ⩽ 1.

A4. P_{α,1} is a reproducing operator on L^{2}(I, μ_{α}). It suffices to prove that

∫_{I} G_{α}(1, y, z)h(z)dμ_{α} = h(y), for all h ∈ L^{2}(I, μ_{α}).

This identity is true for all h ∈ 𝔞_{α}. Indeed, if h = ∑_{n⩾0} a_{n}Q_{n}^{α} with ∑_{n⩾0} |a_{n}| < +∞, then since Q_{n}^{α}(1) = 1,

∫_{I} G_{α}(1, y, z)h(z)dμ_{α}(z) = ∑_{n⩾0}a_{n} ∫_{I} G_{α}(1, y, z)Q_{n}^{α}(z)dμ_{α}(z)

= ∑_{n⩾0}a_{n}Q_{n}^{α}(y) = h(y).

The density of 𝔞_{α} in L^{2}(I, μ_{α}) yields the conclusion.

References

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