ELEMENTARY PROOFS OF PALEY–WIENER THEOREMS FOR THE DUNKL TRANSFORM ON THE REAL LINE

NILS BYRIAL ANDERSEN AND MARCEL DE JEU

Abstract. We give an elementary proof of the Paley–Wiener theorem for smooth functions for the Dunkl transforms on the real line, establish a similar theorem for L^{2}-functions and prove identities in the spirit of Bang for L^{p}-functions. The proofs seem to be new also in the special case of the Fourier transform.

1. Introduction and overview

The Paley–Wiener theorem for the Dunkl transform D_{k} with multiplicity k (where Re k ≥ 0) on the real line states that a smooth function f has support in the bounded interval [−R, R] if, and only if, its transform D_{k}f belongs to the space of rapidly decreasing entire functions f of exponential type R. Various proofs of this result are known, all of which use explicit formulas available in this one-dimensional setting (see Remark 6 for more details). In this paper, however, we present an alternative proof which does not use such explicit expressions, being based almost solely on the formal properties of the transform. Along the same lines, we also obtain a Paley–Wiener theorem for L^{2}-functions for k ≥ 0. The case k = 0 specializes to the Fourier transform, and to our knowledge the proofs of both Paley–Wiener theorems are new even in this case.

In addition, we establish two identities in the spirit of Bang [4, Theorem 1]. These results could be called real Paley–Wiener theorems (although terminology is not yet well-established), since they relate certain growth rates of a function on the real line to the support of its transform. The approach at this point is inspired by similar techniques in [1, 2, 3]. Our results in this direction partially overlap with [6, 7], but the new proofs are considerably simpler, as they are again based almost solely on the formal properties of the transform. We will comment on this in more detail later on, as these results will have been established. For k = 0, one retrieves Bang’s result; we feel that the present method of proof, which, e.g., does not use the Paley–Wiener theorem for smooth functions, but rather implies it, is then more direct than that in [4].

The rather unspecific and formal structure of the proofs suggests that the methods can perhaps be put to good use for other integral transform with a symmetric kernel, both for the Paley–Wiener theorems and the equalities in the spirit of Bang (cf. Remark 6). The structure of the proof is also such that, if certain combinatorial problems can be surmounted, a proof of the Paley–Wiener theorem for the Dunkl transform for invariant balanced compact convex sets in arbitrary dimension might be possible. This would be further evidence for the validity of this theorem for compact convex sets (cf. [13, Conjecture 4.1]), but at the time of writing this higher dimensional result has not been established.

This paper is organized as follows. In Section 2 the necessary notations and previous results are given. Section 3 contains the Paley–Wiener theorem for smooth functions and can—perhaps—serve as a model for a proof of such a theorem in other contexts. The rest of the paper is independent of this section. Section 4 is concerned with the real Paley–Wiener theorem in the L^{p}-case and the L^{2}-case is settled in Section 5.

2000 Mathematics Subject Classification. Primary 44A15; Secondary 42A38, 33C52.

Key words and phrases. Dunkl transform, Fourier transform, Paley–Wiener theorem.

The first author is supported by a research grant from the European Commission IHP Network: 2002–2006 Harmonic Analysis and Related Problems (Contract Number: HPRN-CT-2001-00273 - HARP).

2. Dunkl operators and the Dunkl transform on ℝ

The Dunkl operators and the Dunkl transform were introduced for arbitrary root systems by Dunkl [8, 9, 10]. In this section we recall some basic properties for the one-dimensional case of A_{1}, referring to [17, Section 1 and 2] for a comprehensive overview and to [8, 9, 10, 12, 14, 16] for details. We suppress the various explicit formulas which are known in this one-dimensional context (as these are not necessary for the proofs), thus emphasizing the basic structure of the problem which might lead to generalizations to the case of arbitrary root systems.

Let k ∈ ℂ, and consider the Dunkl operator T_{k}

T_{k}f(x) = f′(x) + k \frac{f(x) − f(−x)}{x} (f ∈ C^{∞}(ℝ), x ∈ ℝ).

Rewriting this as

(1)  T_{k}f(x) = f′(x) + k ∫_{−1}^{1} f′(tx) dt,

it follows that T_{k} maps C^{∞}(ℝ), C_{c}^{∞}(ℝ) and the Schwartz space S(ℝ) into themselves.

If Re k ≥ 0, as we will assume for the remainder of this section, then, for each λ ∈ ℂ, there exists a unique holomorphic solution ψ_{λ}^{k} : ℂ → ℂ of the differential-reflection problem

(2)  \begin{cases}T_{k}f = iλf, \\ f(0) = 1. \end{cases}

The map (z, λ) → ψ_{λ}^{k}(z) is entire on ℂ^{2}, and we have the estimate

(3)  |ψ_{λ}^{k}(z)| ≤ e^{|Im λz|}  (λ, z ∈ ℂ).

In view of (3) the Dunkl transform D_{k}f of f ∈ L^{1}(ℝ, |w_{k}(x)|dx), where the complex-valued weight function w_{k} is given by w_{k}(x) = |x|^{2k}, is meaningfully defined by

(4)  D_{k}f(λ) = \frac{1}{c_{k}} ∫_{ℝ} f(x)ψ_{−λ}^{k}(x)w_{k}(x)dx  (λ ∈ ℝ),

where

c_{k} = ∫_{ℝ} e^{− \frac{|x|^{2}}{2}} w_{k}(x) dx ≠ 0.

We note that D_{0} is the Fourier transform on ℝ. The Dunkl transform is a topological isomorphism of S(ℝ) onto itself, the inverse transform D_{k}^{−1} being given by

D_{k}^{−1}f(x) = \frac{1}{c_{k}} ∫_{ℝ} f(λ)ψ_{λ}^{k}(x)w_{k}(λ)dλ = D_{k}f(−x) (f ∈ S(ℝ), x ∈ ℝ),

The operator T_{k} is anti-symmetric with respect to the weight function w_{k}, i.e.,

(5)  ⟨T_{k}f, g⟩_{k} = −⟨f, T_{k}g⟩_{k},

for f ∈ S(ℝ) and g ∈ C^{∞}(ℝ) such that both g and T_{k}g are of at most polynomial growth. Here ⟨f, g⟩_{k} is defined by

⟨f, g⟩_{k} = ∫_{ℝ} f(x)g(x)w_{k}(x)dx,

for functions f and g such that fg ∈ L^{1}(ℝ, |w_{k}(x)|dx). In particular, (5) yields the intertwining identity

D_{k}(T_{k}f)(λ) = iλ(D_{k}f)(λ) (f ∈ S(ℝ), λ ∈ ℝ).

Furthermore, for λ, z, s ∈ ℂ the symmetry properties ψ_{λ}^{k}(z) = ψ_{z}^{k}(λ), and ψ_{sλ}^{k}(z) = ψ_{λ}^{k}(sz) are valid. Using the first of these, an application of Fubini gives

(6)  ⟨D_{k}f, g⟩_{k} = ⟨f, D_{k}g⟩_{k} (f, g ∈ L^{1}(ℝ, |w_{k}(x)|dx)).

If k ≥ 0, the Plancherel theorem states that D_{k} preserves the weighted two-norm on L^{1}(ℝ, w_{k}(x)dx)∩ L^{2}(ℝ, w_{k}(x)dx) and extends to a unitary operator on L^{2}(ℝ, w_{k}(x)dx).

3. Paley–Wiener theorem for smooth functions

The method of proof in this section is inspired by results of Bang [4]. To be more specific, if k ≥ 0, and f is a rapidly decreasing entire function of exponential type R, we will establish that

(7)  sup{|λ| : λ ∈ supp D_{k}f} ≤ lim inf_{n→∞} ∥T_{k}^{n}f∥_{∞}^{1/n} ≤ lim sup_{n→∞} ∥T_{k}^{n}f∥_{∞}^{1/n} ≤ R,

after which the proof of the Paley–Wiener theorem for smooth functions is a mere formality. Starting towards the third of these inequalities, we first use (1) to gain control over repeated Dunkl derivatives.

Lemma 1. Let k ∈ ℂ, f ∈ C^{∞}(ℝ) and n ∈ ℕ. Then

T_{k}^{n}f(x) = (T_{k}^{n−1}(f′))(x) + k ∫_{−1}^{1} t^{n−1}(T_{k}^{n−1}(f′))(tx)dt (x ∈ ℝ).

Proof. For g ∈ C^{∞}(ℝ) and m ∈ ℕ ∪ {0}, let I_{g,m} ∈ C^{∞}(ℝ) be defined by

I_{g,m}(x) = ∫_{−1}^{1} t^{m}g(tx) dt (x ∈ ℝ).

Using (1), we then find

(T_{k}I_{g,m})(x) = ∫_{−1}^{1} t_{1}^{m+1}g′(t_{1}x) dt_{1} + k ∫_{−1}^{1} ∫_{−1}^{1} t_{1}^{m+1}g′(t_{1}t_{2}x) dt_{1} dt_{2}
= ∫_{−1}^{1} t_{1}^{m+1} (g′(t_{1}x) + k ∫_{−1}^{1} g′(t_{1}t_{2}x) dt_{2}) dt_{1}
= ∫_{−1}^{1} t_{1}^{m+1}(T_{k}g)(t_{1}x) dt_{1} = I_{T_{k}g,m+1}(x).

We conclude that T_{k}I_{g,m} = I_{T_{k}g,m+1}. Since (1) can be written as T_{k}f = f′ + kI_{f′,0} one has

T_{k}^{n}f = T_{k}^{n−1}(f′ + kI_{f′,0}) = T_{k}^{n−1}(f′) + kI_{T_{k}^{n−1}(f′),n−1},

which is the statement in the lemma. □

It follows from Lemma 1 that

|T_{k}^{n}f(x)| ≤ (1 + \frac{2|k|}{n}) sup_{y∈[−|x|,|x|]} |(T^{n−1}(f′))(y)| (x ∈ ℝ),

and induction then yields the following basic estimate, which is more explicit than [6, Prop. 2.1].

Corollary 2. Let k ∈ ℂ, f ∈ C^{∞}(ℝ) and n ∈ ℕ. Then

|T_{k}^{n}f(x)| ≤ \frac{Γ(n + 1 + 2|k|)}{n! Γ(1 + 2|k|)} sup_{y∈[−|x|,|x|]} |f^{(n)}(y)| (x ∈ ℝ).

The third inequality in (7) can now be settled.

Proposition 3. Let k ∈ ℂ, and suppose f : ℂ → ℂ is an entire function of exponential type R > 0, i.e.,

|f(z)| ≤ Ce^{R|Im z|} (z ∈ ℂ),

for some positive constant C. Then, for all n ∈ ℕ, T_{k}^{n}f is bounded on the real line, and

lim sup_{n→∞} ∥T_{k}^{n}f∥_{∞}^{1/n} ≤ R.

Proof. We have, for any r > 0,

f^{(n)}(z) = \frac{n!}{2πi} ∮_{|ζ−z|=r} \frac{f(ζ)}{(ζ − z)^{n+1}} dζ (z ∈ ℂ).

If |ζ − z| = r, then

|f(ζ)| ≤ Ce^{R(|Im z|+r)},

implying

|f^{(n)}(z)| ≤ C \frac{n!}{r^{n}} e^{R(|Im z|+r)} (z ∈ ℂ).

Choosing r = n/R, so that r > 0 if n ∈ ℕ, we find

|f^{(n)}(z)| ≤ C \frac{n!e^{n}}{n^{n}} R^{n}e^{R|Im z|} (n ∈ ℕ, z ∈ ℂ),

whence ∥f^{(n)}∥_{∞} ≤ Cn!e^{n}n^{−n}R^{n}, for n ∈ ℕ. Combining this with Corollary 2 yields

∥T_{k}^{n}f∥_{∞} ≤ C \frac{e^{n} Γ(n + 1 + 2|k|)}{n^{n} Γ(1 + 2|k|)}R^{n} (n ∈ ℕ).

The result now follows from Stirling’s formula. □

As to the first inequality in (7), it is actually easy to prove that it holds for the norm ∥ · ∥_{k,p} in L^{p}(ℝ, w_{k}(x)dx) for arbitrary 1 ≤ p ≤ ∞ (not just for p = ∞), as is shown by the following lemma. It should be noted that this result can be generalized—with different proofs—to complex multiplicities (cf. Lemma 7) and to L^{p}-functions for k ≥ 0 (cf. Theorem 10), but we present it here separately nevertheless, in order to illustrate that for the case k ≥ 0, the proof of one of the crucial inequalities (as far as the Paley–Wiener theorem is concerned) is rather elementary and intuitive.

Lemma 4. Let k ≥ 0, 1 ≤ p ≤ ∞ and f ∈ S(ℝ). Then in the extended positive real numbers we have

(8)  lim inf_{n→∞} ∥T_{k}^{n}f∥_{k,p}^{1/n} ≥ sup{|λ| : λ ∈ supp D_{k}f}.

Proof. Suppose 0 ≠ λ_{0} ∈ supp D_{k}f and let 0 < ε < |λ_{0}|. Define

ϕ(λ) = \overline{D_{k}f(λ)} (λ ∈ ℝ).

Then, with q denoting the conjugate exponent and using (6), we find

∥T_{k}^{2n}f∥_{k,p} ∥D_{k}ϕ∥_{k,q} ≥ |⟨T_{k}^{2n}f, D_{k}ϕ⟩_{k}| = |⟨D_{k}(T_{k}^{2n}f), ϕ⟩_{k}|
= |∫_{ℝ}(iλ)^{2n}D_{k}f(λ)ϕ(λ)w_{k}(λ)dλ|
= ∫_{ℝ} λ^{2n}|D_{k}f(λ)|^{2}w_{k}(λ)dλ
≥ (|λ_{0}| − ε)^{2n} ∫_{|λ|≥|λ_{0}|−ε} |D_{k}f(λ)|^{2}w_{k}(λ)dλ.

With ψ(λ) = \overline{D_{k}(T_{k}f)(λ)} we similarly get

∥T_{k}^{2n+1}f∥_{k,p} ∥D_{k}ψ∥_{k,q} ≥ (|λ_{0}| − ε)^{2n+1} ∫_{|λ|≥|λ_{0}|−ε} |λ||D_{k}f(λ)|^{2}w_{k}(λ)dλ.

These two estimates together yield

lim inf_{n→∞} ∥T_{k}^{n}f∥_{k,p}^{1/n} ≥ |λ_{0}| − ε,

and the lemma follows. □

Now that (7) has been established, we come to the Paley–Wiener theorem for smooth functions. Introducing notation, for R > 0, we let C_{R}^{∞}(ℝ) denote the space of smooth functions on ℝ with support in [−R, R]. Let H_{R}(ℂ) denote the space of rapidly decreasing entire functions of exponential type R, i.e., those entire functions f with the property that, for all n ∈ ℕ∪{0}, there exists a constant C_{n} > 0 such that

|f(z)| ≤ C_{n}(1 + |z|)^{−n}e^{R|Im z|} (z ∈ ℂ).

Theorem 5 (Paley–Wiener theorem for smooth functions). Let R > 0 and k ≥ 0. Then the Dunkl transform D_{k} is a bijection from C_{R}^{∞}(ℝ) onto H_{R}(ℂ).

Proof. If f ∈ C_{R}^{∞}(ℝ), then it is easy to see that D_{k}f ∈ H_{R}(ℂ) [12, Corollary 4.10]. Now assume that f ∈ H_{R}(ℂ). Using Cauchy’s integral representation as in the proof of Proposition 3, we retrieve the well-known fact that f ∈ S(ℝ). From (7) we infer that D_{k}f has support in [−R, R]. Since D_{k}^{−1}f(x) = D_{k}f(−x), for x ∈ ℝ, the same is true for D_{k}^{−1}f, as was to be proved. □

Remark 6.

(1) By holomorphic continuation and continuity, cf. [13], one sees that Theorem 5 also holds in the more general case Re k ≥ 0. Alternatively, one can use Lemma 7 below instead of Lemma 4, which establishes (7) also in the case Re k ≥ 0, and then the above direct proof is again valid.

(2) We emphasize that the present proof does not use any explicit formulas for the Dunkl kernel in one dimension, contrary to the alternative methods of proof in [20] (where Weyl fractional integral operators are used), [13] (where asymptotic results for Bessel functions are needed) and [6] (where various integral operators, Dunkl’s intertwining operator and the Paley–Wiener theorem for the Fourier transform all play a role). Also, the fact that a contour shifting argument for the transform is usually not possible (since w_{k} generically has no entire extension) is no obstruction. Given this unspecific nature, it is possible that the present method can be applied to other transforms as well, although the symmetry of the kernel—as reflected in (6), which was used in the proof of Lemma 4 and which will again be used in the proof of the alternative Lemma 7 below—is perhaps necessary. The same suggestion applies to the results in the remaining sections of this paper.

(3) For even functions, Theorem 5 specializes to the Paley–Wiener theorem for the Hankel transform [11].

4. Real Paley–Wiener theorem for L^{p}-functions

We will now consider the real Paley–Wiener theorem for L^{p}-functions in the spirit of Bang [4]. The result is first proved for Schwartz functions in Theorem 8 and subsequently for the general case in Theorem 10.

Let ∥ · ∥_{Re k,p} denote the L^{p}(ℝ, |w_{k}(x)|dx)-norm, for 1 ≤ p ≤ ∞. Then we have the following generalization of Lemma 4 to complex multiplicities.

Lemma 7. Let Re k ≥ 0, 1 ≤ p ≤ ∞ and f ∈ S(ℝ). Then in the extended positive real numbers we have

(9)  lim inf_{n→∞} ∥T_{k}^{n}f∥_{Re k,p}^{1/n} ≥ sup{|λ| : λ ∈ supp D_{k}f}.

Proof. Let 0 ≠ λ_{0} ∈ supp D_{k}f and choose ϵ > 0 such that 0 < 2ε < |λ_{0}|. Also choose ϕ ∈ C_{c}^{∞}(ℝ) such that supp ϕ ⊂ [λ_{0} − ε, λ_{0} + ε], and ⟨D_{k}f, ϕ⟩_{k} ≠ 0. Define ϕ_{n}(λ) = λ^{−n}ϕ(λ) and P_{n}(x) = x^{n} for n ∈ ℕ ∪ {0}. Then

(1 + P_{N}(x))(D_{k}ϕ_{n})(x) = \frac{1}{c_{k}} ∫_{λ_{0}−ε}^{λ_{0}+ε} (1 + (iT_{k})^{N}) (λ^{−n}ϕ(λ))ψ_{x}^{k}(λ)w_{k}(λ)dλ (N ∈ ℕ ∪ \{0\}).

We fix N such that N is even and N > 2Re k + 1.

Corollary 2 and the binomial formula imply that

|(1 + (iT_{k})^{N}) (λ^{−n}ϕ(λ))| ≤ C_{1}n^{N}(|λ_{0}| − ε)^{−n} (n ∈ ℕ ∪ {0}, λ ∈ ℝ),

where C_{1} is a positive constant. This yields the estimates

∥D_{k}ϕ_{n}∥_{Re k,q} ≤ ∥(1 + P_{N})^{−1}∥_{Re k,q}∥(1 + P_{N})D_{k}ϕ_{n}∥_{∞}

≤ \frac{2ε}{|c_{k}|}C_{1}n^{N} (|λ_{0}| − ε)^{−n} ∥(1 + P_{N})^{−1}∥_{Re k,q}

≤ C_{2}n^{N} (|λ_{0}| − ε)^{−n}

for all n > N, where C_{2} is a positive constant and q is the conjugate exponent.

Using (6), the identity D_{k}(P_{n}ϕ_{n}) = (−i)^{n}T_{k}^{n}D_{k}ϕ_{n} and Hölder’s inequality, we therefore get

|⟨D_{k}f, ϕ⟩_{k}| = |⟨D_{k}f, P_{n}ϕ_{n}⟩_{k}| = |⟨f, D_{k}(P_{n}ϕ_{n})⟩_{k}| = |⟨f, T_{k}^{n}(D_{k}ϕ_{n})⟩_{k}|
= |⟨T_{k}^{n}f, D_{k}ϕ_{n}⟩_{k}| ≤ ∥T_{k}^{n}f∥_{Re k,p} ∥D_{k}ϕ_{n}∥_{Re k,q}
≤ C_{2}n^{N} (|λ_{0}| − ε)^{−n} ∥T_{k}^{n}f∥_{Re k,p} ,

whence

lim inf_{n→∞} ∥T_{k}^{n}f∥_{Re k,p}^{1/n} ≥ lim inf_{n→∞} (C_{2}n^{N})^{−1/n} (|λ_{0}| − ε) |⟨D_{k}f, ϕ⟩_{k}|^{1/n} = (|λ_{0}| − ε) ,

establishing the lemma. □

Using the Paley–Wiener theorem, we can extend the inequality in Proposition 3 to the norms ∥ · ∥_{Re k,p}, 1 ≤ p ≤ ∞, for Re k ≥ 0, and we thus have

Theorem 8 (Real Paley–Wiener theorem for Schwartz functions). Let Re k ≥ 0, 1 ≤ p ≤ ∞ and f ∈ S(ℝ). Then in the extended positive real numbers we have

lim_{n→∞} ∥T_{k}^{n}f∥_{Re k,p}^{1/n} = sup{|λ| : λ ∈ supp D_{k}f}.

Proof. In view of Lemma 7 it only remains to be shown that

lim sup_{n→∞} ∥T_{k}^{n}f∥_{Re k,p}^{1/n} ≤ R,

if f ∈ S(ℝ) is such that supp D_{k}f ⊂ [−R, R] for some finite R > 0. Using the inversion formula and the intertwining properties of the transform we have

(10)  x^{N}T_{k}^{n}f(x) = \frac{i^{N+n}}{c_{k}} ∫_{−R}^{R} T_{k}^{N}(P_{n}D_{k}f)(λ)ψ_{λ}^{k}(x)w_{k}(λ) dλ (n, N ∈ ℕ ∪ \{0\}),

where again P_{n}(x) = x^{n}. Now Corollary 2 and the binomial formula imply that

∥T_{k}^{N}(P_{n}D_{k}f)∥_{∞} ≤ C_{1}n^{N}R^{n},

where C_{1} is a constant depending on f and N. Therefore (10) yields that

(11)  ∥(1 + P_{N})T_{k}^{n}f∥_{∞} ≤ C_{2}n^{N}R^{n+1},

where C_{2} is again a constant depending on f and N. We fix N such that N is even and N > 2Re k+1. Then the observation

∥T_{k}^{n}f∥_{Re k,p} ≤ ∥(1 + P_{N})^{−1}∥_{Re k,p}∥(1 + P_{N})T_{k}^{n}f∥_{∞}

and (11) establish the result. □

Remark 9. Theorem 8 is new for complex k. For real k, the result can be found in [6], where it is proved using the Plancherel theorem for the Dunkl transform, the Riesz–Thorin convexity theorem and the theory of Sobolev spaces for Dunkl operators.

For k ≥ 0, we will now generalize Theorem 8 to the L^{p}-case in Theorem 10, using the structure of S(ℝ) as an associative algebra under the Dunkl convolution ∗_{k}. We refer to [19] for details on this subject.

Let k ≥ 0, and define a distributional Dunkl transform D_{k}^{d}f of f ∈ L^{p}(ℝ, w_{k}(x)dx) by transposition

⟨D_{k}^{d}f, ϕ⟩ = ⟨f, D_{k}ϕ⟩_{k} (ϕ ∈ S(ℝ)).

Clearly D_{k}^{d} is injective on L^{p}(ℝ, w_{k}(x)dx). Furthermore, from

|⟨D_{k}^{d}f, ϕ⟩| ≤ ∥f∥_{p}∥D_{k}ϕ∥_{∞} (ϕ ∈ S(ℝ)),

we see that D_{k}^{d}f is a tempered distribution. Note also that for f ∈ L^{p}(ℝ, w_{k}(x)dx) and 1 ≤ p ≤ 2 we have

⟨D_{k}^{d}f, ϕ⟩ = ⟨D_{k}f, ϕ⟩_{k} (ϕ ∈ S(ℝ)).

by (6) and density of S(ℝ) in L^{p}(ℝ, w_{k}(x)dx). Thus, for f ∈ L^{p}(ℝ, w_{k}(x)dx) and 1 ≤ p ≤ 2, D_{k}^{d}f as defined above corresponds to the distribution D_{k}f w_{k}, implying that in this case supp D_{k}^{d}f = supp D_{k}f.

Theorem 10 (Real Paley–Wiener theorem for L^{p}-functions). Let k ≥ 0, 1 ≤ p ≤ ∞ and f ∈ C^{∞}(ℝ) be such that T_{k}^{n}f ∈ L^{p}(ℝ, w_{k}(x)dx), for all n ∈ ℕ ∪ {0}. Then in the extended positive real numbers we have

lim_{n→∞} ∥T_{k}^{n}f∥_{k,p}^{1/n} = sup{|λ| : λ ∈ supp D_{k}^{d}f}.

If in addition f ∈ L^{s}(ℝ, w_{k}(x)dx), for some 1 ≤ s ≤ 2, then the distribution D_{k}^{d}f corresponds to the function D_{k}f w_{k}, and the support of D_{k}^{d}f in the right hand side is equal to the support of D_{k}f as a distribution.

Proof. First we note that (9) also holds for f as above: in the proof of Lemma 7 we just have to change ⟨D_{k}f, ·⟩_{k} into ⟨D_{k}^{d}f, ·⟩. Therefore, it only remains to be shown that

(12)  lim sup_{n→∞} ∥T_{k}^{n}f∥_{k,p}^{1/n} ≤ R,

if f is as in the theorem and such that supp D_{k}^{d}f ⊂ [−R, R] for some finite R > 0. To this end, choose ε > 0, and fix a function ϕ_{ε} ∈ S(ℝ) such that D_{k}^{−1}ϕ_{ε} = 1 on [−R, R] and D_{k}^{−1}ϕ_{ε} = 0 outside [−R − ε, R + ε]. We have from [19, Proposition 3] that D_{k}^{−1}(ϕ ∗_{k} ψ) = (D_{k}^{−1}ϕ) (D_{k}^{−1}ψ), for all ϕ, ψ ∈ S(ℝ). With P_{n}(x) = x^{n}, we thus find for arbitrary ϕ ∈ S(ℝ) that

⟨f, ϕ ∗_{k} T_{k}^{n}ϕ_{ε}⟩_{k} = ⟨D_{k}^{d}f, D_{k}^{−1}(ϕ ∗_{k} T_{k}^{n}ϕ_{ε})⟩ = (−i)^{n}⟨D_{k}^{d}f, P_{n}(D_{k}^{−1}ϕ) (D_{k}^{−1}ϕ_{ε})⟩
= (−i)^{n}⟨D_{k}^{d}f, P_{n}D_{k}^{−1}ϕ⟩ = ⟨f, T_{k}^{n}ϕ⟩_{k}.

Furthermore, from loc.cit., one knows that ∥ϕ ∗_{k} ψ∥_{k,q} ≤ 4∥ϕ∥_{k,q}∥ψ∥_{k,1}, for all ϕ, ψ ∈ S(ℝ), where q is the conjugate exponent of p. If we combine these two results with (5) and Hölder’s reverse inequality, we infer that

∥T_{k}^{n}f∥_{k,p} = sup_{ϕ} |⟨T_{k}^{n}f, ϕ⟩_{k}| = sup_{ϕ} |⟨f, T_{k}^{n}ϕ⟩_{k}| = sup_{ϕ} |⟨f, ϕ ∗_{k} T_{k}^{n}ϕ_{ε}⟩_{k}| ≤ 4∥f∥_{k,p}∥T_{k}^{n}ϕ_{ε}∥_{k,1},

where the supremum is over all functions ϕ ∈ S(ℝ) with ∥ϕ∥_{k,q} = 1. From Theorem 8 we therefore conclude that

lim sup_{n→∞} ∥T_{k}^{n}f∥_{k,p}^{1/n} ≤ R + ε,

proving (12).

The statement on supports was established in the discussion preceding the theorem. □

Remark 11. For even functions, when the Dunkl transform reduces to the Hankel transform, the previous result can already be found in [2, 3]. Also using Dunkl convolution, and closely following the approach in [1, 2, 3], Theorem 10 has previously been established in [7]. Our proof is considerably shorter than the proof in loc.cit.

5. Paley–Wiener theorem for L^{2}-functions (for k ≥ 0)

In this section we assume that k ≥ 0. For R > 0, we define L_{R}^{2}(ℝ, w_{k}(x)dx) to be the subspace of L^{2}(ℝ, w_{k}(x)dx) consisting of those functions with distributional support in [−R, R]. Let H_{R}^{2,k}(ℂ) denote the space of entire functions of exponential type R (cf. Proposition 3) that belong to L^{2}(ℝ, w_{k}(x)dx) when restricted to the real line.

Theorem 12 (Paley–Wiener theorem for L^{2}-functions). Let R > 0 and k ≥ 0. Then the Dunkl transform D_{k} is a bijection from L_{R}^{2}(ℝ, w_{k}(x)dx) onto H_{R}^{2,k}(ℂ).

Proof. Let f ∈ L_{R}^{2}(ℝ, w_{k}(x)dx). Then f ∈ L^{1}(ℝ, w_{k}(x)dx), and (3) and (4) imply that D_{k}f is entire and of exponential type R. Conversely, let f ∈ H_{R}^{2,k}(ℂ). By the Plancherel theorem one has D_{k}^{−1}f ∈ L^{2}(ℝ, w_{k}(x)dx). In addition, Proposition 3 and Theorem 10 (with p = ∞) show that supp D_{k}f ⊂ [−R, R]. The same is then true for D_{k}^{−1}f, and the result follows. □

Remark 13.

(1) In the Fourier transform case, for k = 0, this is the original Paley–Wiener theorem, see [15] or [18, Theorem 19.3]; cf. also [5, 6.2.4 and 6.7.1]. The present proof seems to be more in terms of general principles than other proofs seen in the literature.

(2) Let 1 ≤ p ≤ ∞, and define H_{R}^{p,k}(ℂ) as the space of entire functions of exponential type R that belong to L^{p}(ℝ, w_{k}(x)dx) when restricted to the real line. As above, using Proposition 3 and Theorem 10, one sees that D_{k}^{d}f has support in [−R, R] when f ∈ H_{R}^{p,k}(ℂ). In particular, for 1 ≤ p ≤ 2, with conjugate exponent q, we have that D_{k} maps H_{R}^{p,k}(ℂ) into L_{R}^{q}(ℝ, w_{k}(x)dx), the subspace of L^{q}(ℝ, w_{k}(x)dx) consisting of those functions with distributional support in [−R, R], cf. [5, 6.8.13] for the Fourier case k = 0.

References

[1] N.B. Andersen, Real Paley–Wiener theorems, Bull. London Math. Soc. 36 (2004), 504–508.

[2] N.B. Andersen, Real Paley–Wiener theorems for the Hankel transform, Submitted.

[3] N.B. Andersen, On the range of the Chébli–Trimèche transform, Monatsh. Math. 144 (2005), 193–201.

[4] H.H. Bang, A property of infinitely differentiable functions, Proc. Amer. Math. Soc. 108 (1990), 73–76.

[5] R.P. Boas, Jr., Entire functions, Academic Press Inc., New York, 1954.

[6] C. Chettaoui, K. Trimèche, New type Paley–Wiener theorems for the Dunkl transform on ℝ, Integral Transforms Spec. Funct. 14 (2003), 97–115.

[7] C. Chettaoui, Y. Othmani, K. Trimèche, On the range of the Dunkl transform on ℝ, Math. Sci. Res. J. 8 (2004), 85–103.

[8] C.F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), 167–183.

[9] C.F. Dunkl, Integral kernels with reflection group invariance, Canad. J. Math. 43 (1991), 1213–1227.

[10] C.F. Dunkl, Hankel transforms associated to finite reflection groups, in Hypergeometric functions on domains of positivity, Jack polynomials, and applications (Tampa, FL, 1991), 123–138, Contemp. Math., 138, Amer. Math. Soc., Providence, RI, 1992.

[11] J.L. Griffith, Hankel transforms of functions zero outside a finite interval, J. Proc. Roy. Soc. New South Wales 89 (1955), 109–115.

[12] M.F.E. de Jeu, The Dunkl transform, Invent. Math. 113 (1993), 147–162.

[13] M.F.E. de Jeu, Paley–Wiener theorems for the Dunkl transform. Preprint (2004), to appear in Trans. Amer. Math. Soc. ArXiv: math.CA/0404439.

[14] E.M. Opdam, Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group, Comp. Math. 85 (1993), 333–373.

[15] R. Paley, N. Wiener, The Fourier Transforms in the Complex Domain, Amer. Math. Soc. Colloq. Publ. Ser., Vol. 19, Providence, RI, 1934.

[16] M. Rösler, Positivity of Dunkl’s intertwining operator, Duke Math. J. 98 (1999), 445–463.

[17] M. Rösler, Dunkl operators: theory and applications, in ”Orthogonal polynomials and special functions”, Leuven, 2002, Lecture Notes in Math., 1817 (2003), 93–135.

[18] W. Rudin, Real and complex analysis, McGraw-Hill Book Co., 1966.

[19] F. Soltani, L^{p}-Fourier multipliers for the Dunkl operator on the real line, J. Funct. Anal. 209 (2004), 16–35.

[20] K. Trimèche, The Dunkl intertwining operator on spaces of functions and distributions and integral representation of its dual, Integral Transform. Spec. Funct. 12 (2001), 349–374.

Dipartimento di Matematico, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy

E-mail address: byrial@calvino.polito.it

Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands

E-mail address: mdejeu@math.leidenuniv.nl