{'fraction_non_alphanumeric': 0.06784643071385722, 'fraction_numerical': 0.017636472705458907, 'mean_word_length': 3.6308333333333334, 'pattern_counts': {'":': 0, '<': 5, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 10, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Remarks on the inverse Cherednik-Opdam transform on the real line

We obtain Hausdorff-Young inequalities for the one-dimensional Cherednik--Opdam transform and its inverse, and we establish a real Paley-Wiener theorem for its inverse that generalizes an analogous result by N. B. Andersen for the Jacobi transform.


Introduction
The present paper is concerned with themes in classical harmonic analysis in the framework of the Cherednik-Opdam transform H on the real line. The starting point is the Hausdorff-Young inequality which, among other things, allows an extension of H from L 2 (R, dµ) to L p (R, dµ) for 1 ≤ p ≤ 2; the extension is denoted H p . Similar statements can be established for the inverse transform I = H −1 , which can also be extended from L 2 (R, dν) to a map I q from L q (R, dν) into L p (R, dµ). Our first result states that the inverse of the extended map H p coincides ν-almost everywhere with I p ′ . This result was recently established for commutative hypergroups in [DS13] but the 'Cherednik-Opdam convolution' on R does not give rise to a hypergroup structure so the result does not follow immediately. The strategy of proof is still mostly the same, however, as the hypergroup structure is not needed. More importantly, the interplay between H and the convolution product still persists. Notation and first results are collected in section 2 whereas as the new results concerning the Hausdorff-Young theorems for H and H −1 appear in section 3.
The second topic pertains to Paley-Wiener theorem for H. The classical Paley-Wiener theorem for the Fourier transform describes the image of the space C ∞ c (R n ) of compactly supported smooth functions in R n as a space of entire functions in C n of a specific exponential growth rate. Similar statements hold for various classes of Lie groups and symmetric spaces, as described succintly in several books by S. Helgason, for example. In the closely related framework of Jacobi analysis (in which the Jacobi transform replaces the spherical transform on a Riemannian symmetric space of rank one), a Paley-Wiener theorem of the aforementioned type was obtained in [FJK73], for example. As already summarized in theorem 2.7 there are analogous results for the Cherednik-Opdam transform associated with arbitrary rank root systems in R n . There are comparable results in the framework of Dunkl theory, but the literature is not nearly as coherent and some details still seem to be missing.
Recently a 'dual' question has started to attract attention, namely, to describe those functions in L 2 whose Fourier transform, the Dunkl transform, or Helgason-Fourier transform are smooth and compactly supported. This amounts to a Paley-Wiener theorem for the inverse transform. For Riemannian symmetric spaces such results and closely related so-called real Paley-Wiener theorems were obtained in [And04] and [Pas00], for the Jacobi transform in [And03], for the one-dimensional Dunkl transform in [CT03] and [And06], and recently for the Heckman-Opdam transform in R n in [MT14]. We complement the literature by proving comparable results for the one-dimensional (inverse) Cherednik-Opdam transform, following the approach in [And06] and [And03]. These matters are detailed in section 4.

Notation and first properties
We are concerned with harmonic analysis for the Cherednik-Opdam transform acting on functions on R. As already indicated elsewhere this is a convenient extension of Jacobi analysis on R + . While Opdam gave a brief discussion of the rank one case in [Opd95], we shall rely exclusively on the [AAS12] since it provides a much more detailed investigation of the eigenfunctions, the convolution structure and aspects of harmonic analysis.
In the following we fix parameters α, β subject to the constraints α ≥ β ≥ − 1 2 and α > − 1 2 . Let ρ = α + β + 1 and λ ∈ C. The Opdam hypergeometric functions G that are normalized such that G (α,β) λ (0) = 1. In the notation of Cherednik one would write T (α,β) as Here k 1 is the multiplicity of a simply positive root and k 2 the (possibly vanishing) multiplicity of a multiple of this root. By [Opd95,page 90] or [AAS12, formula 1.2], the eigenfunction G λ is given by is the classical Jacobi function. Definition 2.1. The associated Laplace operator is defined by L = (T (α,β) ) 2 . The operator L is essentially self-adjoint on L 2 (R, dµ), where the measure dµ is defined by Remark 2.2. It is important to point out that the functions G λ are not the so-called Jacobi-Dunkl functions considered, for example, in [BSOAS06], the latter function being defined as eigenfunctions for the operator T (α,β) defined by According to [AAS12, Theorem 3.2] there exists a family of signed measures µ (α,β) x,y such that the product formula x,y (z) holds for all x, y ∈ R and λ ∈ C, where if x, y, z ∈ R\{0} satisfy the triangular inequality ||x|−|y|| < |z| < |x|+|y|, and K α,β (x, y, z) = 0 otherwise. Here and g(x, y, z, χ) = 1 − cosh 2 x − cosh 2 y cosh 2 z + 2 cosh x · cosh y · cosh z · cos χ. The product formula is used to obtain explicit estimates for the generalized translation operators x,y (z), as well as the convolution product of suitable functions f, g on R defined by Since the convolution of functions that are not even is permitted and produces a function on R that is not an even function either it is clear that the Jacobi transform is inadequate for further studies. To this end we need following transform: The inverse transform is given as where one should take note of the asymmetry of the formulae compared with the inversion formula for the Jacobi transform; the factor " 1 − ρ iλ " is not present in Jacobi analysis. The c-function that appears is the same as in Jacobi analysis, however: According to [AAS12, Lemma 4.1] there is a close relation between H and the Jacobi transform F α,β , expressed by the identity where f e is the even part of f , f o the odd part, and The corresponding Plancherel formula was established in [Opd95, Theorem 9.13(3)], to the effect that . We also note that it follows from the defining identity , and therefore -as will be seen in the course of the proof of lemma 2.4 below - . Note that -contrary to Jacobi analysis -there is no ρ-shift, as the ρ has already been included in the definition of T (α,β) . More generally, we record the following useful identity.
Proof. Choose f as in the hypothesis and let g ∈ C ∞ c (R). Since L is essentially self-adjoint on L 2 (R, dµ), it follows from the Plancherel theorem for H that for ν-almost every λ ∈ R and every n ∈ N 0 . Since L n f belongs to L 2 (R, dµ) for every n, the Plancherel theorem implies that Lemma 2.6.
(i) The function G 0 is strictly positive and The following theorem of Paley-Wiener type was established by Opdam and Cherednik for arbitrary root systems in R n , see also [Sch08].
We shall later obtain a Paley-Wiener theorem for the inverse transform I. Since the map H is not self-dual, this is not automatic.

Around the Hausdorff-Young inequality
Define Proof. The first two statements follow as in the case of Jacobi analysis, cf. [FJK73, Lemma 3.1], and the third statement is established by interpolation between the estimates Hf 2,ν f 2,µ (which is the Plancherel theorem) and the uniform estimate Hf ∞,ν f 1,µ that follows from lemma 2.6.
Remark 3.2. It was observed in [EK82] that the aforementioned Hausdorff-Young inequality can be improved considerably, by using the fact that the Fourier transform on G/K -and in our case the Cherednik-Opdam transform -is well-defined for λ ∈ Ω p . The following extension of the Hausdorff-Young inequality is analogous to [NPP14, Lemma 5.3] but we shall not need it in later parts of the present paper. One simply notices that the non-symmetric Plancherel density (1 − ρ/iλ)|c(λ)| −2 decays like |c(λ)| −2 for large |λ|.

Lemma 3.3.
(1) Let f ∈ L p (R, dµ) for some p ∈ (1, 2) and η ∈ Ω p . Then there is a positive constant C η,p such that We recently obtained an extension of the Hausdorff-Young inequality and several versions of the classical Hardy-Littlewood inequalities for the Heckman-Opdam transform acting on W -invariant functions in R n . These were based on the Hausdorff-Young inequalities obtained in [NPP14]. By using the Hausdorff-Young inequalities in lemma 3.1(iii) and lemma 3.3, one can generalize in a straightforward manner [Joh15, Theorem 3.6; 3.9] to the present context of Cherednik-Opdam analysis on R. We leave the details to the interested reader.
If f ∈ L 1 (R, dµ) ∩ L p (R, dµ) for some p ∈ (1, 2), the Hausdorff-Young inequality implies that Hf is well-defined and belongs to L q (R, dν). The transform H therefore extends to a continuous linear map H p : L p (R, dµ) → L q (R, dν) that coincides with H on L 1 (R, dµ) ∩ L p (R, dµ). In particular, by the Plancherel theorem, we may identify H with H 2 . We employ a similar notational convention to define the map I p : L p (R, dν) → L q (R, dµ) as the extension of I = H −1 .
Proof. The argument is a slight modification of the proof for the Jacobi transform, resp. for the Heckman-Opdam transform, cf. [NPP14,Theorem 5.4].
Associate to a fixed function g ∈ C ∞ c (R) the linear functionals These coincide on the space L 1 (R, dµ) ∩ L 2 (R, dµ) which is dense in L p (R, dµ). We claim that T g and T g are continuous, from which it follows that they coincide on L p (R, dµ). In particular, if f ∈ L p (R, dµ) and Hf = 0, then T g (h) = T g (h) = 0, and f, g L 2 (R,dµ) = 0 for all g ∈ C ∞ c (R). But then f = 0. The continuity of T g follows from the obervation that |T g (h)| ≤ h p,µ g q,µ where 1 p + 1 q = 1. Additionally | T g (h)| ≤ Hg p,ν Hh q,ν ≤ C p Hg p,ν h p,ν , so T g is also continuous. For the first inequality we used the Paley-Wiener estimate in theorem 2.7. This completes the proof.

Proof.
(i) Choose a sequence {g n } ∞ n=1 of simple functions on R such that lim n→∞ f − g n p 1 ,µ = lim n→∞ f − g n p 2 ,µ = 0.
Each function Hg n belongs to L p ′ 1 (R, dν) ∩ L p ′ 2 (R, dν) by the Hausdorff-Young inequality, and lim n→∞ H p 1 f − Hg n p ′ 1 ,ν = lim n→∞ H p 2 f − Hg n p ′ 2 ,ν = 0. One can therefore extract subsequences {Hg n k } ∞ k=1 and {Hg n l } ∞ l=1 of {Hg n } ∞ n=1 such that Hg n k → H p 1 f and Hg n l → H p 2 f ν-almost everywhere on R, from which it follows that H p 1 f = H p 2 f ν-almost everywhere on R as claimed.
(ii) Since I q j h ∈ L q ′ j (R, dµ) for j = 1, 2, the claim follows from the injectivity of H p for p ∈ (1, 2] and (i).

Lemma 3.6.
(a) For f, g ∈ L 2 (R, dµ) and h ∈ L 1 (R, dµ) the convolution product f ⋆ g belongs to C 0 (R) and Proof. The first statement follows from an application of Fubini's theorem and commutativity of the convolution product.
As for the statement in (b), we observe that the inverse Cherednik-Opdam transform then H(f ⋆ (I p h)) = Hf · H(I p h) = H p f · h, and it follows from proposition 3.4 that For f ∈ C c (R) and h ∈ L p (R, dν) there is a sequence {h n } of functions h n ∈ C c (R) such that h − h n ∞,ν → 0 as n∞. We have just seen that I(H p f · h n ) = f ⋆ (I p h n ), so several applications of the Hölder inequality, together with Young's inequality, lead to the estimate Lemma 3.7.
(i) For every compact neighborhood C of a given λ ∈ R there exists a net {f i } n of functions f n ∈ C c (R) such that Hf i → 1 C uniformly on compact subsets and dν) for some p ∈ [1, 2] has the property that I p h belongs to L 2 (R, dµ). Then h ∈ L 2 (R, dν) and h = H(I p h) ν-almost everywhere.
Remark 3.8. Statement (i) appears as [DS13, Lemma 2.6] for hypergroups, including a reference with the proof. Although the convolution ⋆ presently does not define a hypergroup structure, it is still sensible to view the G λ as analogues of the characters. In the special case of Jacobi analysis this analogy is indeed correct. We shall therefore merely outline the necessary adjustments to the proof of (i), (ii), and (iii) as they appeared in [DS13] that are required to take into account the minor differences.
Proof of (i). Let λ ∈ R and ǫ > 0 be fixed, and let C ⊂ R be a compact subset. Since λ → G λ (x) is continuous for fixed x, an equicontinuity argument establishes the existence of an open neighborhood U of λ in R such that 0 < ν(U ) < ∞ and by the construction of the set U . Setting k = ν(U ) −1/2 1 U , there exists a function f ∈ C c (R) such that Hf − k 2,ν < ǫ/4. Since k 2,ν = 1, it suffices to consider the case there Hf 2,ν = 1 = f 2,µ . Because f belongs to C c (R), it holds that H(f ⋆ f ) = (Hf ) · (Hf ) = |Hf | 2 , and therefore For arbitrary Proof of (ii). Choose a net dµ) and such that Hf i converges uniformly on compact subsets of R to the identity. Moreover choose By the Hausdorff-Young inequality it holds that H(h i ⋆ f ) also belongs to L p ′ (R, dν). Now Therefore, according to the Plancherel theorem, By using (i) we can choose the compact set C in the beginning of the proof so that In particular k i ⋆ f − I(H p f ) 2,µ → 0, implying that f = I(H p f ) µ-almost everywhere.
Outline of proof for (iii). Let C be a compact neighborhood of a fixed λ ∈ R and use (i) to produce a net then h belongs to L 1 (R, dν) ∩ L 2 (R, dν) by Plancherel and moreover to C c (R). At this point one can repeat the argument in [DS13]. We leave the details to the interested reader.
The following result is the first main theorem and is analogous to [DS13, Theorem 2.8].
Proof. In the first case H p f belongs to L p ′ (R, dν) ∩ L r (R, dν) ⊂ L 2 (R, dν), implying that H p f = Hf . But then f = I(Hf ) = I p ′ (H p f ) according to proposition 3.7.
The second statement is proved analogously.

Paley-Wiener theorems for the inverse Cherednik-Opdam transform
The present section establishes several Paley-Wiener-type results for the inverse transform I. We shall follow [And03, Section 4] closely, although some minor differences occur due to the Plancherel measure dν being non-symmetric and the G λ being more complicated.
Definition 4.1. The support of g ∈ L 2 (R, dµ) is the smallest closed set, on the complement of which g vanishes almost everywhere. The radius of the support of g is defined by Lemma 4.2. Assume g ∈ L 2 (R, dν) has the property that | · | 2n g ∈ L 2 (R, dν) for all n ∈ N 0 . Then Proof. The conclusion being trivial for g = 0 (in which case R g = 0), we assume without loss of generality that g 2,ν = 0.
R (R), then | · | n g belongs to L 1 (R, dν) ∩ L 2 (R, dν) for every n ∈ N 0 and Ig is therefore a smooth function on R. Moreover L n (Ig) = I((−1) n (·) 2n g) ∈ L 2 (R, dµ) for all n ∈ N 0 by definition of I and the identity L G λ = −λ 2 G λ . We conclude from lemma 3.7 that which means that Ig belongs to PW 2 R (R). If on the other hand f ∈ PW 2 R (R), then H(L n f )(λ) = (−1) n λ 2n Hf (λ) ∈ L 2 (R, dν) for all n ∈ N 0 . Therefore that is, Hf has compact support and R Hf = R.
As in [And03, Corollary 8] one obtains the following Corollary 4.5. Let g be a measurable function on R. Then | · | n g belongs to L 2 (R, dν) for all n ∈ N 0 if and only if L n Ig belongs to L 2 (R, dµ) for all n ∈ N 0 . Definition 4.6. The Paley-Wiener space PW(R) consists of all functions f ∈ C ∞ (R) satisfying Theorem 4.7. The inverse Cherednik-Opdam transform I is a bijection of C ∞ c (R) onto PW(R), and I maps C ∞ R (R) onto PW R (R) for every R ≥ 0.
Proof. First note that Ig belongs to PW 2 R (R) whenever g ∈ C ∞ R (R), according to theorem 4.4. Since L n (Ig) = I((−1) n (·) 2n g) ∈ L 2 (R, dµ) for every n ∈ N 0 , it remains to show that Ig satisfies the polynomial decay condition whenever g ∈ C ∞ c (R). Recall that where |c α,β | −2 is the density in the symmetric Plancherel measure that appears in Jacobi analysis (and in the the proof of [And03, Theorem 11]), so one must prove that belongs to L 2 (R, dµ) for every m, n ∈ N 0 . While the argument given for this fact in the proof of [And03, Theorem 11] does not generalize immediately to the present setting, the basic idea is still sound. The issue is that for the Jacobi case one can use that the Jacobi function ϕ (α,β) λ is even in λ and allows for a Harish-Chandra series expansion adapted to the c-function. One then proceeds with an argument involving Gangolli estimates for the coefficients in the Harish-Chandra expansion, much like in the proof for the Paley-Wiener theorem for the Jacobi transform. Due to the asymmetry in dν, such an expansion will necessarily be more complicated, albeit it is still available (as detailed in [Sch08, Section 3]).
Finally observe that a function f ∈ PW R (R) ⊂ PW 2 R (R) has polynomial decay, so that f Hf is smooth. As | G λ (x)| (1 + |x|)e −ρ|x| , it follows from theorem 4.4 that Hf is compactly supported with R Hf = R. This completes the proof.