Radouan Daher

Salah El Ouadih

El Ouadih, Salah

Daher, Radouan

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 355 (2017) 318–324Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comHarmonic analysisLipschitz conditions for the generalized discrete Fourier transform associated with the Jacobi operator on [0, π]Conditions de Lipschitz pour la transformée de Fourier discrète généralisée associée à l’opérateur de Jacobi sur [0, π]Salah El Ouadih, Radouan DaherDepartment of Mathematics, Faculty of Sciences Aïn Chock, University Hassan II, Casablanca, Moroccoa r t i c l e i n f o a b s t r a c tArticle history:Received 4 August 2016Accepted after revision 23 January 2017Available online 20 February 2017Presented by the Editorial BoardDedicated to Professor François Rouvière on the occasion of his 71st birthdayOur aim in this paper is to prove an analog of the classical Titchmarsh theorem on the image under the discrete Fourier–Jacobi transform of a set of functions satisfying a generalized Lipschitz condition in the space L(α,β)2 .© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éL’objectif de cette Note est de prouver un analogue du théorème de Titchmarsh sur l’image sous la transformée de Fourier–Jacobi discrète d’un jeu de fonctions satisfaisant une condition de Lipschitz généralisée dans l’espace L(α,β)2 .© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionIt is well known that many problems for partial differential equations are reduced to a power series expansion of the desired solution in terms of special functions or orthogonal polynomials (such as Laguerre, Hermite, Jacobi, etc., polynomi-als). In particular, this is associated with the separation of variables as applied to problems in mathematical physics (see, e.g., [10,12]).In this article, we obtain an analog of one classical Titchmarsh theorem on description of the image under the discrete Fourier–Jacobi transform of a class of functions satisfying the Lipschitz condition in weighted function spaces on [−1, 1]. We now give the exact statement of this theorem.Suppose that f (x) is a function in the L2(R) space (all functions below are complex-valued), ‖.‖L2(R) is the norm of L2(R), and δ is an arbitrary number in the interval (0, 1).E-mail addresses: salahwadih@gmail.com (S. El Ouadih), rjdaher024@gmail.com (R. Daher).http://dx.doi.org/10.1016/j.crma.2017.01.0171631-073X/© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.S. El Ouadih, R. Daher / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 318–324 319Theorem 1.1. [13, Theorem 85] Let f ∈ L2(R). Then the following conditions are equivalent:(i) ‖ f (x + h) − f (x)‖L2(R) = O (hδ), as h → 0, 0 < δ < 1,(ii)∫|λ|≥r|̂ f (λ)|2dλ = O (r−2δ) as r → ∞,where ̂ f stands for the Fourier transform of f .This theorem has been generalized in the case of noncompact rank-1 Riemannian symmetric spaces [8], and was ex-tended in [4] for the Fourier transform in the space L2(Rn) using a spherical mean operator. The Titchmarsh’s theorem has been generalized recently for a class of functions satisfying the Lipschitz condition for the Jacobi transform in [1] and also for the Dunkl transform in [3].On the other hand, in [14, Theorem 2.17], M. S. Younis characterizes the set of functions in L2([−π, π]) satisfying the Lipschitz condition by means of an asymptotic estimate growth of the norm of their discrete Fourier transform. More precisely, we have the following theorem.Theorem 1.2. [14, Theorem 2.17] Let f ∈ L2([−π, π]). Then the following conditions are equivalent:(i) ‖ f (x + h) − f (x)‖L2([−π,π]) = O (hδ), as h → 0, δ ∈ (0, 1),(ii)∑|n|≥N|̂ f (n)|2 = O (N−2δ) as N → ∞,where ̂ f (n) stands for the n-th Fourier transform coefficient of f .In our present paper, we investigate, among other things, the validity of Theorem 1.2 in case of functions of the wider Lipschitz class in weighted function spaces on [−1, 1]. For this purpose, we use a generalized translation operator, which was defined by Flensted-Jensen and Koornwinder (see [7]).2. PreliminariesThroughout the paper, α and β are arbitrary real numbers with α ≥ β ≥ −1/2 and α = −1/2. We put w(x) = (1 −x)α(1 + x)β and consider problems of the approximation of functions in the Hilbert spaces L2([−1, 1], w(x)dx).Let P (α,β)n (x) be the Jacobi orthogonal polynomials, n ∈ N0 := {0, 1, 2, .....} (see [11] or [6]). The polynomials P (α,β)n (x), n ∈N0, form a complete orthogonal system in the Hilbert space L2([−1, 1], w(x)dx).It is known (see [11], Ch. IV) thatmax−1≤x≤1 |P(α,β)n (x)| = P (α,β)n (1) =(n+αα) = (α + n + 1)n!(α + 1) .The polynomialsR(α,β)n (x) := P(α,β)n (x)P (α,β)n (1)are called normalized Jacobi polynomials.In what follows it is convenient to change the variable by the formula x = cos t, t ∈ I := [0, π]. We use the notationρ(t) = w(cos t) sin t = 2γ(sint2)2α+1 (cost2)2β+1, γ = α + β + 1,ϕn(t) = ϕ(α,β)n (t) := R(α,β)n (cos t),n ∈N0.Let L(α,β)2 denote the space of square integrable functions f (t) on the closed interval I with the weight function ρ(t)and the norm‖ f ‖ =√√√√√π∫0| f (t)|2ρ(t)dt.The Jacobi differential operator is defined as320 S. El Ouadih, R. Daher / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 318–324B := d2dt2+((α + 12)cott2−(β + 12)tant2)ddt.The function ϕn(t) satisfies the differential equationBϕn = −λnϕn, λn = n(n + γ ),n ∈ N0,with the initial conditions ϕn(0) = 1 and ϕ′n(0) = 0.Lemma 2.1. The following inequalities are valid for Jacobi functions ϕn(t):(i) for t ∈ (0, π/2], we have|ϕn(t)| < 1;(ii) for t ∈ [0, π/2], we have1 − ϕn(t) ≤ c1λnt2;(iii) For t ∈ [0, 1] and tn ≤ 2 we have1 − ϕn(t) ≥ c2λnt2.Proof. See [9, Proposition 3.5. and Lemma 3.1]. Recall from [9], the Fourier–Jacobi series of a function f ∈ L(α,β)2 is defined byf (t) =∞∑n=1an( f )ϕ̃n(t), (1)whereϕ̃n = ϕn‖ϕn‖ , an( f ) = 〈 f , ϕ̃n〉 =π∫0f (t)ϕ̃n(t)ρ(t)dt.The sequence {an( f ),n ∈ N0} is called the discrete Fourier–Jacobi transform of f .LetSm f (t) =m−1∑n=1an( f )ϕ̃n(t),be a partial sums of series (1), and letEm( f ) = infPm‖ f − Pm‖,denote the best approximation of f ∈ L(α,β)2 by polynomials of the formPm(t) =m−1∑n=1cnϕ̃n(t), cn ∈ R.It is well known that‖ f ‖ =√√√√ ∞∑n=1|an( f )|2,Em( f ) = ‖ f − Sm f ‖ =√√√√ ∞∑n=m|an( f )|2.The Jacobi generalized translation is defined by the formulaS. El Ouadih, R. Daher / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 318–324 321Th f (t) =π∫0f (θ)K (t,h, θ)ρ(θ)dθ, 0 < t,h < π,where K (t, s, θ) is a certain function (see [2]).Below are some properties (see [9]):(i) Th : L(α,β)2 → L(α,β)2 is a continuous linear operator,(ii) ‖Th f ‖ ≤ ‖ f ‖,(iii) Th(ϕn(t)) = ϕn(h)ϕn(t),(iv) an(Th f ) = ϕn(h)an( f ),(v) ‖Th f − f ‖ → 0, h → 0,(vi) B(Th f ) = Th(B f ).The following lemma will be needed in due course.Lemma 2.2. [5, p. 101] Suppose bn ≥ 0, and 0 < c < d. ThenN∑n=1ndbn = O (Nc) iff∞∑n=Nbn = O (Nc−d).3. Main resultsFor every function f ∈ L(α,β)2 we define the differences h f with step h, 0 < h < π, by the formulah f (t) = (Th − I) f (t),where I is the identity operator in L(α,β)2 .Let W r2, r ∈ N0, be the Sobolev space constructed by the Jacobi operator B, that is:W r2 := { f ∈ L(α,β)2 : B j f ∈ L(α,β)2 , j = 1,2, ..., r},where B0 f = f , B j f = B(B j−1 f ), j = 1, 2, ..., r.Lemma 3.1. If f ∈ W r2 , r ∈N0 , thenan( f ) = (−1)r 1λrnan(Br f ).Proof. Since B is self-adjoint (see [9]), we havean( f ) = 〈 f , ϕ̃n〉 = − 1λn〈 f ,Bϕ̃n〉 = − 1λn〈B f , ϕ̃n〉 = − 1λnan(B f ).This completes the proof of the lemma. Lemma 3.2. Iff (t) =∞∑n=1an( f )ϕ̃n(t),thenTh f (t) =∞∑n=1ϕn(h)an( f )ϕ̃n(t).Here, the convergence of the series on the right-hand side is understood in the sense of L(α,β)2 .Proof. By the definition of the operator Th ,Th(ϕ̃n(t)) = ϕn(h)ϕ̃n(t).322 S. El Ouadih, R. Daher / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 318–324Therefore, for any polynomialQ N(t) =N∑n=1an( f )ϕ̃n(t).Since Th is linear, we haveTh Q N(t) =N∑n=1ϕn(h)an( f )ϕ̃n(t). (2)Since Th is a linear bounded operator in L(α,β)2 and the set of all polynomials Q N(t) is everywhere dense in L(α,β)2 , the passage to the limit in (2) gives the required equality. Remark. SinceTh f (t) − f (t) =∞∑n=1(ϕn(h) − 1)an( f )ϕ̃n(t),the Parseval’s identity gives‖Th f − f ‖2 =∞∑n=1(1 − ϕn(h))2|an( f )|2.If f ∈ W r2, from Lemma 3.1, we have‖h(Br f )‖2 =∞∑n=1(1 − ϕn(h))2λ2rn |an( f )|2. (3)Definition 3.1. Let δ ∈ (0, 1), r ∈ N0. A function f ∈ W r2 is said to be in the (δ, r, 2)-Jacobi–Lipschitz class, denoted by Lip(δ, r, 2), if‖h(Br f )‖ = O (hδ) as h → 0.Theorem 3.1. Let f ∈ W r2 . The following two conditions are equivalent:(a) f ∈ Lip(δ, r, 2),(b)∑n≥Nλ2rn |an( f )|2 = O (N−2δ), as N → ∞.Proof. (a) ⇒ (b) Let f ∈ Lip(δ, r, 2). Then we have‖h(Br f )‖ = O (hδ) as h → 0,It follows from (3) that∞∑n=1(1 − ϕn(h))2λ2rn |an( f )|2 = O (h2δ).If 0 ≤ n ≤ 1h , then nh ≤ 2, and from the third inequality of Lemma 2.1, we obtain1 − ϕn(h) ≥ c2λnh2.Therefore,[ 1h ]∑n=1λ2nh4λ2rn |an( f )|2 = O (h2δ),and, by λn ≥ n2,S. El Ouadih, R. Daher / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 318–324 323[ 1h ]∑n=1n4λ2rn |an( f )|2 = O (h2δ−4).Putting N = 1h , we may write this inequality in the following form:N∑n=1n4λ2rn |an( f )|2 = O (N4−2δ).From Lemma 2.2, we have∞∑Nλ2rn |an( f )|2 = O (N4−2δ−4) = O (N−2δ).Thus, the first implication is proved.(b) ⇒ (a). Suppose now that∑n≥Nλ2rn |an( f )|2 = O (N−2δ), as N → ∞.It follows from Lemma 2.2 thatN∑n=1n4λ2rn |an( f )|2 = O (N4−2δ).According (3), we write‖h(Br f )‖2 ≤N∑n=1(1 − ϕn(h))2λ2rn |an( f )|2 +∑n≥N(1 − ϕn(h))2λ2rn |an( f )|2.Note thatλn ≤ n2(1 + γn) ≤ n2(1 + γ ),n = 1,2, .... (4)It follows from (4) and the second inequality in Lemma 2.1 thatN∑n=1(1 − ϕn(h))2λ2rn |an( f )|2 ≤ c1h4N∑n=1λ2nλ2rn |an( f )|2 ≤ c1(1 + γ )2h4N∑n=1n4λ2rn |an( f )|2 = O (N4−2δ−4) = O (N−2δ).On the other hand, it follows from the first inequality of Lemma 2.1 that∑n≥N(1 − ϕn(h))2λ2rn |an( f )|2 ≤ 4∑n≥Nλ2rn |an( f )|2 = O (N−2δ).Consequently,‖h(Br f )‖ = O (hδ),and this ends the proof of the Theorem. We conclude this work by the following immediate consequence.Corollary 3.1. Let f ∈ W r2 , and letf ∈ Lip(δ, r,2).Then ∑n≥N|an( f )|2 = O (N−2δ−4r), as N → ∞.324 S. El Ouadih, R. Daher / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 318–324References[1] A. Abouelaz, R. Daher, M. El Hamma, Generalization of Titchmarch’s theorem for the Jacobi transform, Facta Univ., Ser. Math. Inform. 28 (2013) 43–51.[2] R. Askey, S. Wainger, A convolution structure for Jacobi series, Am. J. Math. 91 (1969) 463–485.[3] R. Daher, M. El Hamma, Dini Lipschitz functions for the Dunkl transform in the space L2(Rd, wk(x) dx), Rend. Circ. Mat. Palermo 64 (2) (2015) 241–249.[4] R. Daher, M. Boujeddaine, M. El Hamma, Generalization of Titchmarsh’s theorem for the Fourier transform in the space L2(Rn), Afr. Math. (2016), http://dx.doi.org/10.1007/s13370-015-0368-x.[5] P.L. Duren, Theory of H p Spaces, Academic Press, New York, 1970.[6] A. Erdélyi, W. Magnus, F. Oberttinger, F.G. Tricomi, Higher Transcendental Functions, vol. II, McGraw–Hill, New York–Toronto–London, 1953, Russian transl., Nauka, Moscow, 1974.[7] M. Flensted-Jensen, T. Koornwinder, The convolution structure for Jacobi function expansions, Ark. Mat. 11 (1973) 245–262.[8] S.S. 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Lipschitz conditions for the generalized discrete Fourier transform associated with the Jacobi operator on [0,<i>π</i>]

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Lipschitz conditions for the generalized discrete Fourier transform associated with the Jacobi operator on [0,<i>π</i>]

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