In this paper, using a generalized translation operator, we obtain an analog of Younis Theorem 5.2 in [3] for the Cherednik-Opdam transform for functions satisfying the ( ; ; 2)-Cherednik-Opdam Lipschitz condition in the space L2 ;(R)

Daher, Radouan

El Ouadih, Salah

University of Niš: Facta Universitatis (E-Journals) / Универзитет у Нишу

FACTA UNIVERSITATIS (NISˇ)Ser. Math. Inform. Vol. 31, No 4 (2016), 815–823(ψ, γ, 2)-CHEREDNIK-OPDAM LIPSCHITZ FUNCTIONS IN THESPACE L2α,β(R)Radouan Daher and Salah El OuadihAbstract. In this paper, using a generalized translation operator, we obtain an analogof Younis Theorem 5.2 in [3] for the Cherednik-Opdam transform for functions satisfyingthe (ψ, γ, 2)-Cherednik-Opdam Lipschitz condition in the space L2α,β(R).Keywords: Cherednik-Opdam operator; Cherednik-Opdam transform; generalizedtranslation.1. Introduction and PreliminariesVarious investigators such as V.N. Mishra and L.N. Mishra [7], Mishra and al. [5, 6]have determined the degree of approximation of 2pi-periodic signals (functions) be-longing to various classes Lipα, Lip(α, r), Lip(ξ(t), r) and W (Lr, ξ(t)), (r ≥ 1), offunctions through trigonometric Fourier approximation using different summabilitymatrices with monotone rows. In this direction, Younis Theorem 5.2 [3] charac-terized the set of functions in L2(R) satisfying the Cauchy Lipschitz condition bymeans of an asymptotic estimate growth of the norm of their Fourier transforms,namely we haveTheorem 1.1. [3] Let f ∈ L2(R). Then the following are equivalents(i) ‖f(x+ h)− f(x)‖ = O(hδ(log 1h)γ), as h→ 0, 0 < δ < 1, γ ≥ 0,(ii)∫|λ|≥r|f̂(λ)|2dλ = O(r−2δ(log r)2γ), as r →∞,where f̂ stands for the Fourier transform of f .In this paper, we prove the generalization of Theorem 1.1 for the Cherednik-Opdamtransform for functions satisfying the (ψ, γ, 2)-Cherednik-Opdam Lipschitz condi-tion in the space L2α,β(R). For this purpose, we use the generalized translationoperator. We point out that similar results have been established in the JacobiReceived January 02, 2016; Accepted October 10, 20162010 Mathematics Subject Classification. Primary 42B37; Secondary 47B48815816 R. Daher and S. El Ouadihtransform [8].In this section, we develop some results from harmonic analysis related to thedifferential-difference operator T (α,β). Further details can be found in [1] and [2].In the following we fix parameters α, β subject to the constraints α ≥ β ≥ − 12 andα > −12 .Let ρ = α+β+1 and λ ∈ C. The Opdam hypergeometric functions G(α,β)λ on R areeigenfunctions T (α,β)G(α,β)λ (x) = iλG(α,β)λ (x) of the differential-difference operatorT (α,β)f(x) = f ′(x) + [(2α+ 1) cothx+ (2β + 1) tanhx]f(x)− f(−x)2− ρf(−x),that are normalized such that G(α,β)λ (0) = 1. In the notation of Cherednik onewould write T (α,β) asT (k1+ k2)f(x) = f′(x)+{2k11 + e−2x+4k21− e−4x}(f(x)− f(−x))− (k1 +2k2)f(x),with α = k1+k2−12 and β = k2−12 . Here k1 is the multiplicity of a simply positiveroot and k2 the (possibly vanishing) multiplicity of a multiple of this root. By [1]or [2], the eigenfunction G(α,β)λ is given byG(α,β)λ (x)=ϕα,βλ (x)−1ρ− iλ∂∂xϕα,βλ (x)=ϕα,βλ (x) +ρ4(α+ 1)sinh(2x)ϕα+1,β+1λ (x),where ϕα,βλ (x) =2 F1(ρ+iλ2 ;ρ−iλ2 ;α+ 1;− sinh2 x) is the classical Jacobi function.Lemma 1.1. [4] The following inequalities are valids for Jacobi functions ϕα,βλ (x)(i) |ϕα,βλ (x)| ≤ 1.(ii) 1− ϕα,βλ (x) ≤ x2(λ2 + ρ2).(iii) there is a constant c > 0 such that1− ϕα,βλ (x) ≥ c,for λx ≥ 1.Denote L2α,β(R), the space of measurable functions f on R such that‖f‖2,α,β =(∫R|f(x)|2Aα,β(x)dx)1/2< +∞,whereAα,β(x) = (sinh |x|)2α+1(cosh |x|)2β+1.The Cherednik-Opdam transform of f ∈ Cc(R) is defined byHf(λ) =∫Rf(x)G(α,β)λ (−x)Aα,β(x)dx for all λ ∈ C.(ψ, γ, 2)-Cherednik-Opdam Lipschitz Functions... 817The inverse transform is given asH−1g(x) =∫Rg(λ)G(α,β)λ (x)(1−ρiλ) dλ8pi|cα,β(λ)|2,herecα,β(λ) =2ρ−iλΓ(α+ 1)Γ(iλ)Γ(12 (ρ+ iλ))Γ(12 (α− β + 1 + iλ)).The corresponding Plancherel formula was established in [1], to the effect that∫R|f(x)|2Aα,β(x)dx =∫ +∞0(|Hf(λ)|2 + |Hfˇ(λ)|2)dσ(λ),where fˇ(x) := f(−x) and dσ is the measure given bydσ(λ) =dλ16pi|cα,β(λ)|2.According to [2] there exists a family of signed measures µ(α,β)x,y such that the productformulaG(α,β)λ (x)G(α,β)λ (y) =∫RG(α,β)λ (z)dµ(α,β)x,y (z)holds for all x, y ∈ R and λ ∈ C, wheredµ(α,β)x,y (z) =Kα,β(x, y, z)Aα,β(z)dz, if xy 6= 0dδx(z), if y = 0dδy(z), if x = 0andKα,β(x, y, z) =Mα,β | sinhx. sinh y. sinh z|−2α∫ pi0g(x, y, z, χ)α−β−1+× [1− σχx,y,z + σχx,z,y + σχz,y,x +ρβ + 12cothx. coth y. coth z(sinχ)2]× (sinχ)2βdχif x, y, z ∈ R\{0} satisfy the triangular inequality ||x| − |y|| < |z| < |x| + |y|, andKα,β(x, y, z) = 0 otherwise. Here∀x, y, z ∈ R, χ ∈ [0, 1], σχx,y,z =cosh x+cosh y−cosh z cosχsinhx sinh y , if xy 6= 00, if xy = 0and g(x, y, z, χ) = 1− cosh2 x− cosh2 y. cosh2 z + 2 coshx. cosh y. cosh z. cosχ.Lemma 1.2. [2] For all x, y ∈ R, we have(i) Kα,β(x, y, z) = Kα,β(y, x, z).(ii) Kα,β(x, y, z) = Kα,β(−x, z, y).(iii) Kα,β(x, y, z) = Kα,β(−z, y,−x).818 R. Daher and S. El OuadihThe product formula is used to obtain explicit estimates for the generalized trans-lation operatorsτ (α,β)x f(y) =∫Rf(z)dµ(α,β)x,y (z).It is known from [2] thatHτ (α,β)x f(λ) = G(α,β)λ (x)Hf(λ),(1.1)for f ∈ Cc(R).2. Main ResultIn this section we give the main result of this paper. We need first to define (ψ, γ, 2)-Cherednik-Opdam Lipschitz class.Denote Nh byNh = τ(α,β)h + τ(α,β)−h − 2I,where I is the unit operator in the space L2α,β(R).Definition 2.1. Let γ ≥ 0. A function f ∈ L2α,β(R) is said to be in the (ψ, γ, 2)-Cherednik-Opdam Lipschitz class, denoted by Lip(ψ, γ, 2), if‖Nhf(x)‖2,α,β = O(ψ(h)(log 1h )γ)as h→ 0,where(a) ψ is a continuous increasing function on [0,∞),(b) ψ(0) = 0 , ψ(ts) = ψ(t)ψ(s) for all t, s ∈ [0,∞),(c) and ∫ 1/h0sψ(s−2)(log s)−2γds = O(h−2ψ(h2)(log1h)−2γ), h→ 0.Lemma 2.1. If f ∈ Cc(R), thenHτˇ (α,β)x f(λ) = G(α,β)λ (−x)Hfˇ (λ).(2.1)Proof. For f ∈ Cc(R), we haveHτˇ (α,β)x f(λ) =∫Rτ (α,β)x f(−y)G(α,β)λ (−y)Aα,β(y)dy=∫Rτ (α,β)x f(y)G(α,β)λ (y)Aα,β(y)dy=∫R[∫Rf(z)Kα,β(x, y, z)Aα,β(z)dz]G(α,β)λ (y)Aα,β(y)dy=∫Rf(z)[∫RG(α,β)λ (y)Kα,β(x, y, z)Aα,β(y)dy]Aα,β(z)dz.(ψ, γ, 2)-Cherednik-Opdam Lipschitz Functions... 819Since Kα,β(x, y, z) = Kα,β(−x, z, y), it follows from the product formula thatHτˇ (α,β)x f(λ) = G(α,β)λ (−x)∫Rf(z)G(α,β)λ (z)Aα,β(z)dz= G(α,β)λ (−x)∫Rf(−z)G(α,β)λ (−z)Aα,β(z)dz= G(α,β)λ (−x)Hfˇ(λ).Lemma 2.2. For f ∈ L2α,β(R), then‖Nhf(x)‖22,α,β = 4∫ +∞0|ϕα,βλ (h)− 1|2(|Hf(λ)|2 + |Hfˇ(λ)|2)dσ(λ).Proof. From formulas (1.1) and (2.1), we haveH(Nhf)(λ) = (G(α,β)λ (h) +G(α,β)λ (−h)− 2)H(f)(λ),andH(Nˇhf)(λ) = (G(α,β)λ (−h) +G(α,β)λ (h)− 2)H(fˇ)(λ).SinceG(α,β)λ (h) = ϕα,βλ (h) +ρ4(α+ 1)sinh(2h)ϕα+1,β+1λ (h),and ϕα,βλ is even, thenH(Nhf)(λ) = 2(ϕα,βλ (h)− 1)H(f)(λ)andH(Nˇhf)(λ) = 2(ϕα,βλ (h)− 1)H(fˇ)(λ).Now by Plancherel Theorem, we have the result.Theorem 2.1. Let f ∈ L2α,β(R). Then the following are equivalents(a) f ∈ Lip(ψ, γ, 2),(b)∫ +∞r(|Hf(λ)|2 + |Hfˇ(λ)|2)dσ(λ) = O(ψ(r−2)(log r)2γ), as r →∞.Proof. (a)⇒ (b) Let f ∈ Lip(ψ, γ, 2). Then we have‖Nhf(x)‖2,α,β = O(ψ(h)(log 1h )γ)as h→ 0.From Lemma 2.2, we have‖Nhf(x)‖22,α,β = 4∫ +∞0|1− ϕα,βλ (h)|2(|Hf(λ)|2 + |Hfˇ(λ)|2)dσλ.820 R. Daher and S. El OuadihIf λ ∈ [ 1h ,2h ], then λh ≥ 1 and (iii) of Lemma 1.1 implies that1 ≤1c2|1− ϕα,βλ (h)|2.Then∫ 2h1h(|Hf(λ)|2 + |Hfˇ(λ)|2)dσ(λ) ≤1c2∫ 2h1h|1− ϕα,βλ (h)|2(|Hf(λ)|2 + |Hfˇ(λ)|2)dσ(λ)≤1c2∫ +∞0|1− ϕα,βλ (h)|2(|Hf(λ)|2 + |Hfˇ(λ)|2)dσ(λ)≤14c2‖Nhf(x)‖22,α,β= O(ψ(h2)(log 1h)2γ).We obtain ∫ 2rr(|Hf(λ)|2 + |Hfˇ(λ)|2)dσ(λ) ≤ Cψ(r−2)(log r)2γ, r →∞,where C is a positive constant. Now,∫ +∞r(|Hf(λ)|2 + |Hfˇ(λ)|2)dσ(λ) =∞∑i=0∫ 2i+1r2ir(|Hf(λ)|2 + |Hfˇ(λ)|2)dσ(λ)≤ C(ψ(r−2)(log r)2γ+ψ((2r)−2)(log 2r)2γ+ψ((4r)−2)(log 4r)2γ+ · · ·)≤ Cψ(r−2)(log r)2γ(1 + ψ(2−2) + (ψ(2−2))2 + (ψ(2−2))3 + · · ·)≤ Kψψ(r−2)(log r)2γ,where Kψ = C(1 − ψ(2−2))−1 since ψ(2−2) < 1.Consequently∫ +∞r(|Hf(λ)|2 + |Hfˇ(λ)|2)dσ(λ) = O(ψ(r−2)(log r)2γ), as r →∞.(b)⇒ (a). Suppose now that∫ +∞r(|Hf(λ)|2 + |Hfˇ(λ)|2)dσ(λ) = O(ψ(r−2)(log r)2γ), as r →∞,and write‖Nhf(x)‖22,α,β = 4(I1 + I2),(ψ, γ, 2)-Cherednik-Opdam Lipschitz Functions... 821whereI1 =∫ 1h0|1− ϕα,βλ (h)|2(|Hf(λ)|2 + |Hfˇ(λ)|2)dσλ,andI2 =∫ +∞1h|1− ϕα,βλ (h)|2(|Hf(λ)|2 + |Hfˇ(λ)|2)dσλ.Firstly, we use the formula |ϕα,βλ (h)| ≤ 1 andI2 ≤ 4∫ +∞1h(|Hf(λ)|2 + |Hfˇ(λ)|2)dσ(λ) = O(ψ(h2)(log 1h)2γ), as h→ 0.To estimate I1, we use the inequalities (i) and (ii) of Lemma 1.1I1 =∫ 1h0|1− ϕα,βλ (h)|2(|Hf(λ)|2 + |Hfˇ(λ)|2)dσλ≤ 2∫ 1h0|1− ϕα,βλ (h)|(|Hf(λ)|2 + |Hfˇ(λ)|2)dσλ≤ 2h2∫ 1h0(λ2 + ρ2)(|Hf(λ)|2 + |Hfˇ(λ)|2)dσλ.Now, we apply integration by parts for a functionφ(s) =∫ +∞s(|Hf(λ)|2 + |Hfˇ(λ)|2)dσ(λ)to getI1 ≤ −2h2∫ 1/h0(s2 + ρ2)φ′(s)ds≤ −2h2∫ 1/h0s2φ′(s)ds≤ h2(−1h2φ(1h) + 2∫ 1/h0sφ(s)ds)≤ −φ(1h) + 2h2∫ 1/h0sφ(s)ds≤ 2h2∫ 1/h0sφ(s)ds.Since φ(s) = O(ψ(s−2)(log s)2γ), we have sφ(s) = O(sψ(s−2)(log s)2γ)and∫ 1/h0sφ(s)ds = O(∫ 1/h0sψ(s−2)(log s)2γds)= O(h−2ψ(h2)(log 1h )2γ),822 R. Daher and S. El Ouadihso thatI1 = O(ψ(h2)(log 1h )2γ).Consequently,‖Nhf(x)‖2,α,β = O(ψ(h)(log 1h )γ)as h→ 0,and this ends the proof of the theorem.3. ConclusionIn this work we have succeded to generalise the theorem in [3] for the Cherednik-Opdam transform in the space L2α,β(R). We proved that f(x) belong to Lip(ψ, γ, 2).Then ∫ +∞r(|Hf(λ)|2 + |Hfˇ(λ)|2)dσ(λ) = O(ψ(r−2)(log r)2γ), as r →∞.AcknowledgementsThe authors would like to thank the referee for his valuable comments and sugges-tions.REFERENCES1. E. M. Opdam, Harmonic analysis for certain representations of graded Heckealgebras, Acta Math. Vol. 175, no. 1, (1995), 75-121.2. J. P. Anker, F. Ayadi and M. Sifi, Opdams hypergeometric functions: productformula and convolution structure in dimension 1, Adv. Pure Appl. Math. Vol. 3,no. 1, (2012), 11-44.3. M. S. Younis, Fourier transforms of Dini-Lipschitz functions. Int. J. Math. Math.Sci. Vol. 9, no. 2,(1986), 301-312. doi:10.1155/S0161171286000376.4. S. S. Platonov, Approximation of functions in L2-metric on noncompact rank1 symmetric space. Algebra Analiz . Vol. 11, no. 1, (1999), 244-270.5. L. N. Mishra, V. N. Mishra, K. Khatri and Deepmala, On the trigonometricapproximation of signals belonging to generalized weighted Lipschitz W (Lr, ξ(t)),(r ≥ 1) class by matrix (C1.Np) Operator of conjugate series of its Fourier series,Applied Mathematics and Computation, Vol. 237, (2014), 252-263.6. V. N. Mishra, K. Khatri, L. N. Mishra and Deepmala; Trigonomet-ric approximation of periodic signals belonging to generalized weighted LipschitzW ′(Lr, ξ(t)), (r ≥ 1)− class by No¨rlund-Euler (N, pn)(E, q) operator of conjugateseries of its Fourier series, Journal of Classical Analysis, Vol. 5, no. 2 (2014),91-105. doi:10.7153/jca-05-08.(ψ, γ, 2)-Cherednik-Opdam Lipschitz Functions... 8237. V. N. Mishra and L. N. Mishra, Trigonometric approximation of signals (func-tions) in Lp(p ≥ 1)-norm, International Journal of Contemporary MathematicalSciences, Vol. 7, no. 19, (2012), 909-918.8. A. Abouelaz, R. Daher and M. El Hamma, Generalization of Titchmarshstheorem for the Jacobi transform, Ser. Math. Inform. Vol. 28, no. 1, (2013), 43-51.Radouan DaherDepartment of MathematicsFaculty of Sciences Aı¨n ChockUniversity Hassan II, Casablanca, Moroccorjdaher024@gmail.comSalah El OuadihDepartment of MathematicsFaculty of Sciences Aı¨n ChockUniversity Hassan II, Casablanca, Moroccosalahwadih@gmail.com

( ;  ; 2)-CHEREDNIK-OPDAM LIPSCHITZ FUNCTIONS IN THE SPACE L2 ;(R)

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( ; ; 2)-CHEREDNIK-OPDAM LIPSCHITZ FUNCTIONS IN THE SPACE L2 ;(R)

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