In this paper, we obtain an analog of Youniss Theorem 5.2 in [5] forthe generalized Fourier-Bessel transform on the real line for functions satisfying the Fourier-Bessel Lipschitz condition in the space L

Daher, Radouan

El Hamma, Mohamed

El Ouadih, Salah

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

FACTA UNIVERSITATIS (NISˇ)Ser. Math. Inform. Vol. 31, No 2 (2016), 363–372(η, γ)-FOURIER-BESSEL LIPSCHITZ FUNCTIONSIN THE SPACE Lpα,nRadouan Daher, Salah El Ouadih and Mohamed El HammaAbstract. In this paper, we obtain an analog of Theorem 5.2 in Younis [5] for thegeneralized Fourier-Bessel transform on the real line for functions satisfying the (η, γ)-Fourier-Bessel Lipschitz condition in the space Lpα,n,1 < p ≤ 2.Keywords: Singular differential operator, Generalized Fourier-Bessel transform, Gen-eralized translation operator1. Introduction and PreliminariesVarious investigators such as Mittal and Mishra [7], Mishra et al. [8]-[12] andMishra and Mishra [3] have determined the degree of approximation of 2pi-periodicsignals (functions) belonging to various classes Lipα, Lip(α, r), Lip(ξ(t), r) andW (Lr, ξ(t)), (r ≥ 1), of functions through trigonometric Fourier approximation us-ing different summability matrices with monotone rows. In this direction, Theorem5.2 of Younis [5] characterized the set of functions in L2(R) satisfying the Dini Lip-schitz condition by means of an asymptotic estimate growth of the norm of theirFourier transforms, namely, we haveTheorem 1.1. ([5]) Let f ∈ L2(R). Then, the following statements are equivalent(a) ‖f(x+ h)− f(x)‖ = O(hη(log 1h)γ), as h→ 0, 0 < η < 1, γ ≥ 0,(b)∫|λ|≥r|f̂(λ)|2dλ = O(r−2η(log r)2γ), as r →∞,where f̂ stands for the Fourier transform of f .In this paper, we consider a second-order singular differential operator B on the half-line which generalizes the Bessel operator Bα, we obtain an analog of Theorem 1.1 forthe generalized Fourier-Bessel transform associated to B in Lpα,n,1 < p ≤ 2. For thispurpose, we use a generalized translation operator. Some interesting applicationsReceived July 15, 2015; accepted November 20, 20152010 Mathematics Subject Classification. Primary 42B37; Secondary 47B48363364 R. Daher, S. El Ouadih and M. El Hammaof manuscript can be seen in ([3],[9]-[14]).We briefly overview the theory of generalized Fourier-Bessel transform and relatedharmonic analysis (see [1],[6]):Consider the second-order singular differential operator on the half line defined byBf(x) =d2f(x)dx2+(2α+ 1)xdf(x)dx−4n(α+ n)x2f(x),where α > −12 and n = 0, 1, 2, .... For n = 0, we obtain the classical Bessel operatorBαf(x) =d2f(x)dx2+(2α+ 1)xdf(x)dx.Let M be the map defined byMf(x) = x2nf(x), n = 0, 1, ..Let Lpα,n, 1 < p ≤ 2, be the class of measurable functions f on [0,∞[ for which‖f‖p,α,n = ‖M−1f‖p,α+2n <∞,where‖f‖p,α =(∫ ∞0|f(x)|px2α+1dx)1/p.If p = 2, then we have L2α,n = L2([0,∞[, x2α+1dx).For α ≥ −12 , we introduce the Bessel normalized function of the first kind jα definedbyjα(z) = Γ(α+ 1)∞∑n=0(−1)n( z2 )2nn!Γ(n+ α+ 1), z ∈ C,(1.1)where Γ(x) is the gamma-function (see [4]). The function y = jα(z) satisfies thedifferential equationBαy + y = 0,with the initial conditions y(0) = 0 and y′(0) = 0. The function jα(z) is infinitelydifferentiable, even, and, moreover entire analytic.From (1.1) we see thatlimz→0jα(z)− 1z26= 0,(1.2)hence, there exists c > 0 and ν > 0 satisfying|z| ≤ ν ⇒ |jα(z)− 1| ≥ c|z|2.(1.3)For λ ∈ C, and x ∈ R, putϕλ(x) = x2njα+2n(λx).(1.4)From [1], [6] recall the following properties.(η, γ)-Fourier-Bessel Lipschitz Functions in Lpα,n 365Proposition 1.1. (c) ϕλ satisfies the differential equationBϕλ = −λ2ϕλ.(d) For all λ ∈ C, and x ∈ R|ϕλ(x)| ≤ x2ne|Imλ||x|.The generalized Fourier-Bessel transform that we call it the integral transform de-fined byFBf(λ) =∫ ∞0f(x)ϕλ(x)x2α+1dx, λ ≥ 0, f ∈ L1α,n.From [1], we haveFB = Fα+2noM−1,(1.5)where Fα is the Bessel transform defined by formula (see [4], [2])Fα(f)(λ) =∫ ∞0f(x)jα(λx)x2α+1dx.Let f ∈ L1α,n such that FB(f) ∈ L1α+2n = L1([0,∞[, x2α+4n+1dx). Then the inversegeneralized Fourier-Bessel transform is given by the formula (see [1])f(x) =∫ ∞0FBf(λ)ϕλ(x)dµα+2n(λ),wheredµα+2n(λ) = aα+2nλ2α+4n+1dλ, aα =14α(Γ(α+ 1))2.From [1], [6] we haveProposition 1.2. (e) For every f ∈ L1α,n ∩ L2α,n we have the Plancherel formula∫ +∞0|f(x)|2x2α+1dx =∫ +∞0|FBf(λ)|2dµα+2n(λ).(f) The generalized Fourier-Bessel transform FB extends uniquely to an isometricisomorphism from L2α,n onto L2([0,+∞[, µα+2n).We have the Young inequality‖Fα(f)‖q,α ≤ K‖f‖p,α(1.6)where 1p +1q = 1 and K is positive constant.From (1.5) and (1.6) we have‖FB(f)‖q,α+2n ≤ K‖f‖p,α,n(1.7)366 R. Daher, S. El Ouadih and M. El HammaDefine the generalized translation operator T h, h ≥ 0 by the relationT hf(x) = (xh)2nτhα+2n(M−1f)(x), x ≥ 0,where τhα+2n is the Bessel translation operators of order α+ 2n defined byτhαf(x) = cα∫ pi0f(√x2 + h2 − 2xh cos t) sin2α tdt,wherecα =(∫ pi0sin2α tdt)−1=Γ(α+ 1)Γ(pi)Γ(α + 12 ).For f ∈ Lpα,n, we haveFB(Thf)(λ) = ϕλ(h)FB(f)(λ),(1.8)FB(Bf)(λ) = −λ2FB(f)(λ),(1.9)(see [1],[6] for details).Let f ∈ Lpα,n. We define the differences of the orders k(k = 1, 2, ..) with a steph > 0 by∆khf(x) = (Th − h2nI)kf(x),(1.10)where I is the unit operator in Lpα,n.Let W kp,α,n, 1 < p ≤ 2, be the Sobolev space constructed by the singular differentialoperator B, i.e.,W kp,α,n ={f ∈ Lpα,n,Bmf ∈ Lpα,n,m = 1, 2, ..., k}.2. Fourier-Bessel Dini Lipschitz ConditionDefinition 2.1. Let f ∈W kp,α,n, and define‖∆khBmf(x)‖p,α,n ≤ Chη+2nk(log 1h )γ, γ ≥ 0i.e.,‖∆khBmf(x)‖p,α,n = O(hη+2nk(log 1h )γ),for all x in R+ and for all sufficiently small h,C being a positive constant. Thenwe say that f satisfies a Fourier-Bessel Dini Lipschitz of order η, or f belongs toLip(η, γ, p).(η, γ)-Fourier-Bessel Lipschitz Functions in Lpα,n 367Definition 2.2. If however‖∆khBmf(x)‖p,α,nhη+2nk(log 1h)γ→ 0, as h→ 0, γ ≥ 0i.e.,‖∆khBmf(x)‖p,α,n = O(hη+2nk(log 1h )γ),then f is said to be belong to the little Fourier-Bessel Dini Lipschitz class lip(η, γ, p).Remark. It follows immediately from these definitions thatlip(η, γ, p) ⊂ Lip(η, γ, p).Theorem 2.1. Let η > 1. If f ∈ Lip(η, γ, p), then f ∈ lip(1, γ, p).Proof. For x ∈ R+ and h small, f ∈ Lip(η, γ, p) we have‖∆khBmf(x)‖p,α,n ≤ Chη+2nk(log 1h )γ.Then(log1h)γ‖∆khBmf(x)‖p,α,n ≤ Chη+2nk.Therefore(log 1h )γh1+2nk‖∆khBmf(x)‖p,α,n ≤ Chη−1,which tends to zero with h→ 0. Thus(log 1h )γh1+2nk‖∆khBmf(x)‖p,α,n → 0, h→ 0.Then f ∈ lip(1, γ, p).Theorem 2.2. If η < ν, then Lip(η, 0, p) ⊃ Lip(ν, 0, p) and lip(η, 0, p) ⊃ lip(ν, 0, p).Proof. We have 0 ≤ h ≤ 1 and η < ν, then hν ≤ hη.Then the proof of the theorem is immediate.3. New Results on Fourier-Bessel Dini Lipschitz ClassLemma 3.1. For f ∈W kp,α,n, we have(h2qnk∫ ∞0λ2qm|jα+2n(λh)− 1|qk|FBf(λ)|qdµα+2n(λ)) 1q≤ K‖∆khBmf(x)‖p,α,n,where 1p +1q = 1 and m = 0, 1, 2..., k.368 R. Daher, S. El Ouadih and M. El HammaProof. From formula (1.8), we obtainFB(Bmf)(λ) = (−1)mλ2mFBf(λ);m = 0, 1, ...(3.1)By using the formulas (1.4), (1.8) and (3.1), we conclude that(3.2) FB(ThBmf)(λ) = (−1)mh2njα+2n(λh)λ2mFBf(λ).From the definition of finite difference (1.10) and formula (3.2), the image ∆khBrf(x)under the generalized Fourier-Bessel transform has the formFB(∆khBmf)(λ) = (−1)mh2nk(jα+2n(λh)− 1)kλ2mFBf(λ).Now by formula (1.7), we have the result.Theorem 3.1. Let η > 2k. If f belongs to the Fourier-Bessel Dini Lipschitz class,i.e.,f ∈ Lip(η, γ, p), η > 2k, γ ≥ 0.Then f is equal to the null function in R+.Proof. Assume that f ∈ Lip(η, γ, p). Then we have‖∆khBmf(x)‖p,α,n ≤ Chη+2nk(log 1h )γ, γ ≥ 0.From Lemma 3.1, we geth2qnk∫ ∞0λ2qm|jα+2n(λh)− 1|qk|FBf(λ)|qdµα+2n(λ) ≤ KqCqhqη+2qnk(log 1h )qγ.Therefore∫ ∞0λ2qm|jα+2n(λh)− 1|qk|FBf(λ)|qdµα+2n(λ) ≤ KqCqhqη(log 1h )qγ.Then ∫ ∞0λ2qm|jα+2n(λh)− 1|qk|FBf(λ)|qdµα+2n(λ)h2qk≤ KqCqhqη−2qk(log 1h )qγ,Since η > 2k we havelimh→0hqη−2qk(log 1h )qγ= 0.Thuslimh→0∫ ∞0(|1− jα(λh)|λ2h2)qkλ2qk+2qm|FBf(λ)|qdµα+2n(λ) = 0.and also from the formula (1.2) and Fatou theorem, we obtain∫ ∞0λ2qk+2qm |FBf(λ)|qdµα+2n(λ) = 0.Hence λ2k+2mFBf(λ) = 0 for all λ ∈ R+, and so f(x) is the null function.(η, γ)-Fourier-Bessel Lipschitz Functions in Lpα,n 369Analog of the Theorem 3.1, we obtain this theorem.Theorem 3.2. Let f ∈ W kp,α,n. If f belong to lip(2, 0, p), i.e.,‖∆khBmf(x)‖p,α,n = O(h2), as h→ 0,Then f is equal to null function in R+.Now, we give another main result of this paper analog of Theorem 1.1.Theorem 3.3. Let f belong to Lip(η, γ, p). Then∫ ∞rλ2qm|FBf(λ)|qdµα+2n(λ) = O(r−qη(log r)qγ), as r →∞where 1p +1q = 1 and m = 0, 1, 2, ..., k.Proof. Let f ∈ Lip(η, γ, p) . Then‖∆khBmf(x)‖p,α,n = O(hη+2nk(log 1h )γ), as h→ 0From Lemma 3.1, we have∫ ∞0λ2qm|jα+2n(λh) − 1|qk|FBf(λ)|qdµα+2n(λ) ≤Kqh2qnk‖∆khBmf(x)‖qp,α,n.By formula (1.3), we get∫ νhν2hλ2qm|jα+2n(λh)−1|qk|FBf(λ)|qdµα+2n(λ) ≥cqkν2qk22qk∫ νhν2hλ2qm|FBf(λ)|qdµα+2n(λ).Note that there exists a positive constant C such that∫ νhν2hλ2qm|FBf(λ)|qdµα+2n(λ) ≤ C∫ νhν2hλ2qm|jα+2n(λh)− 1|qk|FBf(λ)|qdµα+2n(λ)≤CKqh2qnk‖∆khBmf‖qp,α,n= O(hqη(log 1h )qγ).So we obtain ∫ 2rrλ2qm|FBf(λ)|qdµα+2n(λ) ≤ C′ r−qη(log r)qγ370 R. Daher, S. El Ouadih and M. El Hammawhere C′ is a positive constant. Now, we have∫ ∞rλ2qm|FBf(λ)|qdµα+2n(λ) =∞∑i=0∫ 2i+1r2irλ2qm|FBf(λ)|qdµα+2n(λ)≤ C′(r−qη(log r)qγ+(2r)−qη(log 2r)qγ+(4r)−qη(log 4r)qγ+ · · ·)≤ C′(r−qη(log r)qγ+(2r)−qη(log r)qγ+(4r)−qη(log r)qγ+ · · ·)≤ C′r−qη(log r)qγ(1 + 2−qη + (2−qη)2 + (2−qη)3 + · · ·)≤ Kηr−qη(log r)qγ,where Kη = C′(1− 2−qη)−1 since 2−qη < 1.Consequently∫ ∞rλ2qm|FBf(λ)|qdµα+2n(λ) = O(r−qη(log r)qγ), as r →∞.Corollary 3.1. Let f ∈ W kp,α,n. If‖∆khBmf(x)‖p,α,n = O(hη+2nk(log 1h )γ), as h→ 0,then ∫ ∞r|FBf(λ)|qdµα+2n(λ) = O(r−2qm−qη(log r)qγ), as r →∞,where 1p +1q = 1 and m = 0, 1, 2..., k.AcknowledgementsThe authors would like to thank the referee for his valuable comments and sugges-tions.REFERENCES1. R. F. Al Subaie and M. A. Mourou: The continuous wavelet transform for aBessel type operator on the half line, Mathematics and Statistics 1(4): 196-203,2013.2. S. M. Nikol’skii: Aproximation of functions of several variables and embeddingtheorems, (Nauka,Moscow.1977)[in Russian].3. 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, pp. 909-918.(η, γ)-Fourier-Bessel Lipschitz Functions in Lpα,n 3714. B. M. Levitan: Expansion in Fourier series and integrals over Bessel functions,Uspekhi Math.Nauk, 6,No.2,102-143,1951.5. M. S. Younis: Fourier transforms of Dini-Lipschitz functions, Int. J. Math.Math. Sci. 1986, 9 (2), 301312. doi:10.1155/S01611712860003766. R. F. Al Subaie and M. A. Mourou: Transmutation operators associated witha Bessel type operator on the half line and certain of their applications, Tamsii.oxf. J. Inf. Math. Scien 29 (3) (2013), pp. 329-349.7. M. L. Mittal and V. N. Mishra: Approximation of signals (functions) belongingto the Weighted W (Lp, ξ(t)), (p ≥ 1)-Class by almost matrix summability methodof its Fourier series, Int. J. of Math. Sci. and Engg. Appls. Vol. 2 (2008), No. IV,1- 9.8. V. N. Mishra, K. Khatri and L. N. 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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 Fourierseries, Applied Mathematics and Computation, Vol. 237 (2014) 252-263.13. 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 conju-gate series of its Fourier series, Journal of Classical Analysis, Volume 5, Number2(2014), 91-105. doi:10.7153/jca-05-08.14. Deepmala, L. N. Mishra and V. N. Mishra: Trigonometric approximationof signals (functions) belonging to the W (Lr, ξ(t)), (r ≥ 1)− class by (E, q)(q >0)- means of the conjugate series of its Fourier series, GJMS Special Issue forRecent Advances in Mathematical Sciences and Applications-13, Global Journalof Mathematical Sciences, Vol 2. No. 2, pp. 61-69, (2014).Radouan DaherDepartment of Mathematics and InformaticsFaculty of Science Aı¨n ChockUniversity Hassan II, Casablanca, Moroccorjdaher024@gmail.com372 R. Daher, S. El Ouadih and M. El HammaSalah El OuadihDepartment of Mathematics and InformaticsFaculty of Science Aı¨n ChockUniversity Hassan II, Casablanca, Moroccosalahwadih@gmail.comMohamed EL HammaDepartment of Mathematics and InformaticsFaculty of Science Aı¨n ChockUniversity Hassan II, Casablanca, Moroccom-elhamma@yahoo.fr

FOURIER-BESSEL LIPSCHITZ FUNCTIONS IN THE SPACE LP

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FOURIER-BESSEL LIPSCHITZ FUNCTIONS IN THE SPACE LP

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