Tow estimates for the Generalized Fourier-Bessel Transform in the Space L 2 α,n

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1 Palestine Journal of Mathematics Vol. 6(1)(17), Palestine Polytechnic University-PPU 17 Tow estimates for the Generalized Fourier-Bessel Transform in the Space L α,n R. Daher, S. El ouadih and M. EL hamma Communicated by Ayman Badawi MSC 1 Classications: Primary 4B1; Secondary 4B37. Keywords and phrases: Singular differential operator; Generalized Fourier-Bessel transform; Generalized translation operator. L α,n Abstract. Two estimates are proved for the generalized Fourier-Bessel transform in the space on certain classes of functions characterized by the generalized continuity modulus. 1 Introduction In [5], Abilov et al. proved two estimates for the Fourier transform in the space of square integrable functions on certain classes of functions characterized by the generalized continuity modulus, using a translation operator. In this paper, we consider a second-order singular differential operator B on the half line which generalizes the Bessel operator B α, we prove two estimates in certain classes of functions characterized by a generalized continuity modulus and connected with the generalized Fourier-Bessel transform associated to B in L α,n analogs of the statements proved in [5]. For this purpose, we use a generalized translation operator. In section, we give some denitions and preliminaries concerning the generalized Fourier- Bessel transform. Two estimates are proved in section 3. Preliminaries on the generalized Fourier-Bessel transform Integral transforms and their inverses the Bessel transform are widely used to solve various problems in calculus, mechanics, mathematical physics, and computational mathematics (see [3, 4]). We briey overview the theory of generalized Fourier-Bessel transform and related harmonic analysis (see [1, 6]) Consider the second-order singular differential operator on the half line dened by where α > 1 Let M be the map dened by Bf(x) = d f(x) dx + (α + 1) df(x) x dx 4n(α + n) f(x), x and n =, 1,,... For n =, we obtain the classical Bessel operator B α f(x) = d f(x) dx + (α + 1) df(x) x dx. Mf(x) = x n f(x), n =, 1,.. Let L p α,n, 1 p <, be the class of measurable functions f on [, [ for which f p,α,n = M 1 f p,α+n <, where ( 1/p f p,α = f(x) p x dx) α+1.

2 196 R. Daher, S. El ouadih and M. EL hamma If p =, then we have L α,n = L ([, [, x α+1 ). For α > 1, we introduce the normalized spherical Bessel function j α dened by j α (x) = α G(α + 1)J α (x) x α, (.1) where J α (x)is the Bessel function of the rst Kind and G(x) is the gamma-function (see [7]). The function y = j α (x) satises the differential equation B α y + y = with the condition initial y() = and y () =. The function j α (x) is innitely differentiable, even and moreover entire analytic. In the terms of j α (x), we have (see []) 1 j α (x) = O(1), x 1. (.) 1 j α (x) = O(x ), x 1. (.3) hxjα (hx) = O(1), hx. (.4) For λ C, and x R, put φ λ (x) = x n j α+n (λx). (.5) From [1, 6] recall the following properties. Proposition.1. (1) φ λ satises the differential equation Bφ λ = λ φ λ. () For all λ C, and x R φ λ (x) x n e Imλ x. The generalized Fourier-Bessel transform that we call it the integral transform dened by F B f(λ) = f(x)φ λ (x)x α+1 dx, λ, f L 1 α,n, (see [1]). Let f L 1 α,n such that F B(f) L 1 α+n = L1 ([, [, x α+4n+1 dx). Then the inverse generalized Fourier-Bessel transform is given by the formula f(x) = F B f(λ)φ λ (x)dµ α+n (λ), where dµ α+n (λ) = a α+n λ α+4n+1 dλ, a α = 1 4 α (G(α + 1)), (see [1]). Proposition.. [1] (1) For every f L 1 α,n L α,n we have the Plancherel formula + f(x) x α+1 dx = + F B f(λ) dµ α+n (λ). () The generalized Fourier-Bessel transform F B extends uniquely to an isometric isomorphism from L α,n onto L ([, + [, µ α+n ).

3 Generalized Fourier-Bessel Transform 197 Dene the generalized translation operator T h, h by the relation T h f(x) = (xh) n τα+n(m h 1 f)(x), x, where τα+n h is the Bessel translation operators of order α + n dened by π ταf(x) h = c α f( x + h xh cos t) sin α tdt, where ( π c α = ) 1 sin α tdt = G(α + 1) G(π)G(α + 1 ). For f L α,n, we have F B (T h f)(λ) = φ λ (h)f B (f)(λ), (.6) F B (Bf)(λ) = λ F B (f)(λ), (.7) (see [1, 6] for details). Let f L α,n. The quantity w(f, δ),α,n = sup T h f(x) h n f(x),α,n, δ >, <hδ is called the modulus of continuity of the function f. Let W,ϕ r (B), r =, 1,..., denote the class of functions f L α,n that have generalized derivatives satisfying the estimate ω(b r f, δ),α,n = O(ϕ(δ)), δ, where ϕ(x) is any nonnegative function given on [, ), and B f = f, B r f = B(B r 1 f), r = 1,,... i.e., W r,ϕ(b) = {f L α,n, B r f L α,n and ω(br f, δ),α,n = O(ϕ(δ)), δ }. 3 Main Results The goal of this work is to prove several new estimates for the integral J (f) = in certain classes of functions in L α,n. Lemma 3.1. For f W,ϕ r (B), we have F B f(λ) dµ α+n (λ), T h B r f(x) h n B r f(x),α,n = h 4n λ 4r j α+n (λh) 1 F B f(λ) dµ α+n (λ). where r =, 1,,... Proof. From formula (.7), we obtain By using the formula (.6), we conclude that F B (B r f)(λ) = ( 1) r λ r F B f(λ); r =, 1,... (3.1) F B (T h f h n f)(λ) = h n (j α+n (λh) 1)F B f(λ). (3.) ow by formulas (3.1), (3.) and Plancherel equality, we have the result.

4 198 R. Daher, S. El ouadih and M. EL hamma Theorem 3.. Given r and f W,ϕ r (B). Then there exist constant c > such that, for all >, J (f) = O( r+n ϕ(c/)). Proof. Firstly, we have J (f) j dµ + 1 j dµ, (3.3) with j = j p (λh), p = α + n and dµ = F B f(λ) dµ α+n (λ). The parameter h > will be chosen in an instant. In view of formulas (.1) and (.4), there exist a constant c 1 > such that j c 1 (λh) p 1. Then j dµ c 1 (h) p 1 J (f). Choose a constant c such that the number c 3 = 1 c 1 c p 1 is positif. Setting h = c / in the inequality (3.3), we have c 3 J (f) By Hölder inequality the second term in (3.4) satises 1 j dµ = From Lemma 3.1, we conclude that Therefore 1 j dµ. (3.4) 1 j.1.dµ ( ) 1/ ( ) 1/ 1 j dµ dµ ( 1/ λ 4r 1 j λ dµ) 4r J (f) ( 1/ r 1 j λ dµ) 4r J (f). 1 j λ 4r dµ h 4n T h B r f(x) h n B r f(x),α,n. 1 j dµ r h n T h B r f(x) h n B r f(x),α,n J (f). For f W r,ϕ (L) there exist a constant c 4 > such that For h = c /, we obtain Hence T h B r f(x) h n B r f(x),α,n c 4 ϕ(h). c 3 J (f) c n n r c 4 ϕ(c /)J (f). c n c 3 J (f) c 4 r+n ϕ(c /). for all >. The theorem is proved with c = c.

5 Generalized Fourier-Bessel Transform 199 Theorem 3.3. Let ϕ(t) = t ν, then where, r =, 1,...; < ν <. J (f) = O( r ν+n ) f W r,ϕ(b), Proof. We prove sufciency by using Theorem 3. let f W,ϕ r (B) then To prove necessity let i.e. J (f) = O( r ν+n ). J (f) = O( r ν+n ) F B f(λ) dµ α+n (λ) = O( 4r ν+4n ) It is easy to show, that there exists a function f L α,n such that Br f L α,n and B r f(x) = ( 1) r λ r F B f(λ)φ λ (x)dµ α+n (λ). (3.5) From formula (3.5) and Plancherel equality, we have T h B r f(x) h n B r f(x),α,n = h 4n λ 4r j α+n (λh) 1 F B f(λ) dµ α+n (λ). This integral is divided into two = + = I 1 + I, where = [h 1 ], We estimate them separately. From (.), we have the estimate I c 5 λ 4r F B f(λ) dµ α+n (λ) λ = c 5 l= +l λ +l+1 c 5 a l (u l u l+1 ), l= with a l = ( + l + 1) 4r and u l = λ +l For all integers m 1, the Abel transformation shows l= λ 4r F B f(λ) dµ α+n (λ) F B f(λ) dµ α+n (λ). m m a l (u l u l+1 ) = a u + (a l a l 1 )u l a m u m+1 a u + n (a l a l 1 ) u l, because a m u m+1. Moreover by the nite increments theorem, we have a l a l 1 4r( + l + 1) 4r 1 Furthermore by the hypothesis of f there exists c 6 > such that, for all > J (f) c 6 4r ν+4n,

6 R. Daher, S. El ouadih and M. EL hamma For 1, we have m ( (a l a l 1 ) u l c ) 4r m ( ν+4n + 4rc ) 4r 1 ( + l) 1 ν+4n + l m r c 6 ν+4n + 4r 4r 1 c 6 ( + l) 1 ν+4n. Finally, by the integral comparison test we have m ( + l) 1 ν+4n x 1 ν+4n dx = 1 ν 4n ν+4n. Letting m we see that, for r and ν >, there exists a constant c 7 such that, for all 1 and for h >, I c 7 ν+4n. ow, we estimate I 1. From formula (.3), we have I 1 c 8 h 4 λ 4r+4 F B f(λ) dµ α+n (λ) with v l = λ l λ 1 = c 8 h 4 l= l λ l+1 1 c 8 h 4 (l + 1) 4r+4 (v l v l+1 ), l= F B f(λ) dµ α+n (λ). λ 4r+4 F B f(λ) dµ α+n (λ) Using an Abel transformation and proceeding as with I we obtain ) 1 I 1 c 8 h (v 4 + ((l + 1) 4r+4 l 4r+4 )v l ( ) 1 c 8 h 4 v + (4r + 4)c 6 (l + 1) 4r+3 l 4r ν+4n, since v l c 6 l 4r ν+4n by hypothesis. From the inequality l + 1 l we conclude ( ) 1 I 1 c 8 h 4 v + c 9 l 3 ν+4n. As a consequence of a series comparison for µ 1 and µ < 1 we have the inequality, 1 µ l µ 1 < µ, for µ > and. If µ = 4 ν + 4n > for ν < then we obtain I 1 c 8 h 4 ( v + c 1 4 ν+4n) c 8 h 4 ( v + c 1 h 4+ν 4n), since 1/h. If h is sufciently small then v c 1 h 4+ν 4n. Then we have Combining the estimates for I 1 and I gives The necessity is proved. I 1 c 11 h ν 4n T h B r f(x) h n B r f(x),α,n = O(h ν ),

7 Generalized Fourier-Bessel Transform 1 References [1] R. F. Al Subaie and M. A. Mourou, The continuous wavelet transform for a Bessel type operator on the half line, Mathematics and Statistics 1(4): 196-3, (13). [] V. A. Abilov and F. V. Abilova, Approximation of functions by Fourier-Bessel sums, Izv. Vyssh. Uchebn. Zaved., Mat., o. 8, 3-9 (1). [3] V. S. Vladimirov, Equations of mathematical physics, Marcel Dekker, ew York,1971,auka, Moscow, (1976). [4] A. G. Sveshnikov, A.. Bogolyubov and V. V. Kratsov, Lectures on mathematical physics, auka, Moscow,(4)[in Russian]. [5] V. A. Abilov, F. V. Abilova and M. K. Kerimov, Some remarks concerning the Fourier transform in the space L (R) Zh. Vychisl. Mat. Mat. Fiz. 48, 939â AŞ945 (8) [Comput. Math. Math. Phys. 48, 885â AŞ891]. [6] R. F. Al Subaie and M. A. Mourou, Transmutation operators associated with a Bessel type operator on the half line and certain of their applications, Tamsii. oxf. J. Inf. Math. Scien 9 (3) (13), pp [7] B. M. Levitan, Expansion in Fourier series and integrals over Bessel functions, Uspekhi Math.auk, 6,o.,1-143,(1951). Author information R. Daher, S. El ouadih and M. EL hamma, Department of Mathematics, Faculty of Sciences Aïn Chock, University of Hassan II, Casablanca, Morocco. salahwadih@gmail.com Received: October 3, 15. Accepted: March, 16.

Direct and inverse theorems of approximation theory in L 2 (R d, w l (x)dx)

Direct and inverse theorems of approximation theory in L 2 (R d, w l (x)dx) MATEMATIKA, 2017, Volume 33, Number 2, 177 189 c Penerbit UTM Press. All rights reserve Direct an inverse theorems of approximation theory in L 2 (, w l (x)x) 1 aouan Daher, 2 Salah El Ouaih an 3 Mohame