Asia Pacific Journal of Mathematics, Vol. 4, No. 2 (2017), 110-115 ISSN 2357-2205
GENERALIZATION OF TITCHMARSH’S THEOREM FOR THE FOURIER TRANSFORM
MOHAMED EL HAMMA∗ AND RADOUAN DAHER
Department of Mathematics, Faculty of Sciences A¨ın Chock, University of Hassan II, Casablanca, Morocco
∗Corresponding author: m elhamma@yahoo.fr Received Jul 30, 2017
Abstract. Using a spherical mean operator, we obtain a generalization of Titchmarsh’s Theorem for the Fourier transform for functions satisfying theψ-Fourier Lipschitz condition in L2(Rn).
2010 Mathematics Subject Classification. 42B45.
Key words and phrases. Fourier transform; spherical mean operator.
1. Introduction and preliminaries
It is well know that integral Fourier transforms are widely used in mathematical physics.
Titchmarsh’s [5, Therem 85] characterized the set of functions in L2(R) satisfying the Cauchy Lipschitz condition by means of an asymptotic estimate growth of the norm of their Fourier transform:
Theorem 1.1. [5] Let α ∈ (0,1) and assume that f ∈ L2(R). Then the following are eguivalent:
(1) kf(t+h)−f(t)kL2(R) =O(hα) as h−→0, (2) R
|λ|≥r|g(λ)|2dλ =O(r−2α) as r−→ ∞.
where g stands for the Fourier transform of f.
In this paper, we prove the generalization of this Theorem 1.1 for the Fourier transforn in Rn for functions satisfying the ψ-Fourier Lipschitz class in the space L2(Rn). For this purpose, we use the spherical mean operator.
Recall that L2(Rn) is the space of integrable function f with the norm
kfk2 = Z
Rn
|f(x)|2dx 1/2
c2017 Asia Pacific Journal of Mathematics
The Fourier transform for a function f ∈L2(Rn) defined by
f(ξ) =b Z
Rn
f(x)e−ix.ξdx.
The inverse Fourier transform defined by formula
f(x) = Z
Rn
fb(ξ)eix.ξdξ.
The Plencherel equality [4] holds Z
Rn
|f(x)|2dx= Z
Rn
|f(ξ)|b 2dξ.
We denote jp(x) for the Bessel normalized function of the first kind, i.e
jp(x) = 2pΓ(p+ 1)Jp(x) xp
where Jp(x) is the Bessel function of the first kind and Γ(x) is the gamma-function(see[3]).
For p≥ −12, we introduce the Bessel normalized function of the first kind jp defined also by
(1) jp(x) = Γ(p+ 1)
∞
X
n=0
(−1)n n!Γ(n+p+ 1)
x 2
2n
Moreover, from (1) we see that
x−→0lim
jp(x)−1 x2 6= 0 consequently, there exist c >0 and η >0 satisfying
(2) |x| ≤η=⇒ |jp(x)−1| ≥c|x|2 In [1], we have
(3) |jp(x)| ≤1.
and we see that
(4) |1−jn−2
2 (|x|)| ≤ |x|, ∀x∈Rn. In L2(Rn) consider the spherical mean operator (see [2])
Mhf(x) = 1 wn−1
Z
Sn−1
f(x+hw)dw,
whereSn−1 is the unit sphere inRn,wn−1 its total surface measure with respect to the usual induced measure dw.
Proposition 1.2. Let f ∈L2(Rn), then
(M\hf)(ξ) =jn−2
2 (h|ξ|)fb(ξ)
Proof. (See [2, Proposition 4]).
2. Main Result
In this section we give the main result of this paper, We need first to define ψ-Fourier Lipschitz class.
Definition 2.1. A functionf ∈L2(Rn)is said to be in theψ-Fourier Lipschitz class, denoted by Lip(ψ, n,2), if
kMhf(x)−f(x)k2 =O(ψ(h)), as h−→0, where
(1) ψ(t) is a continuous increasing function on [0,∞), (2) ψ(0) = 0 and ψ(ts)≤ψ(t)ψ(s) for all t, s∈[0,∞) (3) R1/h
0 sψ(s−2)ds=O(h12ψ(h2)) as h −→0
Theorem 2.2. Let f ∈L2(Rn). Then the following are equivalent (1) f ∈Lip(ψ, n,2).
(2) R
|ξ|≥r|f(ξ)|b 2dξ=O(ψ(r−2)) as r−→+∞.
Proof. 1 =⇒2: Suppose thatf ∈Lip(ψ, n,2). Then we obtain
kMhf(x)−f(x)k2 =O(ψ(h)), as h−→0.
Plancherel’s identity and Proposition 1.2 give
kMhf(x)−f(x)k22 = Z
Rn
|1−jn−2
2 (|ξ|h)|2|fb(ξ)|2dξ.
From formula (2), we have
Z
η 2h≤|ξ|≤η
h
|1−jn−2
2 (|ξ|h)|2|fb(ξ)|2dξ ≥ c2η4 16
Z
η 2h≤|ξ|≤η
h
|f(ξ)|b 2dξ.
There exists then a positive constant C such that Z
η 2h≤|ξ|≤ηh
|f(ξ)|b 2dξ ≤ C Z
Rn
|1−jn−2
2 (|ξ|h)|2|f(ξ)|b 2dξ
≤ Cψ(h2).
We obtain
Z
r≤|ξ|≤2r
|f(ξ)|b 2dξ ≤Cψ(2−2η2r−2).
Thus there exists then a positive constant K such that Z
r≤|ξ|≤2r
|f(ξ)|b 2dξ ≤Kψ(r−2), f or all r >0.
So that Z
|ξ|≥r
|fb(ξ)|2dξ = Z
r≤|ξ|≤2r
+ Z
2r≤|ξ|≤4r
+ Z
4r≤|ξ|≤8r
....
|fb(ξ)|2dξ
= O(ψ(r−2) +ψ(2−2r−2)...)
≤ K(1 +ψ(2−1) +ψ2(2−1) +....)ψ(r−2) We have
Z
|ξ|≥r
|f(ξ)|b 2dξ ≤Cψ(r−2), where C =K(1−ψ(2−1)−1
This proves that
Z
|ξ|≥r
|fb(ξ)|2dξ =O(ψ(r−2)) as r −→+∞
2 =⇒1: Suppose now that Z
|ξ|≤r
|fb(ξ)|2dξ =O(ψ(r−2)) as r −→+∞
We have to show that Z ∞
0
sn−1|1−jn−2
2 (sh)|2φ(s)ds =O(ψ(h2)) as h−→0,
where
φ(s) = Z
Sn−1
|f(sy)|b 2dy.
We write
Z ∞
0
sn−1|1−jn−2
2 (sh)|2φ(s)ds= I1+ I2, where
I1 = Z 1h
0
sn−1|1−jn−2
2 (sh)|2φ(s)ds and
I2 = Z ∞
1 h
sn−1|1−jn−2
2 (sh)|2φ(s)ds We now estimate the summands I1 and I2.
Firstly, from (3) we have
I2 ≤4 Z ∞
1 h
sn−1φ(s)ds =O(ψ(h2) as h−→0 Set
ϕ(r) = Z ∞
r
sn−1φ(s)ds From (4), an integration by parts yields
I1 ≤ −h2 Z 1h
0
r2ϕ0(r)dr
≤ −ϕ(1
h) + 2h2 Z h1
0
rϕ(r)dr
≤ Ch2 Z 1h
0
rψ(r−2)dx
≤ Ch2 1 h2ψ(h2)
≤ Cψ(h2).
and this end the proof.
References
[1] V. A. Abilov and F. V. Abilova, Approximation of Functions by Fourier-Bessel Sums, Izv. Vyssh.
Uchebn. Zaved., Mat., 8 (2001), 3-9.
[2] W.O. Bray, M.A. PinskyGrowth properties of Fourier transforms via moduli of continuity , J. Funct.
Anal. 255 (2008), 2265-2285.
[3] B.M. Levitan,Expansion in Fourier series and integrals over Bessel, Uspekhi Mat. Nauk, 6 (2) (1951), 102-143.
[4] M. Plancherel,Contribution a l’etude de la representation d’une fonction arbitraire par des integrales definies, Rend. Circolo Mat. di Palermo 30 (1910), 289-335.
[5] E.C. Titchmarsh,Introduction to the theory of Fourier Integrals , Claredon, Oxford. 1948, Komkniga, Moscow. 2005.
