A.G. RAMM
LetG(k) =R1
0 g(x)ekxdx,g∈L1(0,1). The main result of this paper is
Theorem 1. There existsg6≡0, g∈C0∞(0,1), such thatG(kj) = 0, kj< kj+1,
j→∞lim kj = ∞, lim
k→∞|G(k)| does not exist, lim sup
k→+∞
|G(k)| = ∞. This g oscillates infinitely often in any interval[1−δ,1], however smallδ >0is.
AMS 2010 Subject Classification: 30D15, 42A38, 42A63.
Key words: Fourier integral, entire function, uniqueness theorem.
1. INTRODUCTION Let g∈ L1(0,1) andG(k) = R1
0 g(x)ekxdx. The function G is an entire function of the complex variable k. It satisfies for allk∈Cthe estimate (1) |G(k)| ≤c0e|k|, c0 :=
Z 1
0
|g(x)|dx, it is bounded for <k≤0
(2) |G(k)| ≤c0, <k≤0, and it is bounded on the imaginary axis
(3) sup
τ∈R
|G(iτ)| ≤c0, where Rdenotes the real axis.
It is not difficult to prove that ifgkeeps sign in any interval [1−δ,1], and R1
1−δ|g(x)|dx >0 then lim
k→∞|G(k)|=∞. By∞ we mean +∞ in this paper.
However, there exists ag 6≡0,g ∈C0∞(0,1), such that lim
k→∞|G(k)| does not exist, but lim sup
k→∞
|G(k)|= ∞. This g changes sign (oscillates) infinitely often in any interval [1−δ,1], however small δ >0 is, and the corresponding G(k) changes sign infinitely often in any neighborhood of +∞, i.e., it has infinitely many isolated positive zeros kj, which tend to ∞.
From the known results the following claim follows (see [1], [2]).
REV. ROUMAINE MATH. PURES APPL.,55(2010),6, 515–519
Claim. If c1 := lim sup
k→∞
|G(k)|<∞, then g= 0.
We prove this Claim for convenience of the reader.
Our main result is
Theorem 1. There exists g6≡0, g∈C0∞(0,1), such that lim sup
k→∞
|G(k)|=∞,
k→∞lim |G(k)| does not exist, G(kj) = 0, where kj < kj+1, lim
j→∞kj =∞, and g oscillates infinitely often in any interval [1−δ,1], however small δ >0 is.
In Section 2 proofs are given. The main tool in the proof of the Claim is a Phragmen-Lindel¨of (PL) theorem (see, e.g., [2], [5]). In the proof of Theorem 1 we use a construction, developed in [3], p. 282, (see also [4]), for a study of resonances in quantum scattering on a half line. The conclusion of the Claim that g= 0, one can also derive from a version of Paley-Wiener theorem given in [1], p. 36.
For convenience of the reader we formulate the Phragmen-Lindel¨of (PL) theorem, used in Section 2. A proof of this theorem can be found, e.g., in [2].
By ∂Qthe boundary of the set Qis denoted, M >0 is a constant, and 0< α <2π.
Theorem (PL). Let G(k) be holomorphic in an angle Q of opening πα and continuous up to its boundary. If sup
k∈∂Q
|G(k)| ≤ M, and the order ρ of G(k) is less than α, i.e.,
ρ:= lim sup
r→∞
ln ln max
|k|=r|G(k)|
lnr < α, then sup
k∈Q
|G(k)| ≤M.
In Section 2 Theorem (PL) will be used forα= π2 and ρ= 1< α.
2. PROOFS
The proof of Theorem 1 is based on two lemmas, and the conclusion of Theorem 1 follows immediately from these lemmas. The first lemma is identical with the Claim.
Lemma 1. If
(4) lim sup
k→∞
|G(k)|<∞,
then
(5) g= 0.
Proof. The entire function G in the first quadrant Q1 of the complex plane k, that is, in the region 0 ≤ argk ≤ π2,|k| ∈ [0,∞), satisfies estimate (1) and is bounded on the boundary of Q1
(6) sup
k∈∂Q1
|G(k)| ≤max(c0, c1) :=c2.
By a Phragmen-Lindel¨of theorem (see, e.g., [5], p. 276) one concludes that
(7) sup
k∈Q1
|G(k)| ≤c2. A similar argument shows that
(8) sup
k∈Qm
|G(k)| ≤c2, m= 1,2,3,4,
where Qm are quadrants of the complex k-plane. Consequently, |G(k)| ≤ c2 for all k ∈ C, and, by the Liouville theorem, G(k) = const. This const. is equal to zero, because, by the Riemann-Lebesgue lemma, lim
τ→∞G(iτ) = 0.
IfG= 0, theng= 0 by the injectivity of the Fourier transform. Lemma 1 and, therefore, Claim, are proved.
Lemma 2. There exists g6≡0, g∈C0∞(0,1), such that lim sup
k→∞
|G(k)|=∞,
k→∞lim |G(k)| does not exist, G(kj) = 0, where kj < kj+1, lim
j→∞kj =∞, and g oscillates infinitely often in any interval [1−δ,1], however small δ >0 is.
Proof. Let ∆j = [xj, xj+1], 0< xj < xj+1 <1, lim
j→∞xj = 1, j= 2,3, . . ., fj(x)≥0,fj ∈C0∞(∆j),R
∆jfj(x)dx= 1, andj >0,j+1 < j, be a sequence of numbers such that f(x)∈C0∞(0,1), where
(9) f(x) :=
∞
X
j=2
jfj(x).
Define
(10) g(x) :=
∞
X
j=2
(−1)jjfj(x), g6≡0.
The functiongoscillates infinitely often in any interval [1−δ,1], however small δ >0 is.
Let
(11) G(k) :=
∞
X
j=2
(−1)jjGj(k), Gj(k) :=
Z
∆j
fj(x)ekxdx.
Note thatGj(k)>0 for all j= 2,3, . . . and allk >0, and
(12) lim
k→∞
Gm(k)
Gj(k) =∞, m > j, because intervals ∆j and ∆m do not intersect if m > j.
Fix an arbitrary small number ω > 0. Let us construct a sequence of numbers
bj >0, bj+1> bj, lim
j→∞bj =∞, such that
(13) G(b2m)≥ω, G(b2m+1)≤ −ω, m= 1,2, . . . . This implies the existence of kj ∈(b2m, b2m+1), such that
G(kj) = 0, kj < kj+1, lim
j→∞kj =∞.
Moreover,
(14) lim sup
k→∞
|G(k)|=∞,
because otherwiseG= 0 by Lemma 1, sog= 0, contrary to the construction, see (10).
Let us chooseb2 >0 arbitrary. ThenG2(b2)>0. Making3 >0 smaller, if necessary, one gets
(15) 2G2(b2)−3G3(b2)≥ω.
Choosing b3 > b2 sufficiently large, one gets
(16) 2G2(b3)−3G3(b3)≤ −ω.
This is possible because of (12).
Suppose thatb2, . . . , b2m+1 are constructed, so that (17)
2m
X
j=2
(−1)jjGj(b2p)≥ω, p= 1,2, . . . , m, and
(18)
2m+1
X
j=2
(−1)jjGj(b2p+1)≤ −ω, p= 1, . . . , m.
Let m→ ∞. Then
(19) lim
m→∞
2m
X
j=2
(−1)jjGj(k) = lim
m→∞
2m+1
X
j=2
(−1)jjGj(k) =G(k),
whereG(k) is defined in (11), and the convergence in (19) is uniform on com- pact subsets of [0,∞). Therefore inequalities (13) hold. Lemma 2 is proved.
The conclusion of Theorem 1 follows from Lemmas 1 and 2.
Theorem 1 is proved.
REFERENCES
[1] P. Koosis,The Logarithmic Integral, Vol. I. Cambridge Univ. Press, Cambridge, 1988.
[2] G. Polya and G. Szeg¨o, Problems and Theorems in Analysis. Springer-Verlag, Berlin, 1983.
[3] A.G. Ramm,Multidimensional Inverse Scattering Problems.Longman/Wiley, New York, 1992.
[4] A.G. Ramm and B.A. Taylor,Example of a potential in one-dimensional scattering prob- lem for which there are infinitely many purely imaginary resonances. Phys. Lett. A124 (1987), 313–319.
[5] W. Rudin,Real and Complex Analysis. McGraw Hill, New York, 1974.
Received 10 September 2010 Kansas State University
Department of Mathematics Manhattan, KS 66506-2602, USA
ramm@math.ksu.edu
