PALEY-WIENER-TYPE THEOREM FOR A CLASS OF INTEGRAL TRANSFORMS ARISING FROM A SINGULAR DIRAC SYSTEM AHMED I. ZAYED Department of Mathematics University of Central Florida Orlando, FL 32816, U.S.A. E-mail address:
[email protected] and VU KIM TUAN Department of Mathematics and Computer Science Faculty of Science, Kuwait University P.O. Box 5969, Safat 13060, Kuwait E-mail address:
[email protected] Abstract A characterization of weighted L2 (I) spaces in terms of their images under various integral transformations is derived, where I is a finite interval. The class of integral transformations considered is related to certain singular Dirac systems on a half line.
1. INTRODUCTION The Paley-Wiener theorem [5] gives a characterization of the space L2 [−σ, σ] in terms of its image under the Fourier transformation by showing that a function f (x) is in L2 [−σ, σ] if and only if its Fourier transform fˆ(ω) can be continued analytically to the whole complex plane as an entire function of exponential type at most σ whose restriction to the real axis belongs to L2 (R). This characterization uses complex-variable techniques and as a result it does not lend itself very naturally to other more complicated integral transformations. Alternative approaches using real analysis techniques have been developed in the last few years to characterize the images of spaces of the form L2 [I, dρ], for some measure dρ and an interval I (finite or infinite), under various integral transformations, such as the Mellin [11], Hankel [10], Y [9], and Airy transforms [12]. One of the first and simplest of such results was discovered by H. Bang [2]. It can be rephrased as follows: let f ∈ C ∞ (R) be such that all its derivatives belong to L2 (R). Then 1/n
lim kf (n) k2
n→∞
n
= sup |y| :
o
y ∈ suppfˆ .
In particular, a function f is the Fourier transform of a square integrable function vanishing outside the interval [−σ, σ] if and only if 1/n
lim kf (n) k2
n→∞ 1
≤ σ.
1991 Mathematics Subject Classifications. Primary 44A15, 34B24; Secondary 42B10, 33C45 Key words and phrases. Paley-Wiener Theorem, Singular Dirac System.
1
In a recent paper [13], we proposed a unified approach to handle a large class of integral transforms that includes the Fourier sine and cosine- transforms, the Hankel, the Weber, the Kontorovich-Lebedev and the Jacobi transforms. This class of integral transforms arises from two types of singular Sturm-Liouville problems: singular on a half line and singular on the whole line. The main idea of this approach goes as follows: Let L be a differential operator, with a continuous spectrum Ω1 , and φ(x, λ) be an eigenfunction with corresponding eigenvalue λ: Lφ = −λφ. In addition, suppose that T : L2 (Ω1 ; dρ1 ) → L2 (Ω2 ; dρ2 ) Z
f (x) = (T F )(x) =
F (λ)φ(x, λ)dρ1 (λ) Ω1
is a unitary transformation Z
|f (x)|2 dρ2 (x) =
Ω2
Z
|F (λ)|2 dρ1 (λ).
Ω1
In that case, if λn F (λ) ∈ L2 (Ω1 ; dρ1 ) for any n = 0, 1, 2 · · · , we have Ln f (x) =
Z
(−λ)n F (λ)φ(x, λ)dρ1 (λ),
Ω1
and
Z
n
2
|L f (x)| dρ2 (x) =
Ω2
Z
|λn F (λ)|2 dρ1 (λ).
(1)
Ω1
Raising both sides of (1) to the power 1/(2n) and taking the limit as n → ∞ we get [8] 1/n
lim kLn f (x)kL2 (Ω2 ;dρ2 ) =
n→∞
|λ|,
sup
(2)
λ∈supp F
where supp F denotes the support of F , the smallest closed set, outside which the function F vanishes almost everywhere. From (2) it is obvious that if 1/n
lim kLn f (x)kL2 (Ω2 ;dρ2 ) < ∞,
n→∞
(3)
then F has a compact support. Hence, formula (2) plays a decisive role in studying integral transforms of functions with compact supports. It can be shown that under some ”extra conditions” on f inequality (3) gives the necessary and sufficient condition for a function f to be a T -transform of a function F ∈ L2 (Ω1 ; dρ1 ) with compact support. Formula (3) yields Bang’s d formula when Ω1 = (−∞, ∞) , φ(x, λ) = eixλ , dρ1 (λ) = dλ, and L = dx . In this paper we extend this technique to study integral transforms arising from singular Dirac systems of differential equations [4]. The image of the space of functions with compact supports under those transforms is fully described. As examples, the Hartley transform and some transforms with the Airy and the Bessel functions as kernels are considered. 2. SINGULAR DIRAC SYSTEM Consider the one-dimensional singular Dirac system on the half line y20 + p11 (x)y1 + p12 (x)y2 = λy1 , −y10
+ p21 (x)y1 + p22 (x)y2 = λy2 , 2
0 < x < ∞, 0 < x < ∞,
(4)
where pij , i, j = 1, 2, are real-valued functions and λ is a parameter. Using matrix notation, we can write the system (4) in the form BY 0 + P (x)Y = λY, where B=
0 1 −1 0
!
,
p11 (x) p12 (x) p21 (x) p22 (x)
P (x) =
!
,
y1 (x) y2 (x)
Y =
!
.
In the case when p12 (x) = p21 (x) = 0, p11 (x) = V (x)+m, p22 (x) = V (x)−m, V (x) is a potential function, and m is the mass of a particle, System (4) is called a one-dimensional stationary Dirac system in relativistic quantum theory. Under the assumption that p12 (x) = p21 (x), we can transform the system (4) into the canonical form BZ 0 + Q(x)Z = λZ, (5) where Y
= H(x)Z,
Q(x) = H −1 (x)B
H(x) =
cos φ(x) − sin φ(x) sin φ(x) cos φ(x)
!
,
d H(x) + H −1 (x)P (x)H(x). dx
In addition, if we choose (
φ(x) =
1 −1 2 tan π 4,
2p12 (x) p11 (x)−p22 (x)
, if p11 = 6 p22 , if p11 = p22 ,
then Q(x) takes the simpler form Q(x) =
q1 (x) 0 0 q2 (x)
!
,
(6)
for some continuous functions q1 (x) and q2 (x). Thus (5) takes the canonical form y20 + q1 (x)y1 = y10
,
0 < x < ∞,
− q2 (x)y2 = −λy2 ,
0 < x < ∞.
λy1
In the sequel we will be dealing mainly with the equivalent system y10 − q2 (x)y2 = y20
,
0 < x < ∞,
+ q1 (x)y1 = −λy1 ,
0 < x < ∞,
λy2
(7)
where q1 (x) and q2 (x) are continuous on [0, ∞). Let Φ(x, λ) =
φ1 (x, λ) φ2 (x, λ)
!
,
Θ(x, λ) =
θ1 (x, λ) θ2 (x, λ)
!
,
(8)
be the solutions of (7) that satisfies φ1 (0, λ) = − cos α,
φ2 (0, λ) = sin α,
θ1 (0, λ) = sin α,
θ2 (0, λ) = cos α. 3
(9)
Let
!
f1 (x) f2 (x)
f (x) =
(10)
be a vector-valued function such that f ∈ L2 (R+ ), i.e., kf k2L2 (R+ ) =
∞
Z
|f (x)|2 dx =
∞h
Z
0
i
|f1 (x)|2 + |f2 (x)|2 dx < ∞.
(11)
0
It is known [4] that for any non-real λ, there exists a function m(λ), analytic in the upper and lower half planes that are not necessarily analytic continuation of each other so that Θ(x, λ) + m(λ)Φ(x, λ), as a vector-valued function of x, is in L2 (R+ ) for non-real λ. Moreover, 1 lim π δ→0+
Z
λ
= m(u + iδ) du = ρ(λ) − ρ(µ),
µ
where ρ is a monotone increasing function (see [3]). If f ∈ L2 (R+ ), then Z
∞
F (λ) =
Z
f T (x). Φ(x) dx =
∞
[f1 (x)φ1 (x, λ) + f2 (x)φ2 (x, λ)] dx,
(12)
0
0
the generalized Fourier transform of f (x) , belongs to L2 (R, dρ) and the Parseval equality Z
∞
|f (x)|2 dx =
Z
∞
|F (λ)|2 dρ(λ)
(13)
−∞
0
holds. In addition, if the integrals Z
∞
−∞
Z
F (λ)φ1 (x, λ) dρ(λ)
∞
and −∞
F (λ)φ2 (x, λ) dρ(λ)
(14)
converge absolutely and uniformly in x in each finite interval, then ∞
Z
f1 (x) =
−∞ ∞
F (λ)φ1 (x, λ) dρ(λ),
Z
f2 (x) =
−∞
F (λ)φ2 (x, λ) dρ(λ).
(15)
We also have the following estimate Z
∞
−∞
dρ(λ) < ∞. λ2 + 1
(16)
Let us denote by L the operator d −Q= L=B dx
0 −1 1 0
!
d − dx
q1 (x) 0 0 q2 (x)
!
,
(17)
so that System (7) takes the form LY = λY. 4
(18)
Hence Φ(x, λ) is a solution of the system LΦ = λΦ,
(19)
AT . Φ(0, λ) = 0,
(20)
and where
!
sin α cos α
A=
,
!
φ1 (0, λ) φ2 (0, λ)
Φ(0, λ) =
.
(21)
We shall call a vector-valued function f (x) the Φ-transform of F (λ) if Z
∞
f (x) =
F (λ)Φ(x, λ) dρ(λ),
(22)
−∞
where Φ satisfies (19) and (20). We shall assume that Q is infinitely differentiable, such that the spectrum of the system (19) is either R or R+ . 3. PALEY-WIENER-TYPE THEOREM Lemma 1. Let F (λ) be such that λn F (λ) ∈ L2 (R, dρ) for all n = 0, 1, · · · . A vector-valued function f (x) is the Φ-transform of F as given by (22) if and only if 1) f (x) is infinitely differentiable on R+ , 2) (Ln f )(x) ∈ L2 (R+ ) for all n, 3) limx→0 AT . (Ln f )(x) = 0, 4) limx→∞ (Ln f )(x) = 0. Proof. Necessity: Let λn F (λ) ∈ L2 (R, dρ) for all n = 0, 1, 2, . . . . It is easy to see that λn F (λ) ∈ L1 (R, dρ). Indeed, applying formula (16) and the Cauchy-Schwartz inequality, we get Z
∞
2
|λn F (λ)| dρ(λ)
Z
∞
≤
−∞
Z
(λ2n+2 + λ2n )|F (λ)|2 dρ(λ)
−∞
∞
−∞
dρ(λ) λ2 + 1
< ∞.
1) Since Q(x) is infinitely differentiable, so are φ1 (x, λ) and φ2 (x, λ) with respect to the variable n n 1 (x,λ) 2 (x,λ) x. It also follows as in the Sturm-Liouville case [7] that ∂ φ∂x and ∂ φ∂x are of order λn n n as λ → ∞. Therefore, Z ∞ ∂ n Φ(x, λ) F (λ) f (n) (x) = dρ(λ), ∂xn −∞ exists for all n, where (n)
f
(n)
!
f1 (x) (n) f2 (x)
(x) =
.
(23)
2) By applying the differential operator L to (22) n times, we have (Ln f )(x) =
Z
∞
F (λ)Ln Φ(x, λ) dρ(λ) =
−∞
Z
∞
λn F (λ)Φ(x, λ) dρ(λ),
−∞
and since λn F (λ) ∈ L2 (R, dρ) we have Ln f ∈ L2 (R+ ). 3) By taking the matrix product with AT , we get T
n
Z
∞
lim A . (L f )(x) = lim
x→0+
x→0+ −∞
5
h
i
λn F (λ) AT . Φ(x, λ) dρ(λ),
(24)
∞
Z
h
i
λn F (λ) AT . Φ(0, λ) dρ(λ) = 0.
= −∞
Taking the limit inside the integral is justified by the Lebesgue dominated convergence theorem. 4) Let ! ! ! 0 −1 −f2 (x) f1 (x) ˜ = . f (x) = Bf (x) = 1 0 f2 (x) f1 (x) We have N
Z
T f˜ (x). Lf (x) dx =
Z
0
N
n
0
h
0
io
+f 1 (x) f1 (x) − q2 (x)f2 (x) Z
N
0
nh
0
h
i
f 2 (x) f2 (x) + q1 (x)f1 (x)
i N
h
dx = |f1 (x)|2 + |f2 (x)|2
i
0
h
0
i
o
−f 1 (x) + f 2 (x)q1 (x) f1 (x) + −f 2 (x) − f 1 (x)q2 (x) f2 (x) dx
+ 0
N
i N Z = |f1 (x)| + |f2 (x)| + h
2
2
0
f T (x) . Lf˜(x) dx.
0
Hence, |f1 (N )|2 + |f2 (N )|2 = |f1 (0)|2 + |f2 (0)|2 +
Z
N
T f˜ (x) . Lf (x) dx
0
−
N
Z
f T (x) . Lf˜(x) dx.
(25)
0
Since f (x) and Lf (x) are in L2 (R+ ), it follows that f˜(x) and Lf˜(x) are also in L2 (R+ ). Therefore, the right-hand side of (25) approaches a limit as N → ∞; hence so is the left handside of (25). But because f1 and f2 belong to L2 (R+ ), the limit must be zero, i.e., lim f1 (N ) = lim f2 (N ) = 0.
N →∞
N →∞
Similarly, one can show that limx→∞ (Ln f )(x) = 0 for all n. Sufficiency: Let f satisfy conditions 1)-4) of the Lemma. Since f ∈ L2 (R+ ) (condition 1) with n = 0), there exists F (λ) ∈ L2 (R, dρ) such that ∞
Z
F (λ) =
f T (x). Φ(x, λ)dx.
0
Then,
∞
Z
n
λ F (λ) =
Z
n T
∞
λ f (x). Φ(x, λ)dx = 0
f T (x). Ln Φ(x, λ)dx.
0
We show by induction on n that n
∞
Z
λ F (λ) =
(Ln f )T (x). Φ(x, λ)dx.
0
For n = 1, Z
λF (λ) =
∞
T
Z
f (x). LΦ(x, λ) dx = 0
0
∞n
h
0
i
−f1 (x) φ2 (x, λ) + q1 (x)φ1 (x, λ)
6
0
h
io
dx = [−f1 (x)φ2 (x, λ) + f2 (x)φ1 (x, λ)]|∞ 0
+f2 (x) φ1 (x, λ) − q2 (x)φ2 (x, λ) ∞ nh
Z
+
0
0
i
o
−φ1 (x, λ)f2 (x) + φ2 (x, λ)f1 (x) − [φ1 (x, λ)f1 (x)q1 (x) + φ2 (x, λ)f2 (x)q2 (x)] dx
0
= [−f1 (x)φ2 (x, λ) + f2 (x)φ1 (x, λ)]|∞ 0 ∞ nh
Z
+
0
i
h
0
i
o
−f2 (x) − f1 (x)q1 (x) φ1 (x, λ) + f1 (x) − f2 (x)q2 (x) φ2 (x, λ) dx
0
= [−f1 (x)φ2 (x, λ) + f2 (x)φ1 (x, λ)]|∞ 0 +
∞
Z
[(Lf )(x)]T . Φ(x, λ) dx.
0
Since f (x) and Φ(x, λ) satisfy the same initial condition (20) it follows that f1 (0)φ2 (0, λ) − f2 (0)φ1 (0, λ) = 0. As x → ∞, from condition 4) (n = 0) it follows that limx→∞ f (x) = 0, and because Φ(x, λ) is bounded in x for each fixed λ we have lim [f1 (x)φ2 (x, λ) − f2 (x)φ1 (x, λ)] = 0.
x→∞
Hence, Z
∞
[(Lf )(x)]T . Φ(x, λ) dx.
λF (λ) = 0
As Lf (x) ∈ L2 (R+ ), we have λF (λ) ∈ L2 (R, dρ). Now assume that Z ∞ [(Ln f )(x)]T . Φ(x, λ) dx. λn F (λ) = 0
Then λ
n+1
Z
∞
F (λ) =
[(Ln f )(x)]T . LΦ(x, λ) dx
0
=
[−g1 (x)φ2 (x, λ) +
g2 (x)φ1 (x, λ)]|∞ 0
where g1 (x) g2 (x)
g(x) =
Z
∞
+
[(Lg)(x)]T . Φ(x, λ) dx,
0
!
= (Ln f )(x).
For g(x) and Φ(x, λ) satisfy the same initial condition (20), then by conditions 3) and 4) we have [−g1 (x)φ2 (x, λ) + g2 (x)φ1 (x, λ)]|∞ 0 = 0. Thus, λn+1 F (λ) =
Z
∞
[(Ln+1 f )(x)]T . Φ(x, λ) dx,
0
and since
(Ln+1 f )(x)
∈ L2
(R+ ),
it follows that λn+1 F (λ) ∈ L2 (R, dρ).
Lemma 2. Let f (x) be the Φ-transform of a function F (λ) as given by (22). Let λn F (λ) ∈ L2 (R, dρ) for n = 0, 1, 2, . . . Then
1/n
lim Ln f
n→∞
L2 (R+ )
7
=
sup λ∈supp F
|λ|.
(26)
Proof. From the relation ∞
Z
n
λn F (λ)Φ(x, λ)dρ(λ),
(L f )(x) = −∞
and the Parseval equation (13) we have L2 (R+ )
∞
λ2n |F (λ)|2 dρ(λ).
= −∞
supλ∈supp F |λ| = δ < ∞. Then
First, let F have a compact support: Z
∞
Z
n 2
L f
Z
λ2n |F (λ)|2 dρ(λ) =
δ
λ2n |F (λ)|2 dρ(λ) ≤ δ 2n
δ
|F (λ)|2 dρ(λ).
−δ
−δ
−∞
Z
Hence,
1/n
lim sup Ln f
≤ δ lim sup
L2 (R+ )
n→∞
(Z
n→∞
)1/(2n)
δ
2
|F (λ)| dρ(λ)
= δ.
−δ
On the other hand, because δ is the supremum of the support of F , for any with 0 < < δ we have Z |F (λ)|2 dρ(λ) > 0. δ− 0. Thus dρ(λ) =
1 π
!
∈ L2 (R+ )
dλ, and ∞
Z
{− cos(λx + α)f1 (x) + sin(λx + α)f2 (x)} dx,
F (λ) =
(29)
0
f1 (x) = − f2 (x) =
1 π
1 π
Z
∞
cos(λx + α) F (λ) dλ,
Z
(30)
−∞ ∞
sin(λx + α) F (λ) dλ.
(31)
−∞
The Paley-Wiener-type theorem for the transforms (30) and (31) takes the form Corollary 1. A vector-valued function f (x) is the Φ-transform (30) and (31) of a function F (λ) ∈ L2 (R) with compact support if and only if f (x) satisfies conditions 1) f (x) is infinitely differentiable on R+ , !n d 0 − dx 2) f (x) ∈ L2 (R+ ) for all n, d 0 dx 9
0
3) limx→0 (sin α, cos α). 4) limx→∞
0 d dx
d − dx 0
!n
d dx
!n
d − dx 0
f (x) = 0,
f (x) = 0,
and
lim n→∞
d − dx 0
0 d dx
!n
1/n
f (x)
< ∞.
(32)
L2 (R+ )
Let α = 0 and 2f (x) = −f1 (|x|) + i sign x f2 (|x|),
x ∈ R.
Then we obtain the pair of the Fourier transforms Z
∞
e−iλx f (x) dx,
F (λ) = −∞
1 2π
f (x) =
Z
∞
eiλx F (λ) dλ.
−∞
If we put α = 0 and 2f (x) = −f1 (|x|) + sign x f2 (|x|),
x ∈ R,
then we arrive at the pair of the Hartley transforms ∞
Z
[cos λx + sin λx] f (x) dx,
F (λ) = −∞
f (x) =
1 2π
Z
∞
[cos λx + sin λx] F (λ) dλ. −∞
Example 2. Airy transform. Consider the system (7)-(9) with q1 (x) = 0, q2 (x) = −x, and α = 0: y1 0 + xy2 = y2
0
λy2 ,
0 < x < ∞,
= −λy1 ,
0 < x < ∞.
(33)
Eliminating y1 from the system (33) yields 00
y2 − (λx − λ2 )y2 = 0. Setting ˜ = λ4/3 − λ1/3 x , t+λ we obtain
d 2 y2 ˜ 2 = 0, and + (t + λ)y dt2 The general solution of the first equation is
y1 = λ−2/3
dy2 . dt
˜ 1/2 J1/3 (X) + B(t + λ) ˜ 1/2 Y1/3 (X) , y2 (t) = A(t + λ) where
2 ˜ 3/2 = 2 λ1/2 (λ − x)3/2 , X = (t + λ) 3 3
10
and Jµ (z) , Yµ (z) are the Bessel functions of the first and second kinds respectively. Hence n
o
y2 (x) = λ1/6 A(λ − x)1/2 J1/3 (X) + B(λ − x)1/2 Y1/3 (X) , and A 0 Aλ1/2 (λ − x)J1/3 (X) + √ J1/3 (X) 2 λ−x B 0 Y1/3 (X) . Bλ1/2 (λ − x)Y1/3 (X) + √ 2 λ−x
y1 (x) = λ +
−5/6
Since at x = 0, X = (2/3)λ2 , we have 2 2 y2 (0) = Aλ2/3 J1/3 ( λ2 ) + Bλ2/3 Y1/3 ( λ2 ) = 0 3 3 2 2 2 2 A 1 2 2 B 1 2 2 3/2 0 3/2 0 λ J1/3 ( λ ) + √ J1/3 ( λ ) + 5/6 λ Y1/3 ( λ ) + √ Y1/3 ( λ ) = 1. y1 (0) = 3 3 3 3 λ5/6 λ 2 λ 2 λ Solving this system for A and B, we obtain φ1 and φ2 . The explicit solution for φ1 and φ2 can be found in [7, p.92-93] and therefore the kernels of the integrals in (15) can be expressed in terms of the Bessel functions Jµ (z) and Yµ (z). However, since the argument of the Bessel functions, 2 X = λ1/2 (λ − x)3/2 3 maybe complex or real depending on whether x is greater or less than λ, and hence the integrals have to be split into different regions with different kinds of Bessel functions as kernels, it is more convenient to use the Airy functions. In so doing, we can express f1 and f2 as single integrals extending over the whole real line; see (43) and (44) below. So, let us make the change of variable λx−λ2 = λ2/3 z to obtain the following Airy differential equation d 2 y2 − zy2 = 0, dz 2 whose general solution is y2 = aAi(z) + bBi(z). Here Ai(z) and Bi(z) are the Airy functions [1]. Hence, y2 (x, λ) = aAi(λ1/3 x − λ4/3 ) + bBi(λ1/3 x − λ4/3 ).
(34)
0
Since y1 = −λ−1 y2 we have y1 (x, λ) = −aλ−2/3 Ai0 (λ1/3 x − λ4/3 ) − bλ−2/3 Bi0 (λ1/3 x − λ4/3 ).
(35)
Thus, (35) and (34) are the general solution of the system (33). The solution Φ(x, λ) satisfies the initial condition φ1 (0, λ) = −1, φ2 (0, λ) = 0. Thus aAi(−λ4/3 ) + bBi(−λ4/3 ) = 0, aλ−2/3 Ai0 (−λ4/3 ) + bλ−2/3 Bi0 (−λ4/3 ) = 1. 11
Using the Wronskian equality for the Airy functions [1] W (Ai(x), Bi(x)) := Ai(x)Bi0 (x) − Ai0 (x)Bi(x) =
1 , π
(36)
we get a = −πλ2/3 Bi(−λ4/3 ),
b = πλ2/3 Ai(−λ4/3 ).
Thus, φ1 (x, λ) = πBi(−λ4/3 )Ai0 (λ1/3 x − λ4/3 ) − πAi(−λ4/3 )Bi0 (λ1/3 x − λ4/3 ), and φ2 (x, λ) = −πλ2/3 Bi(−λ4/3 )Ai(λ1/3 x − λ4/3 ) + πλ2/3 Ai(−λ4/3 )Bi(λ1/3 x − λ4/3 ). Similarly, the solution Θ(x, λ) satisfies the initial condition θ1 (0, λ) = 0,
θ2 (0, λ) = 1,
that yields aAi(−λ4/3 ) + bBi(−λ4/3 ) = 1, aλ−2/3 Ai0 (−λ4/3 ) + bλ−2/3 Bi0 (−λ4/3 ) = 0. Solving this system we get a = πBi0 (−λ4/3 ),
b = −πAi0 (−λ4/3 ).
Thus, θ1 (x, λ) = −πλ−2/3 Bi0 (−λ4/3 )Ai0 (λ1/3 x − λ4/3 ) + πλ−2/3 Ai0 (−λ4/3 )Bi0 (λ1/3 x − λ4/3 ), and θ2 (x, λ) = πBi0 (−λ4/3 )Ai(λ1/3 x − λ4/3 ) − πAi0 (−λ4/3 )Bi(λ1/3 x − λ4/3 ). We have h
i
θ1 (x, λ) + m(λ)φ1 (x, λ) = −λ−2/3 Bi0 (−λ4/3 ) + m(λ)Bi(−λ4/3 ) πAi0 (λ1/3 x − λ4/3 ) h
i
+ λ−2/3 Ai0 (−λ4/3 ) − m(λ)Ai(−λ4/3 ) πBi0 (λ1/3 x − λ4/3 ), and h
i
θ2 (x, λ) + m(λ)φ2 (x, λ) = Bi0 (−λ4/3 ) − m(λ)λ2/3 Bi(−λ4/3 ) πAi(λ−1/3 x − λ4/3 ) h
i
+ −Ai0 (−λ4/3 ) + m(λ)λ2/3 Ai(−λ4/3 ) πBi(λ1/3 x − λ4/3 ). If we take m(λ) = λ−2/3
Ai0 (−λ4/3 ) + iBi0 (−λ4/3 ) , Ai(−λ4/3 ) + iBi(−λ4/3 )
then θ1 (x, λ) + m(λ)φ1 (x, λ) = −λ−2/3
Ai0 (λ1/3 x − λ4/3 ) + iBi0 (λ1/3 x − λ4/3 ) , Ai(−λ4/3 ) + iBi(−λ4/3 ) 12
(37)
and θ2 (x, λ) + m(λ)φ2 (x, λ) =
Ai(λ1/3 x − λ4/3 ) + iBi(λ1/3 x − λ4/3 ) . Ai(−λ4/3 ) + iBi(−λ4/3 ) (2)
The relations between the Airy functions and the Hankel functions Hν (z) [1] r
2 3/2 3 z = e−πi/6 [Ai(−z) + iBi(−z)] , 3 z √ (2) 2 3/2 −πi/6 3 0 H2/3 z = e Ai (−z) + iBi0 (−z) , 3 z
(2) H−1/3
(38)
yield (2)
θ1 (x, λ) + m(λ)φ1 (x, λ) = −
2 1/2 (λ − 3λ (2) λ H−1/3 23 λ2
(λ − x) H2/3
and
(2)
θ2 (x, λ) + m(λ)φ2 (x, λ) =
(λ − x)1/2 H−1/3
(2)
2 1/2 (λ 3λ
λ1/2 H−1/3
2 2 3λ
x)3/2
− x)3/2
,
.
Using the asymptotic expansion for the Hankel function [1] Hν(2) (z)
r
=
2 −i(z−νπ/2−π/4) e 1 + O(z −1 ) , πz
it is clear that (λ −
(2) x) H2/3
(λ − x)
−2π < arg z < π,
(39)
2 1/2 λ (λ − x)3/2 ∈ L2 (R+ ), 3
and 1/2
z → ∞,
(2) H−1/3
(40)
2 1/2 λ (λ − x)3/2 ∈ L2 (R+ ), 3
(41)
for any non-real λ with = λ > 0. Hence, θ1 (x, λ) + m(λ)φ1 (x, λ),
θ2 (x, λ) + m(λ)φ2 (x, λ) ∈ L2 (R+ ).
We have
Ai(−λ4/3 )Ai0 (−λ4/3 ) + Bi(−λ4/3 )Bi0 (−λ4/3 ) + iW Ai(−λ4/3 ), Bi(−λ4/3 ) m(λ) =
.
λ2/3 Ai2 (−λ4/3 ) + Bi2 (−λ4/3 )
Thus, dρ(λ) =
1 1 dλ. = m(λ) dλ = 2 π πλ2/3 Ai (−λ4/3 ) + Bi2 (−λ4/3 )
We arrive at the following triple of integral transforms Z
F (λ) = π +
λ
∞ nh
0 h 2/3
i
Bi(−λ4/3 )Ai0 (λ1/3 x − λ4/3 ) − Ai(−λ4/3 )Bi0 (λ1/3 x − λ4/3 ) f1 (x) i
o
−Bi(−λ4/3 )Ai(λ1/3 x − λ4/3 ) + Ai(−λ4/3 )Bi(λ1/3 x − λ4/3 ) f2 (x) dx, (42)
13
and ∞
Bi(−λ4/3 )Ai0 (λ1/3 x − λ4/3 ) − Ai(−λ4/3 )Bi0 (λ1/3 x − λ4/3 )
−∞
λ2/3 Ai2 (−λ4/3 ) + Bi2 (−λ4/3 )
Z
f1 (x) =
Z
f2 (x) =
∞
−∞
h
i
F (λ) dλ,
(43)
−Bi(−λ4/3 )Ai(λ1/3 x − λ4/3 ) + Ai(−λ4/3 )Bi(λ1/3 x − λ4/3 ) F (λ) dλ. (44) Ai2 (−λ4/3 ) + Bi2 (−λ4/3 )
Theorem 1 holds for the transforms (43) and (44) if A=
0 1
!
,
L=
0 d dx
d − dx x
!
.
Corollary 2. A vector-valued function f (x) is the Φ-transform (43) and (44) of a function F (λ) ∈ L2 (R, dρ) with compact support if and only if f (x) satisfies the conditions 1) f (x) is infinitely differentiable on R+ , !n d 0 − dx 2) f (x) ∈ L2 (R+ ) for all n, d x dx 3) limx→0 (0, 1). 4) limx→∞
0 d dx
0 d dx d − dx
x
d − dx x
!n
f (x) = 0,
!n
f (x) = 0,
and
lim n→∞
0 d dx
d − dx x
!n
1/n
f (x)
L2
< ∞.
(45)
(R+ )
Example 3. Second Airy transform. Consider the same system (33), but with α = π/2. The solution Φ(x, λ) satisfying the initial condition φ1 (0, λ) = 0,
φ2 (0, λ) = 1,
is φ1 (x, λ) = −πλ−2/3 Bi0 (−λ4/3 )Ai0 (λ1/3 x − λ4/3 ) + πλ−2/3 Ai0 (−λ4/3 )Bi0 (λ1/3 x − λ4/3 ), and φ2 (x, λ) = πBi0 (−λ4/3 )Ai(λ1/3 x − λ4/3 ) − πAi0 (−λ4/3 )Bi(λ1/3 x − λ4/3 ). Similarly, the solution Θ(x, λ) satisfying the initial condition θ1 (0, λ) = 1,
θ2 (0, λ) = 0,
is θ1 (x, λ) = −πBi(−λ4/3 )Ai0 (λ1/3 x − λ4/3 ) + πAi(−λ4/3 )Bi0 (λ1/3 x − λ4/3 ), and θ2 (x, λ) = πλ2/3 Bi(−λ4/3 )Ai(λ1/3 x − λ4/3 ) − πλ2/3 Ai(−λ4/3 )Bi(λ1/3 x − λ4/3 ). We have h
i
θ1 (x, λ) + m(λ)φ1 (x, λ) = −Bi(−λ4/3 ) − m(λ)λ−2/3 Bi0 (−λ4/3 ) πAi0 (λ1/3 x − λ4/3 ) h
i
+ Ai(−λ4/3 ) + m(λ)λ−2/3 Ai0 (−λ4/3 ) πBi0 (λ1/3 x − λ4/3 ), 14
and h
i
θ2 (x, λ) + m(λ)φ2 (x, λ) = λ2/3 Bi(−λ4/3 ) + m(λ)Bi0 (−λ4/3 ) πAi(λ−1/3 x − λ4/3 ) h
i
+ −λ2/3 Ai(−λ4/3 ) − m(λ)Ai0 (−λ4/3 ) πBi(λ1/3 x − λ4/3 ). If we take
λ2/3 Ai(−λ4/3 ) + iBi(−λ4/3 ) m(λ) = −
Ai0 (−λ4/3 ) + iBi0 (−λ4/3 )
,
(46)
then the relations between the Airy functions and the Hankel functions (38) yield (2)
θ1 (x, λ) + m(λ)φ1 (x, λ) =
2 1/2 (λ − 3λ (2) 2 2 λ H2/3 3 λ
(λ − x) H2/3
and
(2)
θ2 (x, λ) + m(λ)φ2 (x, λ) = −
(λ − x)1/2 H−1/3
x)3/2
2 1/2 (λ 3λ (2) λ1/2 H2/3 23 λ2
,
− x)3/2
.
From (40) and (41) it is clear that θ2 (x, λ) + m(λ)φ2 (x, λ) ∈ L2 (R+ ).
θ1 (x, λ) + m(λ)φ1 (x, λ), We have
m(λ) = −λ2/3
Ai(−λ4/3 )Ai0 (−λ4/3 ) + Bi(−λ4/3 )Bi0 (−λ4/3 ) − iW Ai(−λ4/3 ), Bi(−λ4/3 )
.
Ai0 2 (−λ4/3 ) + Bi0 2 (−λ4/3 )
Thus, dρ(λ) =
1 λ2/3 dλ. = m(λ) dλ = π π Ai0 2 (−λ4/3 ) + Bi0 2 (−λ4/3 )
We arrive at the following triple of integral transforms Z
F (λ) = π
∞n
h
i
λ−2/3 −Bi0 (−λ4/3 )Ai0 (λ1/3 x − λ4/3 ) + Ai0 (−λ4/3 )Bi0 (λ1/3 x − λ4/3 ) f1 (x)
h0
i
o
+ Bi0 (−λ4/3 )Ai(λ1/3 x − λ4/3 ) − Ai0 (−λ4/3 )Bi(λ1/3 x − λ4/3 ) f2 (x) dx, Z
f1 (x) =
−∞
Z
f2 (x) =
∞
∞
−∞
(47)
−Bi0 (−λ4/3 )Ai0 (λ1/3 x − λ4/3 ) + Ai0 (−λ4/3 )Bi0 (λ1/3 x − λ4/3 ) F (λ) dλ, Ai0 2 (−λ4/3 ) + Bi0 2 (−λ4/3 )
(48)
Bi0 (−λ4/3 )Ai(λ1/3 x − λ4/3 ) − Ai0 (−λ4/3 )Bi(λ1/3 x − λ4/3 ) F (λ) λ2/3 dλ. Ai0 2 (−λ4/3 ) + Bi0 2 (−λ4/3 )
(49)
Theorem 1 holds for transforms (48) and (49) if A=
1 0
!
,
L=
0 d dx
d − dx x
!
.
Corollary 3. A vector-valued function f (x) is the Φ-transform (48) and (49) of a function F (λ) ∈ L2 (R, dρ) with compact support if and only if f (x) satisfies conditions 15
1) f (x) is infinitely differentiable on R+ , !n d 0 − dx f (x) ∈ L2 (R+ ) for all n, 2) d x dx 3) limx→0 (1, 0). 4) limx→∞
0 d dx
0 d dx d − dx
x
d − dx x
!n
f (x) = 0,
!n
f (x) = 0,
and
lim n→∞
0 d dx
d − dx x
!n
1/n
f (x)
L2
< ∞.
(50)
(R+ )
4. ACKNOWLEDGEMENT This work was accomplished while Dr. Zayed was visiting the Department of Mathematics and Computer Science, Faculty of Science, Kuwait University, and he would like to take this opportunity to thank the department for its hospitality. The work of the second named author was supported by the Kuwait University Research Administration under the research grant SM 187.
References [1] Abramowitz M. and Stegun I.A. Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables. Dover Publication, New York, 1972. [2] Bang H.H. A property of infinitely differentiable functions. Proc. Amer. Math. Soc. 108 (1990), pp.73-76. [3] Hinton D. and Shaw J. Hamiltonian systems of limit point or limit circle type with both end points singular. J. Diff. Eqs. 50(1983), pp. 444-464. [4] Levitan B.M. and Sargsjan I.S. Sturm-Liouville and Dirac Operators. Kluwer Acad. Publ., Dordrecht, 1991. [5] Paley R.E.A.C. and Wiener N. Fourier Transforms in the Complex Domain. Colloq. Publ. Amer. Math. Soc., 1934. [6] Titchmarsh E.C. Introduction to the Theory of Fourier Integrals. Chelsea Publishing Company, New York, 1986. [7] Titchmarsh E.C. Eigenfunction Expansions Associated with Second-Order Differential Equations. Part 1. Clarendon Press, Oxford, 1962. [8] Vu Kim Tuan. Supports of functions and integral transforms. Proceedings of International Workshop on the Recent Advances in Applied Mathematics (RAAM’ 96), held in Kuwait, May 4-7, 1996, pp. 507-521. [9] Vu Kim Tuan. On the range of the Y-transform. Bull Austral. Math. Soc. 54(1996), pp. 329-345. [10] Vu Kim Tuan. On the range of the Hankel and extended Hankel transforms. J. Math. Anal. Appl. 209(1997), pp. 460-478. [11] Vu Kim Tuan. New type Paley-Wiener theorems for the modified multidimensional Mellin transform. J. Fourier Anal. Appl. 4(1998), no. 3., pp. 317-328.
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[12] Vu Kim Tuan. Airy integral transform and the Paley-Wiener theorem. Proceedings of the 2nd International Workshop on Transform Methods and Special Functions, Varna, Bulgaria, 23-30 August 1996. Eds.: Rusev D., Dimovski I., and Kiryakova V., IMI-BAS, Sofia, 1998, pp. 523-531. [13] Vu Kim Tuan and Zayed A.I. Paley-Wiener-type theorems for a class of integral transforms (submitted).
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