New Convolutions for QuadraticPhase Fourier Integral Operators and their Applications
02 Jan 2018Mediterranean Journal of Mathematics (Springer International Publishing)Vol. 15, Iss: 1, pp 117
TL;DR: In this article, the authors obtained new convolutions for quadraticphase Fourier integral operators (which include, as subcases, e.g., the fractional Fourier transform and the linear canonical transform).
Abstract: We obtain new convolutions for quadraticphase Fourier integral operators (which include, as subcases, e.g., the fractional Fourier transform and the linear canonical transform). The structure of these convolutions is based on properties of the mentioned integral operators and takes profit of weightfunctions associated with some amplitude and Gaussian functions. Therefore, the fundamental properties of that quadraticphase Fourier integral operators are also studied (including a Riemann–Lebesgue type lemma, invertibility results, a Plancherel type theorem and a Parseval type identity). As applications, we obtain new Young type inequalities, the asymptotic behaviour of some oscillatory integrals, and the solvability of convolution integral equations.
New Convolutions for QuadraticPhase Fourier
Integral Operators and their Applications
1
L.P. Castro, L.T. Minh and N.M. Tuan
Abstract
We obtain new convolutions for quadraticphase Fourier integral operators (which
include, as subcases, e.g., the fractional Fourier transform and the linear canonical
transform). The structure of these convolutions is based on properties of the mentioned
integral operators and takes proﬁt of weightfunctions associated with some amplitude
and Gaussian functions. Therefore, the fu ndamental properties of that qu ad ratic
phase Fourier integral operators are also studied (in cluding a RiemannLebesgue type
lemma, invertibility results, a Plancherel type theorem and a Parseval type identity).
As applications, we obtain new Young type inequalities, the asymptotic behaviour of
some oscillatory integrals, and the solvability of convolution integral equations.
Keywords: convolution; Young inequality; oscillatory integral; convolution integral
equation; fractional Fourier transform; lin ear canonical transform.
AMS Subject Classiﬁcation: Primary 44A35; Secondary 42A38; 42A85; 43A32;
44A20; 45E10.
1 Introduction
The interest in having new convolutions associated with integral operators is wide and based
on both theoretical and applied aspects. Outside mathematics, the use of diﬀerent types of
convolutions is very diverse, ranging, e.g., from signal processing to neural networks. Within
mathematics, it is also very proﬁtable to construct new convolutions that will facilitate the
identiﬁcation of new factorization properties to decouple t he convolutions into a (weighted)
product of integrals. Typically, this decoupling has strong consequences and originates new
results in diﬀerent branches of mathematics (e.g., in harmonic analysis and diﬀerential equa
tions). This last aspect will be exhibited in here, for a large class of integral opera t ors, with
1
Accepted author’s manuscript (AAM) of [L.P. Castro, L .T. Minh, N.M. Tuan: Mediterr. J. Math.
(2018) 15:1 3. DOI: 10.1007/s000090 171063y].
The ﬁnal publica tion is available at Springer via https://doi.org/10.1007/s000090171063y and
http://rdcu.be/DWRU
1
consequences that will be exempliﬁed for three diﬀerent topics: (i) Young type inequalities,
(ii) asymptotic behaviour of oscillatory integrals, and (iii) the solvability of classes of integral
equations.
A diversity of convolutions which are suitable for other integral operators can be found
in several r ecent publications. For other convolutions and integral operators, while not being
exhaustive, we refer the reader to [1, 2, 3, 8, 12, 13, 14, 15, 16, 17, 18, 22, 25, 26, 28]. In
addition, it is relevant to have in mind that the fa cto r izat ion property of convolutions is
crucial in solving corresponding convolution type equations [6, 7, 11, 25]. It is a lso clear that
convolution type equations are very often used in the modelling of a broad range of diﬀerent
problems (cf. [9, 10]), and so additional knowledge on their solvability is very welcome.
Throughout this paper, for parameters a, b, c, d, e ∈ R (with b 6= 0), we take
Q
(a,b,c,d,e)
(x, y) := ax
2
+ bxy + cy
2
+ dx + ey (1.1)
to be the quadraticphase function within the kernel of our integral operator. In what follows,
for shortening fo rmulas, we will also use the notation Q
(a−e)
(x, y) := Q
(a,b,c,d,e)
(x, y). Besides
this, we shall also write Q
(a−c)
(x, y) := Q
(a,b,c,0,0)
(x, y). So, this allows us to introduce the
integral operator Q deﬁned by
(Qf)(x) :=
1
√
2π
Z
R
e
iQ
(a−e)
(x,y)
f(y)dy, (1.2)
where f ∈ L
1
(R) or f ∈ L
2
(R), and that we will denominate by quadraticphase Fourier
integral operator.
Let us make a brief discussion on the integral operator Q by comparing it with other
wellknown operators. In ﬁrst place, we would like to notice that when a = c = d = e = 0
and b = ±1, Q is simply the wellknown Fourier and inverse Fourier integral t r ansforms,
respectively. Secondly, when d = e = 0, the kernel generated by (1.1) includes the kernel
of the linear canonical tr ansform as well as of the one of the fractional Fourier transform.
Typically, it is clear that the constant factors incorporated in the integral operators are
considered in view of the ﬁnal purposes and problems where the operators are used. Still
within in the last comparison, and a s about the constant fa cto r appearing in (1.2), there is
a diﬀerence between our concept of quadraticphase Fo ur ier integral operator and the most
frequent choices of constant factors for the linear canonical transform and fractional Fourier
transform. In our case, the factor 1/
√
2π is chosen intentionally since it ensures consequent
convenient computations involving the quadraticphase Fourier integral operat or, as we shall
see later on. The constant
√
−i typically chosen in the pa r t icular case of linear canonical
2
transform, and
p
(1 − i cot(α))/2π for the fra ctio nal Fourier transform, are more convenient
in view of the particular properties of those cases.
The paper is divided into four sections and organized as fo llows. Section 2 presents some
basic theorems for the integral operator Q such as the R iemann L ebesgue lemma, inversion
formula, Plancherel’s extension, Parseval identity. Section 3 provides new convolution theo
rems which, in part icular cases, turn out to be convolution theorems for the linear canonical
transform, fractional Four ier transform and Fourier transform. In Section 4 we apply the
obtained results in order to derive Young’s convolution inequalities for the propo sed convo
lutions, the asymptotic behaviour of a class of oscillatory integrals, a nd the solvability of
classes of convo lut ion integral equations.
2 Basic Properties of the Integral Operator
In this section, as a preliminary step to the main content, we will study some basic pro
perties of the integral operator Q. This will include it s mapping properties, as well as a
corresponding inversion formula, a Plancherel’s extension and a Parseval identity.
Let us denote by S the Schwart z space, and by C
0
(R) the Banach space o f all continuous
functions on R that vanish at inﬁnity, endowed with the supremum norm k ·k
∞
. Moreover,
in L
1
(R) we will be using the norm k · k
1
deﬁned by
kfk
1
:=
1
√
2π
Z
R
f(y)dy,
where the factor 1/
√
2π is here considered just to obtain more direct computations later on.
In the case of 1 < p < ∞, the space L
p
(R) will be endowed with the norm
kfk
p
:=
Z
R
f(y)
p
dy
1
p
.
Lemma 2.1 (RiemannLebesgue lemma). If f ∈ L
1
(R) then Qf ∈ C
0
(R), and kQfk
∞
≤
kfk
1
.
Proo f . Since e
iQ
(a−e)
(x,y)
 = 1, it is clear that
kQfk
∞
= sup
x∈R
(Qf)(x) = sup
x∈R
1
√
2π
Z
R
e
iQ
(a−e)
(x,y)
f(y)dy
≤ sup
x∈R
1
√
2π
Z
R
e
iQ
(a−e)
(x,y)
f(y)dy = kfk
1
.
3
In addition, choosing g(y) := e
i(cy
2
+ey)
f(y) it is clear that g ∈ L
1
(R) if and only if f ∈ L
1
(R).
Therefore, using the classic RiemannLebesgue lemma, we derive that
(Qf)(x) =
e
i(ax
2
+dx)

√
2π
Z
R
e
ibxy
g(y)dy
=
1
√
2π
Z
R
e
ibxy
g(y)dy
→ 0,
as x → ∞, and the proof of the lemma is complete.
The next lemma is known as a version of the NyquistShannon sampling theorem, a nd
will be useful for proving Theorem 2.3.
Lemma 2.2 (cf., e.g., Theorem 12, [24]). The formula
1
2
[f(x + 0) + f (x − 0)] = lim
λ→∞
1
π
Z
+∞
−∞
f(t)
sin λ(x − t)
x − t
dt
holds true if
f(x)
1 + x
belongs to L
1
(R).
Theorem 2.3 (Inversion theorem). If f ∈ L
1
(R) and Qf ∈ L
1
(R), then
f(x) =
b
√
2π
Z
R
(Qf)(y)e
−iQ
(a−e)
(y, x)
dy, (2.1)
for almost every x ∈ R.
Proo f . First, let us prove the inversion formula for f ∈ S. In this case, by Lemma 2.2 and
direct computations, we have
b
√
2π
Z
R
(Qf)(y)e
−iQ
(a−e)
(y, x)
dy
=
b
2π
lim
λ→+∞
Z
λ
−λ
Z
R
e
−iQ
(a−e)
(y, x)
e
iQ
(a−e)
(y, u)
f(u)dudy
=
b
2π
e
−i(cx
2
+ex)
Z
R
f(u)e
i(cu
2
+eu)
du lim
λ→+∞
Z
λ
−λ
e
−iby (x−u)
dy
=
1
π
e
−i(cx
2
+ex)
lim
λ→+∞
Z
R
f(u)e
i(cu
2
+eu)
sin bλ(x − u)
x − u
du
= e
−i(cx
2
+ex)
f(x)e
i(cx
2
+ex)
= f(x).
Thus, Q is a onetoone, linear and continuous operator from S onto S (with a continuous
inverse).
We now assume that f ∈ L
1
(R), a nd let g ∈ S. A direct computation gives us
Z
R
f(x)(Qg)(x)dx =
Z
R
g(y)(Qf)(y)dy.
4
Using this identity and (2.1), for g ∈ S, we have
Z
R
f(x)(Qg)(x)dx =
b
√
2π
Z
R
Z
R
e
−iQ
(a−e)
(y, x)
(Qg)( x)dx
(Qf)(y)dy
=
Z
R
(Qg)( x)
b
√
2π
Z
R
e
−iQ
(a−e)
(y, x)
(Qf)(y)dy
dx
=
Z
R
f
0
(x)(Qg)(x)dx,
where
f
0
(x) :=
b
√
2π
Z
R
(Qf)(y)e
−iQ
(a−e)
(y, x)
dy.
By (2.1) t he function Qg covers all S when g runs in S. Therefore,
R
R
(f
0
(x) −f(x))Φ(x)dx
= 0 for every Φ ∈ S. Since S is dense in L
1
(R), we obtain that f
0
(x) −f(x) = 0 for almost
every x ∈ R, as desired.
The uniqueness property below is an immediate consequence of Theorem 2.3.
Corollary 2.4 (Uniqueness). If f ∈ L
1
(R) and Qf = 0, then f = 0.
In what follows we will be denoting the inverse operator of Q by Q
−1
:
(Q
−1
g)(x) :=
b
√
2π
Z
R
(Qf)(y)e
−iQ
(a−e)
(y, x)
dy
Theorem 2.5 (Plancherel theorem). Th e re is a linear isomorphic operator Q : L
2
(R) →
L
2
(R) which is uniquely determined by the requirement that
Qf = Qf for every f ∈ S.
The inv e rs e operator is also uniquely de term i ned by having
Q
−1
f = Q
−1
f for every f ∈ S.
Proo f . Recall that S is dense in L
2
(R). Thus, as the map f 7→ Qf is continuous (relative
to the L
2
−norm) from the dense subspace S of L
2
(R) onto S, it has a unique continuous
extension
Q : L
2
(R) → L
2
(R). Theorem 2.5 is proved.
Thanks to the uniqueness of the extension opera t or, one can formulate another theorem
in a more detailed way by exhibiting the explicit form of the operator in L
2
.
Theorem 2.6 (Plancherel theorem). Let f be a complexvalued function in L
2
(R) and le t
Q(x, k) :=
1
√
2π
Z
y<k
f(y)e
iQ
(a−e)
(x,y)
dy.
Then, as k → ∞, Q(x, k) converges stro ngly (over R) to a function, say Qf, of L
2
(R);
rec i procally,
f(x, k) :=
b
√
2π
Z
y<k
Qf( y)e
−iQ
(a−e)
(y, x)
dy.
converges strongly to f .
5
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