Inversion of the convolution operators on a rectangular

0.1 MB·

by  Alexander Sakhnovich

Inversion of the convolution operators
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1
0 on a rectangular
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n
a
J Alexander Sakhnovich

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3

] Abstract
A
C Using simultaneously two operator identities, we consider the in-
h. version of the convolution operators on a rectangular. The structure
at of the inverse operators and of some corresponding forms, which are
m
important in signal processing, is derived.
[

1 MSC(2010): 45E10, 47A05, 47A65
v
Keywords: Convolution operator on a rectangular, operator with differ-
9
5 ence kernel, inverse operator, special solutions, structure, operator identity.
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8
0
1 Introduction
.
1
0
7
We consider a convolution operator S (or, equivalently, an operator S with
1
: difference kernel) of the form
v
i
X ∂ ∂
Sf = s(x−t)f(t)dt x = (x , x ), t = (t , t ) , (1.1)
ar ∂x1 ∂x2 ZΩ (cid:0) 1 2 1 2 (cid:1)
where
Ω = {x : 0 < x < ω , 0 < x < ω }, s(x−t) = s(x −t , x −t ). (1.2)
1 1 2 2 1 1 2 2
We assume that s(x) ∈ L2(Ω), where Ω = {x : |x | < ω , |x | < ω }, and so
1 1 2 2
the integrals in (1.1) are well-defined for f ∈ L2(Ω). Moreover, we assume
e e
that the right-hand side of (1.1) is well-defined and that S is bounded in
L2(Ω).
1
In one-dimensional case, the inversion of convolution operators is con-
nected with the names of N. Wiener, E. Hopf, M.G. Krein, I.C. Gohberg,
V.A. Ambartsumian, V. V. Sobolev, L.A. Sakhnovich and many other math-
ematicians and applied scientists. The inversion of convolution operators on
a semi-axis (of Wiener-Hopf operators) was studied in various papers includ-
ing the brilliant works [5,10] (see also, e.g., [2,3,6,11] and references therein).
ThesituationwiththeinversionofconvolutionoperatorsS onafiniteinterval
is more complicated and essentially different from the case of semi-axis. One
ofthefirst works onthesubject was againwrittenbyM.G. Krein[9]. Then, a
procedure to recover the operator S−1, from its action on two functions only,
was published [17]. Further developments as well as various applications and
references can be found in [18,19]. The method of operator identities, which
was introduced in [17] (see also [16]), may be successfully used for the in-
version of various other structured operators. See, for instance, [4,14] and a
more detailed discussion with references in [15, Appendix D].
According to [19], “an important part in the theory of the equations with
a difference kernel (on the interval) is played by the equations with a special
right-hand side”:
Sf(x,λ) = eiλx.
The reflection coefficient in the light scattering theory coincides with the
function
ρ(λ,µ) = S−1eiλx,eiµx (λ,µ ∈ C), (1.3)
ω
(cid:0) (cid:1)
where (·,·) stands for the scalar product in L2(0,ω) and C denotes the
ω
complex plane. The structure of ρ derived in [18] (see also [19]) generalizes
a well-known astrophysics result from [1,7,20].
The inversion of the convolution operators and the structure of the in-
verse operators in the two-dimensional case is much more complicated than
in the one-dimensional case (similar to some other inversion and interpola-
tion problems). In spite of several books (see, e.g., [12,21]) and many papers
on multidimensional convolution operators, the problem to take simultane-
ously into account the dependence of the integral kernel of the operator on
the difference of the arguments (with respect to both variables) remained
2
unsolved. This problem is essential in applications (e.g., in astrophysics and
signal processing [8,13]) and in factorization and interpolation theories.
In our paper we express the ρ-function
ω1 ω2
ρ(λ,µ) = e−iµx S−1eiλx (x ,x )dx dx (µx = µ x +µ x ), (1.4)
1 2 1 2 1 1 2 2
Z Z
0 0 (cid:0) (cid:1)
where x ∈ Ω and λ,µ ∈ C2, in terms of a bounded operator acting from
L2(0,ω ) into L2(0,ω ) (i 6= k). Such an expression is a direct analog of [19,
2 i 2 k
(1.3.12)], where ρ in one-dimensional case (see (1.3)) is expressed via four
functions depending on one variable. Using the structure of ρ, we derive the
structure of the operator, which is inverse to a convolution operator on Ω.
Here, the notation 1 stands for the function which identically equals 1
a
on the corresponding set, and col means column (e.g., col a b = ). As
(cid:20)b(cid:21)
(cid:2) (cid:3)
usual, R denotes the real axis and C denotes the complex plane. The class of
bounded operators acting from the Hilbert space H into the Hilbert space
1
H is denoted by B(H ,H ). When H = H = H, we write B(H) instead
2 1 2 1 2
of B(H,H).
2 Structures of the ρ-function
and of the inverse operator
2.1 Bounded operators on a rectangular
Similarly to the proof of [19, Theorem 1.1.1] on the representation of the op-
eratorsfromB L2(0,ℓ) (one-dimensional case), onecanshow thatoperators
Q ∈ B L2(Ω) (cid:0)admit re(cid:1)presentation
(cid:0) (cid:1) ∂ ∂
Qf = q(x,t)f(t)dt, (2.1)
∂x ∂x Z
1 2 Ω
where the integrals q(x,t)f(t)dt are generating functions of absolutely
Ω
continuous (with respRect to the planar Lebesgue measure) charges on Ω, and
for each fixed x ∈ L2(Ω) (and varying t) we have q(x,t) ∈ L2(Ω). More
precisely, we have
q(x,t) = Q∗χ (t), (2.2)
x
(cid:0) (cid:1)
3
where
χ (t) = 1 for t ∈ U = {t : 0 < t < x , 0 < t < x },
x x 1 1 2 2
χ (t) = 0 for t ∈ Ω\U .
x x
(cid:0) (cid:1)
We consider a slightly more general than (1.1) class of bounded convolution
operators, namely, operators of the form
∂ ∂
Sf = q(x,t)f(t)dt, q(x,t) =s(x−t)−s(x−t ,−t )
1 2
∂x ∂x Z
1 2 Ω
−s(−t ,x −t )+s(−t), (2.3)
1 2 2
where the expression −s(x−t ,−t )−s(−t ,x −t )+s(−t) may be used
1 2 1 2 2
for normalization purposes and disappears (after the differentiation of the
integral in (2.3)) in the cases of comparatively smooth functions s(x−t).
We always assume that s(x) ∈ L2(Ω), for Ω = {x : |x | < ω , |x | < ω }.
1 1 2 2
Without loss of generality we assume also that
e e
ω1 ω2
s(−t ,x )dt = s(x ,−t )dt = 0. (2.4)
1 2 1 1 2 2
Z Z
0 0
Note that if (2.4) does not hold, we may substitute s(x) with
1 ω1 1 ω2 1
s(x) = s(x)− s(−t ,x )dt − s(x ,−t )dt + s(−t)dt,
ω Z 1 2 1 ω Z 1 2 2 ω ω Z
1 0 2 0 1 2 Ω
b
and the function s satisfies (2.4).
b
2.2 Structure of the ρ-function
The ρ-function given by (1.4) admits representation
ρ(λ,µ) = (I −µ A∗)−1(I −µ A∗)−1S−1(I −λ A )−1(I −λ A )−11,1 ,
1 1 2 2 2 2 1 1
(cid:16) (cid:17)Ω
where (·,·) is thescalar product inL2(Ω), 1 is thefunction which identically
Ω
equals 1 (equals 1 on Ω in the formula above), and the operators A ∈
k
B L2(Ω) are given by
(cid:0) (cid:1)
x1 x2
A f = i f(t ,x )dt , A f = i f(x ,t )dt . (2.5)
1 1 2 1 2 1 2 2
Z Z
0 0
4
Our approach is based on the simultaneous usage of two operator identities:
∗
A S −SA = i(M M +M M ) (k = 1,2). (2.6)
k k 1k 2k 3k 4k
More precisely, we use the next easy proposition. (Recall that we always
assume that (2.4) is valid and that s(x) ∈ L2(Ω).)
Proposition 2.1 Let S ∈ B L2(Ω) be an opeerator of the form (2.3).
Then, the operatoridentit(cid:0)ies (2.6(cid:1)), whereM , M ∈ B L2(0,ω ), L2(Ω) ,
1k 3k i
M , M ∈ B L2(Ω), L2(0,ω ) (i 6= k), (cid:0) (cid:1)
2k 4k i
(cid:0) (cid:1)
∂ ω2
M f (x) = s(x ,x −t )f(t )dt , (2.7)
11 ∂x Z 1 2 2 2 2
(cid:0) (cid:1) 2 0
∂ ω1
M f (x) = s(x −t ,x )f(t )dt ; (2.8)
12 ∂x Z 1 1 2 1 1
(cid:0) (cid:1) 1 0
ω1 ω2
M f (x ) = f(t ,x )dt , M f (x ) = f(x ,t )dt ; (2.9)
21 2 1 2 1 22 1 1 2 2
Z Z
(cid:0) (cid:1) 0 (cid:0) (cid:1) 0
∂
M f (x) = f(x ), M f (x ) = − s(−t ,x −t )f(t)dt; (2.10)
31 2 41 2 1 2 2
∂x Z
(cid:0) (cid:1) (cid:0) (cid:1) 2 Ω
∂
M f (x) = f(x ), M f (x ) = − s(x −t ,−t )f(t)dt; (2.11)
32 1 42 1 1 1 2
∂x Z
(cid:0) (cid:1) (cid:0) (cid:1) 1 Ω
are valid.
The operator identities (2.6) may be rewritten in the more traditional form:
M
A S −SA∗ = iΠ Π , Π := M M , Π := 2k , (2.12)
k k k k k 1k 3k k (cid:20)M (cid:21)
(cid:2) (cid:3) 4k
b b
where k = 1,2. Clearly, the operators A∗ in (2.12) have the form
k
ω1 ω2
∗ ∗
A f = −i f(t ,x )dt , A f = i f(x ,t )dt . (2.13)
1 Z 1 2 1 2 Z 1 2 2
x1 x2
We note that the integral kernels of the operators M and M given in
1k 4k
(2.7), (2.8) and in (2.10), (2.11), respectively, depend on the difference of a
one of two variables. Introducing integration operators A ∈ B L2(0,ω ) :
k k
(cid:0) (cid:1)
x1 x2
A f = i f(t )dt , A f = i f(t )dt , (2.14)
1 1 1 2 2 2
Z Z
0 0
5
and operators K ∈ B L2(0,ω ), L2(0,ω ) (k 6= i), K ∈ B C, L2(0,ω ) ,
1i k i 2i i
K ∈ B L2(Ω),C) : (cid:0) (cid:1) (cid:0) (cid:1)
4
(cid:0) (cid:1)
ω2
K f = − s(x ,−t )f(t )dt , K 1 = 1, (2.15)
11 1 2 2 2 2i
Z
0
ω1
K f = − s(−t ,x )f(t )dt , K = s(−t)f(t)dt, (2.16)
12 1 2 1 1 4
Z Z
0 Ω
one easily obtains, for instance, operator identities for M (see the proposi-
4k
tion below).
Proposition 2.2 Let the conditions of Proposition 2.1 hold. Then the fol-
lowing operator identities are valid:
∗
A M −M A = i(K M +K K ) (i,k = 1,2, i 6= k). (2.17)
i 4k 4k i 1i 2i 2i 4
Assuming that S has a bounded inverse operator, we express ρ(λ,µ) in terms
of the operators g (i 6= k):
ik
K
g = 3i I 0 −Π S−1Π , g ∈ B L2(0,ω ), L2(0,ω ) , (2.18)
ik (cid:20)K (cid:21) k i ik 2 k 2 i
1i (cid:2) (cid:3) (cid:0) (cid:1)
b
where the operators Π and Π are given in (2.12), the operators K are
i k 1i
defined in (2.15) and (2.16), and
b
ωk
K = ·dt ∈ B L2(0,ω ), L2(0,ω ) , i 6= k. (2.19)
3i k k i
Z
0 (cid:0) (cid:1)
The operators g and g are connected via the relation
ik ki
0 −I
∗
g = −U J g J U ; U f (x ) = f(ω −x ), J = i , (2.20)
ki k k ik i i i i i i i (cid:20)I 0 (cid:21)
(cid:0) (cid:1)
where J is acting in L2(0,ω ) and U is acting (depending on the context)
i 2 i i
in L2(0,ω ) or in L2(0,ω ). (cid:1)
2 i i
Theorem 2.3 Let S ∈ B L2(Ω) be a convolution operator of the form (2.3)
(where s(x) ∈ L2(Ω)). Ass(cid:0)ume th(cid:1)at S has a bounded inverse operator.
e
6
Then the ρ-function introduced in (1.4) has the forms
ρ(λ,µ) = 1 ωie−iωµ J U ψ (µ,x ) ∗ψ (λ,x )dx , (2.21)
i i i i i i i
µ −λ Z
k k 0 (cid:0) (cid:1)
where i = 1 or i = 2 (k = 1,2; k 6= i), ψ (λ,x ) ∈ L2(0,ω ),
i i 2 i
ψ (λ,x )
ψ(λ,x) = 1 1 = θ(λ)G(λ)−1col 0 1 0 1 , (2.22)
(cid:20)ψ (λ,x )(cid:21)
2 2 (cid:2) (cid:3)
I −λ A 0
1 1 iλ g
2 12
 0 I −λ A 
G(λ) := 1 1 , (2.23)
I −λ A 0
 iλ g 2 2 
 1 21 
 0 I −λ A 
2 2
 
col in (2.22) means column, and the scalar function θ(λ) is the unique func-
tion of the form
θ(λ) = 1+λ λ eiλ(ω−x)h(x)dx, h ∈ L2(Ω), (2.24)
1 2
Z
Ω
such that the expression on the right-hand side of (2.22) does not have poles
or zeros.
Here the operators A are introduced in (2.14), the operators U and J
k i i
are introduced in (2.20), and the operators g are expressed (in terms of
ik
S and S−1) in (2.18). Moreover g is expressed via g and vice versa in
12 21
(2.20), and so the representations (2.21) are completely determined by g or,
12
equivalently, by g .
21
We have also a precise formula for h(x) in (2.24):
h = S−1y, y(x) := s(x −ω ,x −ω ). (2.25)
1 1 2 2
Remark 2.4 Clearly, ρ(λ,µ) determines (see (2.26)) the inverse operator
S−1. According to Theorem 2.3, ρ(λ,µ) (and so the operator S−1) is deter-
mined by g .
ik
Recall that the operator S of the form (2.3) is determined by its inte-
gral kernel or, equivalently, by the four functions s(x), s(−x), s(x ,−x ) and
1 2
s(−x ,x ), where x ∈ Ω. Since g ∈ B L2(0,ω ), L2(0,ω ) , the integral
1 2 ik 2 k 2 i
kernel of the operator g is also determine(cid:0)d by some four func(cid:1)tions on Ω.
ik
7
The considerations above confirm a heuristic principle which, for the case
of convolution operators on the interval, was formulated and proved in [18]
(see also [17]). This principle states that the amount of information (i.e., the
number of functions) which determines S coincides with the minimal amount
of information which is necessary to exactly construct S−1.
2.3 Structure of the inverse operator
In view of (1.4), (2.1) and (2.2), the operator T = S−1 admits representation
1 ∂ ∂
Tf (x) =− l.i.m.r→∞ e−iλt (2.26)
(cid:0) (cid:1) 16π4∂x1 ∂x2 ZΩ(cid:18) ZUr
× (µ µ )−1ρ(λ,µ)(eiµ1x1 −1)(eiµ2x2 −1)dµdλ f(t)dt,
1 2
ZR2 (cid:19)
where l.i.m. stands for the limit in L2-norm and
U = {λ : |λ | < r (i = 1,2)}.
r i
We note that the boundedness of the convolution operator S and of S−1
yields the boundedness of the operator Γ determined by the relation
−1
λ I 0
Γeiλx = −i 2 ψ(λ)−col 0 eiλ1x1 0 eiλ2x2 , (2.27)
(cid:20) 0 λ I(cid:21)
1 (cid:0) (cid:2) (cid:3)(cid:1)
and acting from L2(Ω) into L2(0,ω )⊕L2(0,ω ).
2 1 2 2
Now, we formulate an inverse result.
Theorem 2.5 Let a given operator g (where i = 1, k = 2 or i = 2, k = 1)
ik
belong to B L2(0,ω ),L2(0,ω ) . Assume that the operator T, which is de-
2 k 2 i
termined by(cid:0)g via formula (2.2(cid:1)6) and via the procedure to construct ρ(λ,µ)
ik
from Theorem 2.3, is bounded. Assume that the operator Γ (which is deter-
mined by g via equality (2.27), relation(2.20) and formulas from Theorem
ik
2.3, namely, formulas (2.22)-(2.24)) is bounded as well.
Then, if the inverse operator S = T−1 exists and is bounded, this operator
S is a convolution operator, that is, S admits representation (2.3).
Acknowledgments. TheauthorisgratefultoB.KirsteinandL.Sakhnovich
for useful discussions. The research was supported by the Austrian Science
Fund (FWF) under Grant No. P29177.
8
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