Download General Domain Truncated Correlation and Convolution Operators

On General Domain Truncated Correlation and
Convolution Operators with Finite Rank
Fredrik Andersson and Marcus Carlsson
Abstract. Truncated correlation and convolution operators is a general
operator-class containing popular operators such as Toeplitz (WienerHopf), Hankel and nite interval convolution operators as well as small
and big Hankel operators in several variables. We completely characterize the symbols for which such operators have nite rank, and develop
methods for determining the rank in concrete cases. Such results are
well known for the one-dimensional objects, the rst discovered by L.
Kronecker during the 19th century. We show that the results for the
multidimensional case dier in various key aspects.
Mathematics Subject Classication (2010). 47B35, 15A03, 47A13, 33B10.
Keywords. Hankel, Toeplitz, Truncated convolutions, nite rank, exponential functions.
1. Introduction
Υ we will mean an open, non-empty set in Rd , d ≥ 1,R and we de2
note by L (Υ) the space of functions on Υ such that kf kL2 (Υ) =
|f |2 dx <
Υ
∞. Let Υ, Ξ be two domains and set
By a domain
2
Ω = Ξ + Υ = {x + y : x ∈ Ξ, y ∈ Υ}.
Denition 1.1.
and such that
L2 (Υ) via
φ ∈ L2loc (Ω) be such that φ(x + ·) ∈ L2 (Υ) for all x ∈ Ξ
φ(· + y) ∈ L2 (Ξ) for all y ∈ Υ. We dene Γφ = Γφ,Υ,Ξ on
Let
Z
Γφ (f )(x) =
φ(x + y)f (y) dy,
x ∈ Ξ.
(1.1)
Υ
Γφ
will be called a truncated correlation operator, and the function
referred to as its symbol.
φ
will be
2
Fredrik Andersson and Marcus Carlsson
The above assumptions on
φ will remain xed throughout the paper and will
not be repeated. The word truncated thus refers to that we restrict the
domain of denition of the functions on both sides of the correlation. If we
let
ι : L2 (−Υ) → L2 (Υ)
denote the operator
ι(f )(x) = f (−x),
then
Γ◦ι
is
a truncated convolution operator, since
Z
φ(x − y)f (y) dy,
Γφ ι(f )(x) =
x ∈ Ξ.
(1.2)
−Υ
Hence
Γφ
is unitarily equivalent to the truncated convolution operator
Γφ ι,
and thus all statements concerning the rank of one can easily be transferred
to the other. In the remainder we focus mainly on the truncated correlation
operators.
Γφ is always bounded (in fact, even HilbertL2 (Ξ) as long as Υ and Ξ are bounded domains
We shall see in Section 2 that
Schmidt) as an operator into
and
φ ∈ L2 (Ω),
but this may not be the case otherwise. We shall restrict
attention only to
2
L (Ξ).
φ's
such that
Γφ
is a bounded operator from
Whenever we mention boundedness of
Γφ ,
L2 (Υ)
into
it should be interpreted in
this sense.
1.1. Review
The class of operators of the form
Γφ
and
Γφ ι contain several standard classes
d = 1 and Υ = Ξ =
of Hankel and Toeplitz operators. For example, setting
R+ = (0, ∞)
we obtain the class of Hankel operators on the real line
∞
Z
Γφ (f )(x) =
φ(x + y)f (y) dy,
x > 0,
φ(x − y)f (y) dy,
x > 0,
0
whereas with
Υ = R−
we have
Z
∞
Γφ ι(f )(x) =
0
i.e. we retrieve the class of Toeplitz operators on
R, also known as the WienerR are
Hopf operators. Apart from semi-axes the only connected domains in
open intervals. Taking
tors
{Γφ ι}
Υ = Ξ = (−1, 1)
the corresponding class of opera-
is known by many names, e.g. nite interval convolution opera-
tors, Toeplitz (or Hankel) operators on the Paley Wiener space, Truncated
Wiener-Hopf operators. In the above settings one usually also allows certain
φ = δ1 in the Hankel case
Γδ1 (f )(x) = 1(0,1) (x)f (1 − x), where 1(0,1)
function of (0, 1). However, boundedness of clas-
distributions as symbols. For example, if we set
then it is clearly natural to have
denotes the characteristic
sical Toeplitz and Hankel operators is well understood, which implies that
the classes of distributions that give rise to bounded operators can be characterized. This is also true to a lesser degree for nite interval convolution
operators, [2, 9, 30], but it is not the case in the full generality that we will
consider in this paper. For simplicity, such symbols are excluded from the
analysis of this paper.
On General Domain Truncated Correlation and Convolution.
We briey discuss classes of operators in higher dimensions. Setting
Rd+
3
Υ=Ξ=
we obtain the class of small Hankel operators. The question of criteria
for boundedness of such operators was open for around 50 years and recently
solved in [15] for
d=2
and [27] for
d > 2.
If we switch
Ξ
for
Rd \ (−∞, 0]d ,
the corresponding class of operators are known as the big Hankel operators,
see e.g. [12, 13] (for the unitarily equivalent case on the unit circle).
Kronecker's theorem [26, 29] states that classical Hankel operators (on
R+ )
have nite rank if and only if the symbol is of the form
φ(x) =
J
X
pj (x)eζj x
(1.3)
j=1
where
pj
are polynomials and
the rank of
Γφ
Re ζj < 0. Moreover, unless there is cancelation,
equals
K=
J
X
(deg(pj ) + 1).
(1.4)
j=1
An analogous result holds for nite interval convolution operators, but without the restriction
Re ζj < 0, as shown in [30]. Finally, we recall that there are
no non-trivial nite rank Toeplitz operators. These 3 results are all special
cases of Theorem 1.2 below.
Symbols of the form
φ(x) =
K
X
c k e ζk x ,
ck ∈ C
(1.5)
k=1
are known to be dense in the set of all symbols giving rise to rank
K
Han-
kel operators. Hence, the general form (1.3) is hiding the following simpler
statement:
Observation 1.
φ
is a sum of
Γφ generically has rank
exponential functions.
A Hankel operator
K
K
if and only if
We remark that (discretized) nite rank Hankel operators and nite interval
convolution operators plays an important role in signal processing, control
theory and approximation theory, see for instance [10, 29, 31] and the references therein. In particular, Observation 1 is important for a number of algorithms for numerical integration [5], frequency estimation [1], crack detection
[3] and tomography [6], to name a few. However, most of these applications
are limited to one-dimensional results. One reason for studying truncated
convolution operators on arbitrary domains, as opposed to domains with a
simple geometrical structure, is that data (e.g. in geoseismic imaging) are
often not measured on such domains.
1.2. Results
Before giving a more detailed overview, we summarize our main results:
4
Fredrik Andersson and Marcus Carlsson
•
•
An analogue of (1.3) is true in several variables.
There is no simple expression corresponding to (1.4) that determines
the rank of a given bounded
Γφ .
Nevertheless, we develop methods for
determining the rank.
•
Observation 1 does not generalize to several variables.
d≥1
Let
be xed. Given
x ∈ Rd
Pd
ζ ∈ Cd we set x · ζ = i=1 xi ζi . Pol
Rd and ExpPol the set of functions of
and
will denote the set of polynomials on
the form
N
X
pj (x)ex·ζ j :
ζ j ∈ Cd ,
pj ∈ Pol,
N ∈ N.
(1.6)
j=1
Ω ∈ Rd , we will write φ ∈ ExpPol to mean
that φ can be represented by (1.6) on Ω. If the ζ j 's are such that Re (ζ j )
d
is restricted to lie in some subset Θ ⊂ R , we write ExpPolΘ . ExpSep will
denote the subset of ExpPol of functions where the pj 's are constant, and for
a given K we write ExpSepK for such sums with precisely K terms. Finally,
d
given a domain Ω, we let ΘΩ denote the set of directions θ ∈ R such that
the orthogonal projection of Ω on the half line [0, ∞) · θ is a bounded set,
and we let int(ΘΩ ) denote its interior. We show (see Theorems 4.4 and 9.3)
If
φ
is a function on a domain
Theorem 1.2.
Υ, Ξ ⊂ Rd
Ω = Υ + Ξ.
φ ∈ ExpPolint(ΘΩ ) .
Let
bounded, and set
only if
be connected domains that are either convex or
Then
Γφ
is bounded and has nite rank if and
Note that the three Kronecker type results for the one dimensional case follows directly from the above theorem, since
•
Hankel: Υ = Ξ = R+ implies Ω = R+ , ΘΩ = (−∞, 0] and int(ΘΩ ) =
(−∞, 0).
• Toeplitz: Υ = −Ξ = R+ implies Ω = R, ΘΩ = {0} and int(ΘΩ ) = ∅.
• Finite interval convolution operators: Υ = Ξ = [−1, 1] implies Ω =
[−2, 2] and ΘΩ = int(ΘΩ ) = R.
We also point out the following corollary concerning small Hankel operators,
which seems to be new.
Corollary 1.3.
A small Hankel operator
rank if and only if
This follows since
Γφ,Rd+ ,Rd+
is bounded and has nite
φ ∈ ExpP ol(−∞,0)d .
ΘΩ = (−∞, 0]d .
It it also easy to use Theorem 1.2 to show
that there are no nite rank big Hankel operators.
We now focus on determining the rank of a given
say that (1.6) is reduced if
Proposition 1.4.
Let
φ ∈ ExpPol
Let
ζ j 6= ζ j 0
Υ, Ξ ⊂ Rd
for all
Γφ , φ ∈ ExpPol.
We will
j 6= j 0 .
be connected domains and set
Ω = Υ + Ξ.
be given and assume that the form (1.6) is reduced. Suppose
On General Domain Truncated Correlation and Convolution.
also that
Γφ
is bounded. Then its rank is independent of
Rank Γφ =
X
Υ
and
Ξ.
5
Moreover,
Rank Γpj .
j
We remark that
Γφ , φ ∈ ExpPol,
is always bounded when
Υ, Ξ
are bounded
domains. Due to the above proposition, we will in the remainder of the in-
Υ and Ξ. In one variable, the rank of a
deg(p) + 1. The situation in several variables is
more intricate. In what follows, we will use C[x] to denote all polynomials
d
over C in variables x = (x1 , . . . , xd ). Let y ∈ R denote another independent
variable and note that C[x, y] = C[x] ⊗ C[y].
troduction not explicitly write out
given
Γp (p ∈ Pol)
Proposition 1.5.
p(x + y)
equals
Given
p ∈ C[x] the rank
C[x] ⊗ C[y].
of
Γp
equals the tensor rank of
as an element of
Furthermore, the tensor rank can be computed by computing the rank of a
certain matrix, which is explained in Section 5. It turns out that it is not
possible to determine the rank of a given
p.
Γp
only by knowing the degree of
However, it is possible to say what the rank generically is, cf. Section 7:
Theorem 1.6.
d
be a polynomial on R of degree N . The rank of Γp then
N −1
N
N
2 +d , if N is odd, and 2 2 +d−1 +
2 +d−1 , if N is
generically equals 2
d
d
d−1
even. These numbers are also upper bounds for the rank.
Fix
K ∈ N.
Let
p
We now turn to describing
MK = {φ : Rank Γφ = K}
as a geometrical object in
L2 (Υ) → L2 (Ξ)
ExpPol.
Here,
Γφ
should be interpreted as
for arbitrary bounded domains
Υ
and
Ξ,
Γφ :
since the rank is
independent of this choice by Proposition 1.4. It can be veried that
MK
is a
union of dierentiable manifolds of varying dimensions. We also remark that
Rank Γφ = K whenever φ ∈ ExpSepK , so ExpSepK ⊂ MK . In one variable,
ExpSepK is dense in MK (with respect to uniform convergence on compacts in
C, c.f. Observation 1). Moreover, ExpSepK is a 2K -dimensional manifold and
all other components of MK are manifolds of lower dimension. Surprisingly,
this is not the case in several variables; it will turn out that ExpSepK is often
a manifold of lower dimension than the dimension of MK , (i.e. the maximal
dimension of its various manifold components). In other words, a generic
element of
MK
is not in
ExpSepK ,
showing that Observation 1 is false for
d > 1.
The paper is organized as follows. In Section 2 we give a number of basic results concerning
Γφ 's. In particular, we provide necessary conditions for
boundedness, compactness, and we describe the Takagi factorization for compact
Γφ 's.
The latter is a useful result regarding the structure of the singular
vectors which is interesting in its own right. In Section 3 we prove some rudimentary results concerning the class of functions
ExpPol. Section 4 is devoted
6
Fredrik Andersson and Marcus Carlsson
to proving Theorem 1.2 in the case of bounded domains. In Section 5 we give
Proposition 1.4 and show how to determine the rank of a given
Γp , p ∈ Pol,
and in Section 6 some illustrative examples are given. In Section 7 the notion
of generic is dened and basic results concerning this concept are proved. The
section then proceeds with dealing with the generic rank of
Γp 's, in particular
we state and prove Theorem 1.6. Section 8 describes the manifold structure
MK which concludes that ExpSepK in fact only makes up
MK . In Section 9 we extend most previous results to the case
of
a tiny part of
of unbounded
convex domains, and in particular the full version of Theorem 1.2 is given.
We end the paper by demonstrating how the fact that there are no non-trivial
ΘΩ = ∅
nite rank operators when
2. Basic properties of
Given any domains
1.1. Then
Γφ
Υ
can be circumvented by using weights.
Γφ 's
and
Ξ,
let
Ω = Υ+Ξ
and let
φ
be as in Denition
is a particular case of an integral operator (see e.g. [19]), and
several books devoted to this subject give particular boundedness criteria.
We recall a few basic ones below as well as some other useful observations of
|Υ| for the Lebesgue measure of Υ.
p
p
kΓφ k ≤ |Ξ|kφk2 and kΓφ k ≤ |Ξ||Υ|kφk∞ .
independent interest. We write
Proposition 2.1.
We have
Proof. By the Cauchy-Schwartz inequality we have that
2
Z Z
=
φ(x + y)f (y) dy dx ≤
Υ
Ξ
Z Z
|Ξ|kφk22 kf k22
≤
|φ(x + y)|2 dykf k22 dx ≤
|Ξ||Υ|kφk2∞ kf k22
Ξ Υ
kΓφ (f )k22
φ ∈ L2 (Ω),
the operator Γφ,Υ,Ξ is
Υ and Ξ are bounded domains. In fact, it is even compact,
The above proposition implies that for all
bounded whenever
as will be shown in Proposition 2.3 below. Proposition 2.1 combined with the
next proposition implies that it suces that either
order for
Γφ,Υ,Ξ
to be bounded (when
Proposition 2.2.
If
Γφ,Υ,Ξ
Proof. By Denition 1.1,
2
2
f ∈ L (Υ)
g ∈ L (Ξ)
and
is bounded, then so is
|φ|2
Ξ
or
Υ
be bounded in
φ ∈ L2 (Ω)).
Γφ,Ξ,Υ
and
Γ∗φ,Υ,Ξ = Γφ,Ξ,Υ .
is integrable on compact subsets of
have compact support (in
Υ
and
Ξ
By Fubini's theorem we then have
Z Z
hΓφ,Υ,Ξ f, giL2 (Ξ) =
φ(x + y)f (y) dy g(x) dx =
Ξ
Z
=
Υ
Z
f (y)
Υ
Ξ
Ω.
Let
respectively).
φ(x + y)g(x) dx dy = hf, Γφ,Ξ,Υ giL2 (Υ)
On General Domain Truncated Correlation and Convolution.
7
R
Γ∗φ,Υ,Ξ g = Ξ φ(x + y)g(x) dx for any such g . Since φ(· + y) ∈
y , the general statement follows by a standard approximation
It follows that
L2 (Ξ)
for all
argument.
We recall that the Hilbert-Schmidt norm of a compact operator
2
(σn )∞
n=0 is kAkHS =
we dene kAkHS = ∞.
gular values
compact,
Proposition 2.3.
Let domains
P∞
Υ, Ξ
Then
kΓφ,Υ,Ξ k2HS =
n=0
|σn |2 .
If this equals
be given and set
Z
∞
A
or if
with sin-
A
is not
w(z) = |({z} − Υ) ∩ Ξ|.
|φ(z)|2 w(z)dz.
Ω
k(x, y) = φ(x + y) for (x, y) ∈ Ξ × Υ. Then Γφ equals the integral
k , and it is well known that this is Hilbert-Schmidt if
and only if kkkL2 < ∞, and moreover that this expression equals the HilbertProof. Set
operator with kernel
Schmidt norm, (see e.g. the proof of Theorem 4.5 in [19]). We now consider
φ
to be dened on
Rd
and identically zero outside
Ω.
The result then follows
upon noting that
kkk2L2
Z Z
1Υ
Z Z
|k(x, y)| dxdy =
=
Υ
where
2
Ξ
Υ
1Υ (y)1Ξ (z − y)|φ(z)|2 dzdy
Ξ
denotes the characteristic function of
Υ.
In particular, we have
Corollary 2.4.
If
Υ, Ξ
are bounded domains and
φ ∈ L2 (Ω),
then
Γφ,Υ,Ξ
is
Hilbert-Schmidt.
Corollary 2.5.
For a small Hankel operator
kΓφ,Rd+ ,Rd+ k2HS =
For particular classes of
Z
Γφ,Rd+ ,Rd+
we have
x1 . . . xd |φ(x)|2 dx
Γφ 's, necessary and sucient conditions for compact-
ness are available only in few cases. For one-dimensional Hankel operators,
these are given by the classical theorem of Hartman [20, 25, 29], and for nite
interval convolution operators such conditions are given in [9, 30]. Compact
Wiener-Hopf operators do not exist, since they are unitarily equivalent with
Toeplitz operators [7, 29]. Concerning the multi-variate case, conditions for
small Hankel operators in two dimensions are available in [28].
Ξ = Υ. Let C be
L2 (Υ) given by C(f ) = f . By Proposition 2.2 it
CΓφ = Γ∗φ C , which in the terminology of [16, 17]
We conclude with a few remarks on the particular case
the antilinear involution on
immediately follows that
means that
Γφ
is
C -symmetric. This implies a number of structural properΓφ is also compact, then a generalized form of the Takagi
ties. In particular, if
factorization [23] applies;
8
Fredrik Andersson and Marcus Carlsson
Theorem 2.6.
Let
Υ=Ξ
be any domain and let Γφ be compact. Then there
(un )∞
n=0 of singular vectors such that
exists an orthonormal basis
Γφ (un ) = σn un ,
where
σn
n ∈ N,
are the singular values.
The remainder of this paper will be devoted to the study of nite rank
Γφ 's.
In the next section we introduce the corresponding symbol-class.
3. The class
ExpPol
We recall the terminology
ticular, given
Rd
K∈N
Pol, ExpPol etc. introduced around (1.6). In parExpSepK denoted the set of all functions in
recall that
of the form
K
X
ck ex·ζ k ,
cj ∈ C, ζ k ∈ Cd
k=1
1 ≤ j < l ≤ K and ck 6= 0 for all 1 ≤ k ≤ K . ExpSepK
is an abbreviation of (sums of ) K exponential functions with separated exponents. If we work on a domain Υ, we shall without comment interpret
ExpPol as a set of functions on Υ, which we sometimes denote by ExpPol(Υ)
where
ζ j 6= ζ l
for all
for clarity.
To
ExpPol
Cd
(of the unique holomorphic extensions). More precisely, we give
we associate the topology of uniform convergence on compacts in
ExpPol
the locally convex topology dened by the semi-norms
X
X
pj (x)ex·ζ j = sup pj (z)ez·ζ j ,
Λ
where
Λ
z∈Λ
is an arbitrary compact subset of
Cd .
Clearly,
ExpPol
is a Fréchet
space. We have chosen this topology because the Taylor coecients at a given
point then depend continuously on the function. The closure of e.g.
ExpSepK
with respect to this topology will be denoted by cl(ExpSepK ). It is easy to
see that
∪∞
K=1 cl(ExpSepK ) = ExpPol,
(3.1)
since e.g.
eεx1 − 1
ε→0
ε
x1 = lim
(3.2)
n ∈ Nd , we shall use
n1 n2
standard multi-index notation, i.e. if d = 2 we have x = x1 x2 and |n| =
n1 + n2 . We let PolN denote the set of all polynomials with degree ≤ N . For
and the left hand side of (3.1) is an algebra. Given
n
later reference, we recall the following well known facts, and include proofs
for the convenience of the reader.
On General Domain Truncated Correlation and Convolution.
Proposition 3.1.
let
N ∈N
Let
Υ
be a domain, let
{ζ k }K
k=1
be distinct points in
9
Cd
and
be given. Then
{xn ex·ζ k }1≤k≤K,|n|<N
is a linearly independent set of functions on
Proof. For
d=1
Υ.
the statement is elementary, see e.g. [8, Sec. 3.3]. On prod-
uct domains in several variables, the statement is a direct consequence of the
fact that tensor products of linearly independent elements are linearly independent. Finally, for a general domain, we can consider a product domain
inside it, and since
ExpPol
are real analytic functions they can not vanish on
open subsets unless they are identically zero. (See e.g. [18, Sec. A] for the
necessary holomorphic function theory).
Proposition 3.2.
.
Let
Υ
be a domain. We have
dim(PolN (Υ) PolN −1 (Υ)) =
dim(PolN (Υ)) =
N +d
d
and
N +d−1
.
d−1
N +d−1
.
d−1
By Proposition 3.1 the latter part of this proposition follows. The former can
N +d−1
+d−1
now be deduced by induction and the formula
+ Nd−1
= Nd+d .
d
Proof. The amount of monomials
xα of degree equal to N
is given by
We end this section with a useful technical observation.
Lemma 3.3.
φ be a measurable function on
x ∈ Ω there is an open neighborhood
that φ is Lebesgue a.e. equal to an element in ExpPol(Ξ). Then φ is
equal to an element of ExpPol(Ω).
Ω, and
Ξ such
a.e.
Let
Ω
be a connected domain,
suppose that around each point
x be xed and redene φ near x so that it equals the corresponding
ExpPol. If two dierent points x1 and x2 have overlapping neighborhoods, the corresponding redenitions of φ must agree on the intersection,
Proof. Let
element in
since otherwise their dierence would have a non-zero Taylor expansion at
some point, and such functions can not be a.e. equal to zero, as is seen e.g. by
φ is locally
ExpPol near each point, which makes it a real analytic function. The fact
that φ ∈ ExpPol(Ω) now follows by the uniqueness of continuation of real
analytic functions on connected domains.
the Weierstraÿ preparation theorem. Thus we may assume that
in
4. Finite rank
Γφ 's
on bounded domains
In this section we prove Theorem 1.2 for bounded domains
Ξ, Υ. We restrict
attention to bounded domains in order to keep diculties of rank and boundedness separate. The unbounded case will be considered rst in Section 9. We
will also henceforth restrict attention to connected domains
Ξ, Υ.
10
Fredrik Andersson and Marcus Carlsson
Proposition 4.1.
the operator
Γφ
Given connected bounded domains
has rank
ζ ∈ Cd
Υ, Ξ
and
x ∈ Ξ we have
Z
E
D
Γex·ζ (f )(x) =
e(x+y)·ζ f (y) dy = ex·ζ f, ey·ζ
Proof. Given
and
Υ
by which we conclude that this operator has rank 1. Let
ExpSepK
φ ∈ ExpSepK ,
K.
L2 (Υ)
φ=
,
PK
k=1 ck e
x·ζ k
ΓPK
k=1
ck e
x·ζ k
=
K
X
D
E
ck ex·ζ k f, ey·ζ k
L2 (Υ)
k=1
,
the desired conclusion follows by Proposition 3.1.
Given
where
∈
be given. Since
ζ ∈ Cd , let PolN · ex·ζ
p ∈ PolN .
Lemma 4.2.
Let
denote the set of functions of the form
φ(x) ∈ PolN · ex·ζ .
Then
p(x)ex·ζ ,
Ran Γφ ⊂ PolN · ex·ζ .
Proof. This follows immediately since
x·ζ
Z
p(x + y)ey·ζ f (y) dy
Γφ (f )(x) = e
Υ
p(x + y) can be expanded
with |i| ≤ N and |j| ≤ N .
and
into a sum of coecients of the form
xi y j
Γφ1 +φ2 = Γφ1 + Γφ2 already
φ ∈ ExpPol. To determine the rank of
The above lemma combined with the formula
Γφ
shows that
a given
Γφ
has nite rank for any
with
φ ∈ ExpPol \ ExpSep
is much more complicated, and we will
return to this issue after showing that all nite rank
Γφ 's
have
φ ∈ ExpPol.
Before proceeding, we need the following result:
Theorem 4.3.
Let
set of polynomials
Υ ⊂ Rd be a connected domain and let {p1 , . . . , pd }
∞
on C. If φ ∈ C (Υ) solves pj (∂j )φ = 0, 1 ≤ j ≤ d,
be a
then
φ ∈ ExpPol.
The theorem is a special case of more general theorems on solutions of systems
of constant coecient PDE's, such as Theorem 7.6.14 in [21]. The special
case stated above can be proven by basic means, and we include a proof
for completeness. The exponents involved in
φ
as a function in
ExpPol
are
obviously connected to the zeroes of the polynomial equations, but a precise
statement gets complicated especially when we have multiplicities larger than
1, so we omit it since it is not needed. The relationship between exponents
and zeroes is implicit in the proof.
On General Domain Truncated Correlation and Convolution.
Proof. We proceed by induction. For
d=1
the statement is well known, see
e.g. [8, Sec. 3.5]. Now assume that the statement is true for
3.3 it suces to assume that
Υ
11
d − 1. By Lemma
is a multidimensional cube. By translation
Υ = I × Υ0 where I
0
is an interval and Υ is a cube in R
. Similarly we write x = (x, x ) with
0
d−1
x ∈ R . Now, by the one-dimensional result we have, for each xed x0 ,
invariance, we may also assume that
0
0 ∈ Υ.
We write
d−1
that
φ(x, x0 ) =
mk
K X
X
fk (x0 )xj−1 exλk
(4.1)
k=1 j=1
mk their multiplicity. For notational simmk = 1 for all k , so that (4.1) simplies to
PK
φ(x, x0 ) = k=1 fk (x0 )exλk . Let ε be such that Kε ∈ I and let M be the
K × K -matrix M (n, k) = eεnλk , 1 ≤ n, k ≤ K . Then M is a Vandermonde
where
λk
are the zeroes of
p1
and
plicity, we shall assume that
matrix which is well known to be invertible (see e.g. [23] or prove it us-
Φ and F denote the vectors of
0 K
(φ(εn, x0 ))K
and
(f
(x
))
,
then
(4.1) implies that Φ = M F
k
n=1
k=1
−1
so F = M
Φ. Hence each of the coecients fk solve the equation-system
{pj (fk )}dj=2 , so by the induction hypothesis, fk ∈ ExpPol for 1 ≤ k ≤ d.
Inserting this into (4.1) gives the desired result.
ing the fundamental theorem of algebra). If
functions
We are now in a position to prove Theorem 1.2 for the case of bounded connected domains, the proof follows that of [30]. Given a function
we write
φ ∈ ExpPol
φ.
if there exists a representative in
ExpPol
φ ∈ L2loc (Ω)
for the equiva-
lence class
Theorem 4.4.
Let
Υ
Ξ be
Γφ,Υ,Ξ
and
in Denition 1.1. Then
bounded connected domains and let
φ
be as
is bounded and has nite rank if and only if
φ ∈ ExpPol(Ω).
Proof. The if part has already been established above, so suppose that Γφ
2
K and pick functions {hk }K
k=1 ⊂ L (Υ) and
P
K
2
{gk }K
k=1 ⊂ L (Ξ) such that Γφ =
k=1 gk ⊗ hk , i.e.
is bounded and has nite rank
Γφ (f ) =
K
X
Z
gk
f (y)hk (y) dy
k=1
for all
f ∈ L2 (Υ).
We claim that
φ(x + y) =
K
X
gk (x)hk (y)
(4.2)
k=1
Ξ × Υ. To see this, note that the function φ by Denition 1.1
L2loc (Ω), which implies that |φ|2 is integrable over compact
subsets of Ω. Given compact subsets Ξ̃ ⊂ Ξ and Υ̃ ⊂ Υ we thus have that
PK
φ(x + y) − k=1 gk (x)hk (y) is a function in L2 (Ξ̃ × Υ̃) which annihilates all
2
2
functions of the form e(x)f (y) for e ∈ L (Ξ̃) and f ∈ L (Υ̃). This implies
Lebesgue a.e. in
is a member of
12
Fredrik Andersson and Marcus Carlsson
PK
φ(x + y) = k=1 gk (x)hk (y) Lebesgue a.e. in Ξ̃ × Υ̃ since L2 (Ξ̃ × Υ̃) =
L2 (Ξ̃) ⊗ L2 (Υ̃). (4.2) thus follows since Ξ̃ and Υ̃ were arbitrary.
that
Υ̃ be an open connected domain with cl(Υ̃) ⊂ Υ. Let α ∈ Cc∞ (Rd ) be
a positive radial function with norm 1 and support in Ball(0, dist(∂Υ, ∂ Υ̃)).
2
Given any f ∈ L (Υ) with support in cl(Υ̃), a short computation shows that
Now let
Γφ,Υ̃,Ξ (f ∗ α) = Γφ∗α,Υ̃,Ξ (f ) =
K
X
(gk ⊗ (hk ∗ α))(f ),
k=1
where e.g.
x ∈ Υ.
f ∗α
is to be interpreted as
f ∗ α(x) =
R
Υ
f (y)α(x − y)dy
for
As before this leads to the identity
φ ∗ α(x + y) =
K
X
gk (x)hk ∗ α(y)
k=1
(x, y) ∈ Ξ × Υ̃. But this means that for a.e. x, the identity holds
y , which means everywhere (in y ) since both sides are
n
continuous. Fix such an x and apply ∂1 for 1 ≤ n ≤ K to obtain
for a.e.
a.e. in the variable
∂1n φ ∗ α(x + y) =
K
X
gk (x)(∂1n hk ∗ α)(y).
k=1
y ∈ Υ̃. The above identity then holds a.e. in x, so x 7→ ∂1n φ∗α(x+y)
K
is in the K -dimensional space Span {gk }k=1 . We may thus nd a polynomial
p1 (depending on y and α) such that p1 (∂1 )(φ∗α(·+y)) = 0 a.e. However, φ∗α
Now x
and its derivatives are continuous, so the identity actually holds pointwise.
The same argument can of course be repeated for the other variables, and so
Theorem 4.3 implies that
φ ∗ α(· + y) ∈ ExpPol(Ξ).
Thus
φ ∗ α(· + y) ∈ ExpPol(Ξ) ∩ Span {gk }K
k=1
L2 (Ξ). Now, the space on the right is nite dimensional and
independent of α. Since α can be chosen such that φ ∗ α approximates φ with
2
arbitrary precision in L (Ω), we conclude that φ(· + y) is a.e. identical to an
element of ExpPol(Ξ), for every y ∈ Υ̃. Since Υ̃ was arbitrary, the proof is
as an identity in
complete by Lemma 3.3.
5. Determining the rank.
In comparison with the neat theory in one variable, determining the rank of a
given
Γφ
with
φ ∈ ExpPol
is rather tricky and the theory has a few surprises.
The rst steps to simplify the problem are however rather straightforward.
On General Domain Truncated Correlation and Convolution.
Proposition
5.1.
P
K
k=1
pk ex·ζ k
Υ and Ξ be bounded connected domains.
pk ∈ Pol and the ζ k 's are distinct, we have
Let
where
Rank Γφ =
K
X
Rank Γpk ex·ζk =
k=1
K
X
13
Given
φ =
Rank Γpk
k=1
Ran Γpk ex·ζk ⊂ Pol·ex·ζ k , which by Proposition
the various Γp ex·ζk 's are linearly independent,
k
Proof. By Lemma 4.2 we have
3.1 means that the ranges of
from which the rst equality follows. For the second, suppose that we have
functions
a ∈ L2 (Ξ)
and
b ∈ L2 (Υ)
Γpk (f ) =
such that
J
X
aj (x)hf, bj iL2 (Υ) .
j=1
Then
Γpk ex·ζk (f ) = ex·ζ k Γpk (ey·ζ k f ) =
J
X
ex·ζ k aj (x)hf, ey·ζ k bj iL2 (Υ) ,
j=1
from which the identity
Rank Γpk ex·ζk = Rank Γpk
easily follows.
p ∈ Pol, we need some elements
U and V be linear spaces, and
denote by W the tensor product W = U ⊗ V . The rank of an element w ∈ W
is then the minimal R for which there exists a representation
To determine the rank of a given
Γp
with
from tensor rank theory, see e.g. [11, 24]. Let
w=
R
X
ur ⊗ vr .
r=1
The obvious analogue for multiple tensor products also holds, but is not
needed here. For more than two spaces the determination of the rank is very
complicated [11]. Fortunately, for just two spaces the determination of the
rank can be reduced to linear algebra, as follows (see e.g. [24]):
Proposition 5.2.
be a basis for a
1
{e1j }dj=1
be a basis for a subspace of U and
subspace of V . Then each w can be expressed as
X
w=
ci,j e1i ⊗ e2j .
Let
let
2
{e2j }dj=1
i,j
C = (ci,j )i,j be the corresponding matrix. The (tensor) rank of
equal to the (matrix) rank of C .
Let
w
is then
For the remainder of this section, we will use
C[x] to denote all polynomials
x = (x1 , . . . , xd ) (there is no dierence with the previous
d
notation Pol). Let y ∈ R denote another independent variable and note that
C[x, y] = C[x] ⊗ C[y].
over
C
in variables
Proposition 5.3.
the rank of
Γp
Let
Υ
and
Ξ
p ∈ C[x]
C[x] ⊗ C[y].
be bounded connected domains. Given
equals the rank of
p(x + y)
as an element of
14
Fredrik Andersson and Marcus Carlsson
p(x + y) as an element of C[x] ⊗ C[y], and
u
(x)v
(y)
for some ur ∈ C[x] and vr ∈ C[y]. Then
r
r=1 r
Z
R
X
Γp (f )(x) =
p(x + y)f (y) dy =
ur (x)hf, vr iL2 (Υ) , x ∈ Ξ,
Proof. Let
p(x + y) =
R be
PR
the rank of
Υ
write
(5.1)
r=1
Rank Γp ≤ R. Conversely,
2
and vt ∈ L (Υ) such that
so
Γp (f )(x) =
set
T
X
T = Rank Γp .
Then there are
ut (x)hf, vt iL2 (Υ) ,
ut ∈ L2 (Ξ)
x ∈ Ξ.
t=1
Let
N
be the total degree of
p
and consider
PolN (Υ)
as a subspace of
L2 (Υ).
Clearly
PolN (Υ)
⊥
⊂ Ker Γp ,
which implies that vt ∈ PolN (Υ), t = 1 . . . T , which immediately gives vt ∈
PolN (Υ). The same argument applied to Γ∗p , keeping Proposition 2.2 in mind,
also yields that ut ∈ PolN (Ξ) for all t = 1 . . . T . Clearly
p(x + y) =
T
X
ut (x)vt (y)
t=1
a.e. on
C[y]
Ξ × Υ.
If we let
Ut (x)
and
Vt (y)
denote the elements of
C[x]
and
that coincide a.e. with their lower case counterparts on the respective
domains, we get that
p(x + y) =
T
X
Ut (x)Vt (y)
t=1
holds in the open set
Υ × Ξ.
C[x, y],
R ≤ T , and
This implies that the identity holds in
since polynomials can not vanish on open sets. This proves that
the proof is complete.
Γφ
Υ and Ξ, as claimed in Proposition 1.4. This will be further
Note that Proposition 5.1 and 5.3 together show that the rank of a given
is independent of
elaborated on in Theorem 9.1.
6. Examples
To better understand the propositions in the above section, we develop a few
examples. We let
degree
≤ N.
(C[x])N
the set of polynomials in two variables of total
(This notation is more suitable than
PolN
in what follows, but
the meaning is the same.) We will denote the independent variables of
(u, v)
and those in
y
by
(µ, ν).
bounded connected domains in
In all examples,
R2 .
Υ
and
Ξ
x
by
will be arbitrary
On General Domain Truncated Correlation and Convolution.
Example 6.1.
First consider a general
15
p ∈ (C[u, v])2 ;
p(u, v) = a0,0 + a1,0 u + a0,1 v + a2,0 u2 + a1,1 uv + a0,2 v 2 .
(e11 , . . . , e16 ) = (1, u, v, u2 , uv, v 2 ) for (C[u, v])2 and the basis
2
2
(e1 , . . . , e6 ) = (1, µ, ν, µ2 , µν, ν 2 ) for (C[µ, ν])2 , the rank determining matrix
for p(u + µ, v + ν) in C[u, v] ⊗ C[µ, ν] (see Proposition 5.2) is given by
Using the basis




C=



a0,0
a1,0
a0,1
a2,0
a1,1
a0,2
a1,0
2a2,0
a1,1
a0,1
a1,1
2a0,2
a2,0
a1,1
a0,2




,



(6.1)
where the empty spaces represent zeroes. This is seen by expanding
p(u + µ, v + ν) =
a0,0 +
a1,0 (u + µ) + a0,1 (v + ν)+
a2,0 (u + µ)2 + a1,1 (u + µ)(v + ν) + a0,2 (v + ν)2 .
Clearly this matrix has rank 4 except for degenerate cases, and hence
Γp,Υ,Ξ
Γp =
generically has rank 4 by Propositions 5.2 and 5.3. Each of the 3
rst rows form its own rank 1 matrix by inserting zeroes elsewhere, and
the remaining 3 elements of the rst column also forms a rank 1 matrix by
C in four rank one
p(u + µ, v + ν) becomes
adding zeroes elsewhere. This gives us a decomposition of
matrices. The corresponding decomposition of
p(u + µ, v + ν) =
(a0,0 + a1,0 u + a0,1 v + a2,0 u2 + a1,1 uv + a0,2 v 2 ) · (1)+
(a1,0 + 2a2,0 u + a1,1 v) · (µ)
+ (a0,1 + a1,1 u + 2a0,2 v) · (ν)+
(1) · (a2,0 µ2 + a1,1 µν + a0,2 ν 2 ).
It is clear that this implies an explicit decomposition of
Γp in rank 1 operators,
as in (5.1).
We now look at a degenerate case.
Example 6.2.
By considering (6.1), it is clear that
Rank Γu2 = 3,
which is
also suggested by the 1 dimensional theory. A bit surprisingly, the rank does
not change if we add a linear term in the second variable;
p(u, v) = u2 + v .
To see this just consider (6.1). This can also be seen from the fact that
u2 + v ∈ cl(ExpSep3 ),
since
u2 + v = lim
ε→0
eεu+ε
2
v/2
+ e−εu+ε
ε2
2
v/2
−2
16
Fredrik Andersson and Marcus Carlsson
As mentioned in the introduction, this will not always be the case, (i.e. it
Γp
can happen that
has a certain rank but
ExpSep,
corresponding subset of
p
is not in the closure of the
which we prove in Section 8).
We return to the general case and consider polynomials of degree 3.
Example 6.3.
Let
p ∈ (C[u, v])3
be given;
p(u, v) = a0,0 + a1,0 u + a0,1 v + a2,0 u2 + a1,1 uv + a0,2 v 2 +
a3,0 u3 + a2,1 u2 v + a1,2 uv 2 + a0,3 v 3 .
Using the obvious extension of the ordering in Example 6.1, the rank determining matrix for

∗
∗
∗
∗
∗
∗







C=


 a3,0

 a2,1

 a1,2
a0,3
p(u + µ, v + ν)
∗
∗
∗
3a3,0
2a2,1
a1,2
∗
∗
∗
is obtained by appending
∗
3a3,0
a2,1
∗
2a2,1
2a1,2
∗
a3,0
a2,1
a1,2
a0,3
a1,2
3a0,3
















a2,1
2a1,2
3a0,3
(6.2)
to the matrix in (6.1). Notice that the matrix has 4 distinct blocks or
submatrices, which are separated. This immediately yields a few interesting
observations: As long as each of these 4 blocks has full rank, it is clear that the
full matrix has rank 6, which thus is the generic rank of
Γp .
Moreover, this
conclusion is completely independent of the values of coecients for lower
order terms (marked by
∗). Thus, the only way to obtain a lower rank than 6
is if any of the 4 submatrices would be degenerate, and in this case the lower
order terms can be of importance.
We shall see in the next section that the above example is typical.
7. Generic rank of
Given any
Γp 's
p ∈ (C[x])N , x ∈ Rd ,
we now ask what the generic rank of
K = K(N, d) and
Rank Γp = K whenever p 6∈ V . What
is. More precisely, we want to nd a number
set
V ⊂ (C[x])N
such that
Γp
negligible
negligible
mean depends between various authors and settings, the most common one
being that it has zero measure with respect to the Haar-measure. In the
present setting, we will work with a much stronger condition, whose denition
requires some preliminary denitions. Let
let
k ∈ N.
B
be a Fréchet space over
By a (k−dimensional) dierentiable manifold in
B
R
and
we shall mean
a set which locally is the image of an immersion whose domain is a subset
On General Domain Truncated Correlation and Convolution.
17
Rk . (More precisely, given every x ∈ B there exists an open neighborhood
U 3 x, an open set V ∈ Rk and an injective dierentiable map Ψ : V → U
whose derivative has rank k at all points, such that M ∩ U = Ψ(V ).) If
dim B < ∞, this coincides with the classical denition, see e.g. [4, Thm.
of
2.1.2]. Now, consider a set which is a union of dierentiable manifolds. There
seems to be no standard terminology for such an object, and we have decided
to simply call it a union of manifolds.
Denition 7.1. Let B be a Fréchet space and let M1 , . . . , MJ be a number of dierentiable manifolds in B with possibly dierent dimensions, say
n1 , . . . , nJ . The set M = ∪Jj=1 Mj will then be called a union of manifolds,
and we dene the dimension of M to equal the maximum dimension of the
Mj ,
component manifolds
i.e.
dim M = max(n1 , . . . , nJ ).
With this terminology, real algebraic varieties are unions of manifolds [32],
but the converse is of course not true. Another example of a natural set which
K, d ∈ N, the set of all d×d rank K matrices.
Υ, Ξ) the set
{φ : Rank Γφ ≤ K} is a union of manifolds in the (Fréchet space) ExpPol,
which also implies that {Γφ : Rank Γφ ≤ K} is a union of manifolds in the
2
2
Banach space L(L (Υ), L (Ξ)).
is a union of manifolds is, given
In the next section we shall see that (given bounded domains
Denition 7.2.
P
Let
M
be a union of manifolds (in some space
B)
and let
M.
V ⊂ M with dim V < dim M,
M \ V . Then P is said to hold
be a property which may or may not hold for any given element of
Suppose that there exists a union of manifolds
and suppose that
elements of
generically on
P is true for all
M. The set V will
be referred to as negligible.
In the case when
M
itself is a nite dimensional linear space, note that
the above denition of generic is indeed much stronger than saying that a
property holds a.e. (with respect to the Haar measure). To see this, write
V
locally as the graph of a function ([4, Thm. 2.1.2 (iv)]) and use Fubini's
theorem. The following proposition is often useful in order to conclude that
a given property is generic.
Proposition 7.3.
Let
a given property
P
variety on
M.
M
be a nite dimensional linear space and suppose that
holds outside of a non-trivial real (or complex) algebraic
Then
P
holds generically in
M.
Proof. This follows immediately from [32], whose main result is that real
d
∞
algebraic varieties on
dimension
< d.
R
can be written as a nite union of
C
-manifolds of
The corresponding statement for complex algebraic varieties
is immediate since such can be identied with real algebraic varieties via the
usual identication of
C
with
R2 .
In the remainder of this section, we set
M = (C[x])N . Note that since (C[x])N
C[(C[x])N ], i.e. the set of
is a nite dimensional linear space, we can consider
18
Fredrik Andersson and Marcus Carlsson
(C[x])N . Concretely, this can be realized by considering
{aj }|j|≤N of a given p ∈ (C[x])N as independent variables in
N +d
{j: |j|≤N }
the
-dimensional space C
(see Proposition 3.2), and consider
d
C[(C[x])N ] as all polynomials in these variables. The main result of this
all polynomials on
the coecients
section is the following:
Theorem 7.4.
(C[x])N ,
N
+d−1
2
d
what the
d
be bounded connected domains in R . Given p ∈
N −1
+d
2
the rank of Γp,Υ,Ξ generically equals 2
, if N is odd, and
d
N
+d
+ 2 d , if N is even. These numbers are also upper bounds for
rank can become.
Let
Υ
and
Ξ
Before the proof we need a couple of lemmas. We rst introduce some formalism which is constructed in order to treat matrices with the special structure
that arise in Section 5, see e.g. (6.1) and (6.2). By an ordered partition of
{1, 2, . . . , M }, we mean a sequence I0 , I1 , . . . , IN of disjoint subsets such
N
that ∪n=0 In = {1, 2, . . . , M } and such that the numbers in In are lower than
those in In+1 for 0 ≤ n < N . The partition will be called increasing if the
cardinalities |In | increase with n. Given an M × M matrix C we will write
C(In , In0 ) for the |In | × |In0 |-submatrix
C(In , In0 ) = (ci,j )i∈In ,j∈In0 .
C(In , In0 )
A submatrix
is said to have maximal rank if
Rank C(In , In0 ) = min(|In |, |In0 |).
We are now ready for the lemma. To easier understand it, keep in mind that
with
M = 10
and
I0 = {1}, I1 = {2, 3}, I2 = {4, 5, 6}
and
I3 = {7, 8, 9, 10},
we are describing matrices with the same structure as the one in (6.2).
Lemma 7.5.
of
Fix
{1, 2, . . . , M }.
M ∈ N and consider an increasing partition I0 , I1 , . . . , IN
Let C be an M × M -matrix with the following structure:
i) ci,j = 0 whenever i ∈ In and j > max(IN −n ).
ii) Each submatrix C(In , IN −n ) has maximal rank, n = 0, . . . , N .
Then
(
Rank C =
P(N −1)/2
2 n=0
|In |,
PN/2−1
2 n=0 |In | + |IN/2 |.
if
if
N
N
is odd
is even
(7.1)
These numbers are also the maximum possible rank for matrices with the
structure specied in
i).
Proof. First assume that both
i)
and
ii)
are satised. Let
x ∈ CM
be any
solution to
We will write
C(IN −n , In )
xn
Cx = 0.
x(In ). As
for the subvector
long as
n ≤ N/2, the matrices
|IN −n | ≥ |In |.
are injective, since they have maximal rank and
In particular, since
C(IN , I0 )x0 = 0
On General Domain Truncated Correlation and Convolution.
we conclude that
x0 = 0.
19
With this at hand, we see that
C(IN −1 , I1 )x1 = 0
N −1
N
2 (odd case) or 2 (even case).
We now focus on the odd case. The kernel of C thus has the same dimension
which gives
x1 = 0 and so on until we reach
as that of the submatrix
N −1
2
I ).
In , ∪N
D = C(∪n=0
n= N +1 n
2
However, repeating the above argument for
D∗ ,
it is easy to see that
injective, and hence the dimension of the kernel of
D
D∗
is
equals
N −1
N
2
N −1 X
X
2
dim Ker D = ∪N
I
−
|In | .
I
=
|I
|
−
∪
n
n=0 n
n= N +1 n
2
n=0
n= N2+1
Summing up we have that
N
X
Rank C = M −dim Ker C =

|In |− 
n=0
N
X
|In | −
N −1
2

N −1
2
X
|In | = 2
X
n=0
n= N2+1
|Ia | .
n=0
The proof of the even case is similar, the only dierence is that the central
C I N , I N changes the limits a bit, (compare the central matrix
2
2
2a2,0 a1,1
in (6.1) with the absence of an analogue in (6.2)). Therefore
a1,1 2a0,2
matrix
we set
N
−1
2
In , ∪N
I )
D = C(∪n=0
n= N +1 n
2
and conclude as earlier that
Rank C =
N
X

|In | − 
n=0
as desired. Finally, if only
N
2
N
X
|In | −
n=0
n= N
2 +1
i)
−1
X

N
2
−1
X
|In | = 2
|In | + I N ,
2
n=0
is satised, it is clear that
dim Ker C
can only
get larger. This gives the second statement in the lemma.
Returning once more to (6.2), the following lemma will be applied to the
sub-matrices
C(IN −n , In )
Lemma 7.6.
Let
matrix
i)
ii)
Then
with
In
as described before Lemma 7.5.
a1 , . . . , aK ∈ C be variables and let C = C((ak )K
k=1 )
K
depending on (ak )k=1 in the following way:
be a
ci,j is of the form ni,j aki,j , where ni,j ∈ R \ {0}.
ki1 ,j1 = ki2 ,j2 then either i1 > i2 and j1 < j2 , or reversely, i1 < i2
and j1 > j2 .
Each
If
(ak )K
k=1
can be chosen such that
C
has maximal rank.
20
Fredrik Andersson and Marcus Carlsson
Proof. Since the structure is preserved by submatrices, it clearly suces to
C . Hence let C be an m × m matrix.
m = 1 the statement is obvious. Now let
of det C along the rst row;
prove the result for square matrices
We proceed by induction over
m>1
m.
If
and consider the expansion
det C = c1,1 det(C({2, . . . , m}, {2, . . . , m})) + . . .
(7.2)
We are only interested in the rst term. By the induction hypothesis,
(ak )K
k=1
det(C({2, . . . , m}, {2, . . . , m})) 6= 0. Moreover, we
c1,1 = n1,1 ak1,1 and by ii) we have that the independent variable ak1,1
does not appear in C({2, . . . , m}, {2, . . . , m}) or the remaining terms in the
expansion (7.2), and thus, with the other variables xed, ak1,1 can be chosen
such that the expression in (7.2) in non-zero. Hence C becomes invertible and
thus has maximal rank.
can be chosen such that
have
N +d
which by Proposition 3.2 is the did
(C(x))N . A basis for (C(x))N is given by
Proof of Theorem 7.4. Set
mension of
PolN =
M =
(e1 , . . . , eM ) = (1, x1 , x2 . . . , xd , x21 , x1 x2 , . . . , xN
d ),
(7.3)
where we use the order that lower total degree comes rst, and ties are broken
n = 0, 1, . . . , N ,
In ⊂ {1, . . . , M } be the set of integers j such that the monomial on the j th
position in (7.3) has total degree n. Clearly, I0 , . . . , IN becomes an ordered
by putting exponents of higher lexicographical order rst. For
let
partition. Note that it also is an increasing partition, since
|In | =
n+d−1
d−1
(7.4)
p ∈ (C[x])N . By Proposition 5.3, the rank
p(x+y) considered as a tensor in (C[x])N ⊗(C[y])N . This
equals the rank of the corresponding matrix C as constructed
by Proposition 3.2. Consider any
of
Γp
equals that of
rank in turn
in Proposition 5.2, where we use the monomial basis (7.3) with the ordering
p(x + y) becomes a sum of coecients
|i| + |j| ≤ N , which means that the matrix C
has the structure specied in condition i) of Lemma 7.5. Whether it satises
condition ii) depends on the particular coecients in p, as we saw in Section
specied above. Upon expanding,
multiplied with
xi y j
where
6.
Suppose for the moment that we can nd one
p such that also ii) is satised.
n to be odd,
Then Lemma 7.5 applies and hence, using (7.4) and assuming
we get
(N −1)/2
Rank C = 2
X
(N −1)/2 |In | = 2
n=0
X
n=0
n+d−1
d−1
N −1
2 +d ,
=2
d
where the last identity follows by standard combinatorics. In the even case,
the central matrix
C(IN/2 , IN/2 ) has to be taken into account and a similar
calculation as above gives
Rank C = 2
N −1
2 +d
d
=2
N
2
+d−1
d
+
N
2
+d−1
d−1
=
N
2
+d−1
d
+
N
2
+d
,
d
On General Domain Truncated Correlation and Convolution.
21
as desired. Let us also point out that the statements concerning maximal
rank of
Γp
as well follows by Lemma 7.5.
Finally, we need to show that condition
ii)
of Lemma 7.5 generically is sat-
ised. We split the argument in several steps. First, due to the symmetry
between
x
and
y
in the expansion of
and hence it suces to nd a
ii0 )
Each submatrix
p
p(x + y),
the matrix
C
is symmetric,
such that
C(IN −n , In )
has maximal rank for each
n ≤ N/2.
p(x) = 0≤|j|≤N aj xj be any p ∈ (C[x])N . For each n ≤ N/2
subset J ⊂ IN −n such that |J| = |In |, consider the polynomial
P
Let
and each
qn,J ((aj )j ) = det(C(J, In )).
n ≤ N/2, we can nd a polynomial pn such that
C(IN −n , In ) has maximal rank. This would imply
variety Vn ⊂ (C[x])N dened by the ideal h{qn,J }J i is
Suppose that, given a xed
the corresponding matrix
that the algebraic
non-trivial. But by basic algebraic geometry [14, Ch. 4.3] this implies that
V = ∪n≤N/2 Vn
(aj ) 6∈ V implies that the
ii) of Lemma 7.5, and so the proof would
is non-trivial. Moreover, it is easy to see that
corresponding
C
satises condition
be done upon invoking Proposition 7.3.
n ≤ N/2 be xed, and let us prove that a p such that C(IN −n , In )
C(IN −n , In ) are indexed with (i, j)
where j = 1, 2, . . . , |In | and i = 1, 2, . . . , |IN −n |. We recall that the index set
In corresponds to all monomials y j where |j| = n, and that these are ordered
Hence, let
has maximal rank, exists. The elements in
lexicographically, starting with the largest, i.e.
y1n , y1n−1 y21 , . . . , y1n−1 yd1 , y1n−2 y22 , y1n−2 y21 y31 , . . .
Given
place
make
(7.5)
j(j) such that y j(j) appears on the j th
in (7.5). Likewise, we write j(j) for the reverse correspondence, and
analogous denitions for i ∈ {1, 2, . . . , |IN −n |}. Note that this ordering
j ∈ {1, 2, . . . , |In |},
we dene
is such that
i > i0 ⇐⇒ i(i) <lex i(i0 ),
(where
<lex
(7.6)
refers to smaller with respect to the lexicographical order). We
now verify that
C(IN −n , In )
satises the conditions in Lemma 7.6, thereby
nishing the proof. This is indeed the case for the sub-matrices in (6.2). In
general, it is easy to see that
condition
i)
ci,j
is fullled. To verify
is an integer multiple of
ii),
suppose that
(i0 , j 0 )
ai(i)+j(j) .
Hence
is another index
pair such that
i(i) + j(j) = i(i0 ) + j(j 0 ).
which by (7.7) implies that
are equivalent with
(7.7)
i(i) = i(i ), so suppose for concreteness that i(i) >lex
j(j) <lex j(j 0 ). By (7.6), these inequalities
0
i < i and j > j 0 , and the proof is complete.
Clearly we can not have
i(i0 ),
0
22
Fredrik Andersson and Marcus Carlsson
Given the setting of Theorem 7.4, it is also of some interest to know the
lowest possible rank, which is described by the next proposition.
Proposition 7.7.
Let
Υ
and
p ∈ (C[x])N \ (C[x])N −1 ,
Ξ
be bounded connected domains in
we have
Rd .
Given
Rank Γp,Υ,Ξ ≥ N + 1.
This follows by a slight modication of Lemma 7.5:
Lemma 7.8.
given. Let
C
Let
M ∈ N and a partitioning I0 , I1 , . . . , IN of {1, 2, . . . , M }
M × M -matrix with the following structure:
be
be an
i) ci,j = 0 whenever i ∈ In and j > max(IN −n ).
ii) Each submatrix C(IN −n , In ) is non-zero, n = 0, . . . , N .
Then
Rank C ≥ N + 1.
n pick a row-index rn ∈ IN −n and a column-index sn ∈ In
crn ,sn =
6 0. Set R = {r0 , . . . , rN } and S = {s0 , . . . , sN }, and consider
sub-matrix D = C(R, S). This (N + 1) × (N + 1) matrix has non-zero
Proof. For each
such that
the
elements on its anti-diagonal and zeroes below it, and hence it is invertible.
Since the sub-matrix
Rank C ≥ N + 1,
D
has rank
as desired.
Proof of Proposition 7.7. Let
already established that
C
C
N + 1, it follows by basic linear algebra that
be as in the proof of Theorem 7.4. We have
satises
i) in Lemma 7.5, which is the same condi-
tion as that appearing in Lemma 7.8. By the reasoning above (7.7), it is easy
to see that
ii) is satised as well. Hence the Proposition follows by combining
Propositions 5.2, 5.3 and the above lemma.
8. On the (manifold-)structure of
The set
{φ : Rank Γφ ≤ K} is a union
d = 1, the structure of the
In the case
{φ : Rank Γφ ≤ K}
of manifolds, as we shall prove below.
component manifolds is quite easy to
describe, and one easily sees that a generic element of
is in
ExpSepK .
{φ : Rank Γφ ≤ K}
In other words, the symbols containing polynomial factors
are negligible. Rather surprisingly, the situation in several variables does not
resemble the one-variable case. We rst go through the one variable case.
Let
K
be xed and let
TK
T = (t1 , . . . , tJ )
be all tuples
are positive integers such that
tj+1 ≤ tj
J
X
where
J ≤K
and
tj
and
tj = K.
(8.1)
j=1
Given such a tuple
T,
consider the expression
φ(x) =
tj
J X
X
j=1 l=1
atj ,l xl−1 exζj .
(8.2)
On General Domain Truncated Correlation and Convolution.
23
Let
RT =
J
X
(tj + 1) = K + J
(8.3)
j=1
and order the variables
BT ⊂ C
ζj = ζj 0
RT
atj ,l
and
ζj
so that they are elements of
CRT .
Let
be the subset with the hyper-planes dened by the equations
and
atj ,tj = 0
removed. By Proposition 3.1 it easily follows that
BT to ExpPol. Its image, which we
MT , is then an RT −dimensional manifold by the denitions
in Section 7. Let 1K denote the tuple of length K with all entries equal to 1,
and note that ExpSepK corresponds to M1K . We now collect the information
(8.2) implicitly denes an immersion from
will denote by
on these objects in one theorem.
Theorem 8.1.
Let
Υ
and
Ξ
be nite intervals in
R.
Then
{φ ∈ ExpPol : Rank Γφ,Υ,Ξ = K} = ∪T ∈TK MT .
T 6= 1K
Moreover, given
we have
dim MT < dim ExpSepK
and
∪T 6=1K MT ⊂ cl ExpSepK .
In other words, we have that a generic element of
in
ExpSepK
elements in
{φ : Rank Γφ,Υ,Ξ = K}
is
and those that are not can be approximated arbitrarily well with
ExpSepK .
Proof. The fact that
{φ : Rank Γφ,Υ,Ξ = K} = ∪T ∈TK MT
is the essence
of Rochberg's version of Kronecker's theorem [30, Thm. 3.1]. The statement
concerning dimension is immediate by (8.3) since this expression is maximized
by choosing
J = K,
which forces
t1 = . . . = tK = 1
by (8.1). Finally, the last
equation boils down to showing that an arbitrary function of the form
t
X
al xl−1 exζ ,
t ∈ N, ζ ∈ C, a1 , . . . , at ∈ C
l=1
is in cl(ExpSept ), which follows since
t
X
al xl−1 exζ = exζ lim
l=1
ε→0
uniformly on compacts (with respect to
t
X
al
l=1
eεx − 1
ε
l−1
x).
We now begin the study of the corresponding situation for
Proposition 8.2.
K ∈ N.
Then
d > 1.
Υ and Ξ be connected bounded domains in Rd ,
{φ ∈ ExpPol : Rank Γφ,Υ,Ξ ≤ K} is a union of manifolds.
Proof. Denote
Let
M = {φ : Rank Γφ,Υ,Ξ ≤ K}.
Pick a nite subset
and consider the map
ΦI (ζ, (ai )i∈I ) =
X
i∈I
ai xi ex·ζ .
I
of
and
Nd ,
24
Fredrik Andersson and Marcus Carlsson
ΓPi∈I ai xi ,
which can be determined by constructing a corresponding matrix C via
Proposition 5.2 and 5.3. Let the generic rank of this matrix be tI , (the generic
By Proposition 5.1, the rank of a given
ΓΦI (ζ,(ai )i∈I )
equals that of
rank exists by considering sub-determinants and Proposition 7.3). Note that
tI ≥ max{|i| : i ∈ I},
(t1 , . . . , tJ ) = T ∈ TK ,
by Proposition 7.7. Given a tuple
set such that
tIj = tj ,
(8.4)
let
Ij
be any index
and consider the map
J
X
Ψ ζ 1 , . . . , ζ J , (a1i )i∈I1 , . . . , (aJi )i∈IJ =
ΦIj ζ j , (aji )i∈Ij
(8.5)
j=1
CdJ ×CI1 ×. . . CIJ , with all hyperplanes
j
dened by a
i(j) = 0, where i(j) is the
The domain of this map is taken to be
ζ j = ζ j0
removed, as well as those
largest index in
Ij ,
using the ordering from the proof of Theorem 7.4. By
Ψ denes an injective dierentiable
ExpPol, and hence its image is a dierentiable manifold, which
we can denote by MΨ . Let ΣK be the collection of all possible Ψ as above.
Equation (8.4) implies that ΣK is a nite set. By the above construction and
Proposition 5.1, it follows that any φ ∈ M lies in MΨ for some Ψ ∈ ΣK , and
hence M is a union of manifolds by Denition 7.1.
Proposition 3.1, it is easy to see that each
immersion into
ExpSepK is not, in general,
{φ : Rank Γφ,Υ,Ξ ≤ K}. To get the idea behind the
3
matrices C for a general p(x + y) of degree 3 in R .
Now for the main result of this section, stating that
a dense component of
proof, let us consider
With the same ordering of the monomials as used in Section 7, this looks like











a000
a100
a010
a001
a200
a110
..
.
a100
2a200
a110
a101
3a300
2a210
..
.
a010
a110
2a020
a011
a210
2a120
..
.
a001
a101
a011
2a002
a201
a111
..
.
a200
3a300
a210
a201
a110
2a210
2a120
a111
a101
2a201
a111
2a102
a020
a120
3a030
a021
a011
a111
2a021
2a012
a002
a102
a012
3a003
a300
...











We see that the generic rank is 8, in accordance with Theorem 7.4. Moreover,
increases, it is easy to see that the generic rank, K say,
d, whereas the amount of independent variables aijk grows
much faster. Hence, for d large enough, the amount of independent variables
in the corresponding piece of the union of manifolds {φ : Rank Γφ ≤ K} will
be much greater than that of ExpSepK , which has dimension 2K . Note that
as the dimension
d
grows linearly with
ExpSepK = ExpSep ∩ {φ : Rank Γφ = K}
by Proposition 4.1.
Theorem 8.3.
negligible in
Given any d > 1,
{φ : Rank Γφ = K}.
there are
K ∈ N
such that
ExpSepK
is
On General Domain Truncated Correlation and Convolution.
25
N , let I ⊂ Nd be the index set {i : |i| ≤ N }
MΦI obtained by setting Ψ = ΦI in (8.5). By
Proof. Given an odd integer
and consider the manifold
Proposition 3.2, this manifold has dimension
dim (MΦI ) =
N +d
.
d
for a given Γφ
N −1
+d
2
.
K = tI = 2
d
By Theorem 7.4, the generic rank
Hence
K
MΦI ⊂ {φ : Rank Γφ ≤ K},
dim ({φ : Rank Γφ = K}) ≥
for large enough
N,
φ ∈ MΦI
equals
so
On the other hand,
with
N +d
.
d
ExpSepK is a manifold with dimension 2K .
N −1
2 +d < N +d
4
d
d
Since
the theorem follows.
9. Unbounded domains
The additional diculty when considering nite rank correlation operators
on unbounded domains is that many symbols in
ExpPol
generate unbounded
operators. There are two ways to deal with this situation, either by characterizing the subset of
ExpPol
which generate bounded operators (in analogy
with (1.3) concerning the case of Hankel operators on
weighted
L2 -spaces
R+ ), or by considering
ExpPol generate
with weights such that all symbols in
bounded operators. We will primarily focus on the former setting and briey
treat the second option in Section 9.1. We begin however with a theorem basically stating that, in order to determine the rank of a given
Γφ , the material
of the previous sections apply with minor changes.
Theorem 9.1.
Let
Υ, Ξ ⊂ Rd
φ ∈ ExpPol
Γφ,Υ,Ξ
be sub-domains. Then
be connected domains and suppose that
is bounded and has nite rank. Let
γ⊂Υ
and
ξ⊂Ξ
and
Rank Γφ,Υ,Ξ = Rank Γφ,γ,ξ .
Proof. Let ιγ
dene
ιξ
: L2 (Υ) → L2 (γ)
be the canonical restriction;
ιγ (f ) = f |γ ,
and
analogously. Then
Γφ,γ,ξ = ιξ ◦ Γφ,Υ,Ξ ◦ ι∗γ ,
and hence
By Theorem 4.4, there
Rank Γφ,γ,ξ ≤ Rank Γφ,Υ,Ξ .
exists a ψ ∈ ExpPol such that
(9.1)
φ = ψ a.e.
(9.2)
26
in
Fredrik Andersson and Marcus Carlsson
ξ + γ.
φ ∈ ExpPol. It remains to prove
Rank Γφ,Υ,Ξ = K , and suppose that there
ξ ⊂ Ξ such that Rank Γφ,γ,ξ < K. By (9.1) this is
By Lemma 3.3 we conclude that
the reverse inequality to (9.1). Set
exists some
γ ⊂Υ
and
then true for some bounded connected subsets. By the material in Section
5, this means that Rank Γφ,γ,ξ < K for all bounded connected domains. Set
γk = Υ ∩ {x : |x| < k} for all k ∈ N, and let µγk : L2 (Υ) → L2 (Υ) be
the operator of multiplication by the characteristic function of γk . Also make
analogous denitions for Ξ, and set Γk = µξk ◦ Γφ,Υ,Ξ ◦ µγk . Γk then clearly
∞
has the same rank as Γφ,γk ,ξk , so Rank Γk < K . However, (Γk )k=1 converges
in the weak operator topology to Γφ,Υ,Ξ . Since the subset of rank K operators
is closed in the weak operator topology, it follows that Rank Γφ,Υ,Ξ < K , a
contradiction.
The above proof gives that
Γφ
has nite rank only if
φ ∈ ExpPol, independent
of the domains involved. Due to boundedness issues, the converse is not true,
and it is our next aim to clarify this situation. Given a set
function
d
hΩ : R → (−∞, ∞]
Ω ∈ Rd its indicator
is dened as
hΩ (θ) = sup x · θ.
x∈Ω
We also set
ΘΩ = {θ : hΩ (θ) < ∞},
Ω
(cf. Figure 1). When
(9.3)
is understood from the context we usually omit it
from the subindex. The interior of
Θ = ΘΩ
will be denoted by int(Θ). We
refer to e.g. [22, Sec.7.4] for basic information concerning indicator functions.
In particular we have
Proposition 9.2.
Also
Θ
If
Ω
hΩ is convex and lower semi-continuous.
is continuous on int(Θ).
is convex then
is a convex cone and
hΩ
The last statement is not written explicitly in [22], but follows easily from the
convexity and lower semi-continuity. Given any set
ExpPol
Θ ⊂ Rd
we let
ExpPolΘ ⊂
denote all functions of the form
ExpPolΘ =

X
pj (x)ex·ζ j : pj ∈ Pol, Re (ζ j ) ∈ Θ



.

f inite
We are now ready for the main theorem of this section, which completes the
proof of Theorem 1.2.
Theorem 9.3.
Θ via (9.3).
ExpPolint(Θ) .
Υ, Ξ ⊂ Rd be convex
Then Γφ is bounded and
Let
domains, set
Ω=Υ+Ξ
and dene
has nite rank if and only if
φ ∈
Ω = R+
and
Θ = R− .
retrieve Kronecker's theorem for Hankel operators on
R+ ,
as stated in (1.3).
For example, if
d=1
and
Υ = Ξ = R+ ,
then
We thus
On General Domain Truncated Correlation and Convolution.
27
e2
a⊥
b⊥
r
cos 20
a+
s− cos r20
sin 20
b⊥
a
Ω
s
e1
0
r
Θ
b
Figure 1.
The geometric idea for the proof of Proposition 9.4
Likewise, in the Toeplitz (or Wiener-Hopf ) operator case, we have
and
Ξ = R−
so
Ω=R
and
Θ = {0},
Υ = R+
and hence the theorem implies the well
known fact that there are no nite rank Toeplitz operators. Before proving
the above theorem, we need a few other results. We let
denote the boundary of
Proposition 9.4.
ξ ∈ Rd \ int(Θ),
Ω + R+ y = Ω.
Proof. Let
and
Ω
Let
be the canonical basis in
Θ
Rd .
First assume that
ξ ∈ ∂Θ
The problem is clearly invariant under rotations and dilations.
with
ε < ε0 ,
ξ = e1
and that, for some
0 < ε0 < π/4
and all
we have
cos(ε)e1 + sin(ε)e2 6∈ Θ
as
\ int(Θ)
be a convex domain and dene Θ via (9.3). Given
y ∈ Rd such that y · ξ ≥ 0 and
Hence we can assume that
ε ∈ R+
cl(Θ)
there exists a non-zero vector
e1 , . . . , ed
ξ 6= 0.
∂Θ =
Θ.
is a convex cone. Moreover,
Θ
and
cos(ε)e1 − sin(ε)e2 ∈ Θ,
is unchanged by translations of
Ω,
so we
may assume that
0 ∈ int(Ω).
ε as above to be xed and set a = cos(ε)e1 + sin(ε)e2 , a⊥ =
− sin(ε)e1 + cos(ε)e2 , b = cos(ε)e1 − sin(ε)e2 and b⊥ = sin(ε)e1 + cos(ε)e2 ,
cf. Figure 1. Let r = hΩ (b). Then 0 < r < ∞ and
Now, consider
Ω ⊂ {x : x · b ≤ r}.
On the other hand,
s
and let
y
Ω ∩ {x : x · a = s}
is non-void for all
be such a point. Its projection in
Span (e1 , e2 )
(9.4)
s ∈ R.
We now x
will be denoted
ỹ .
28
Fredrik Andersson and Marcus Carlsson
Since
a · b⊥ = sin(2ε),
this can be written as (see Figure 1)
ỹ =
t ∈ R.
for some
r
s − cos(2ε)
r
a+
b⊥ + ta⊥ ,
cos(2ε)
sin(2ε)
In fact, we must have
t ≥ 0,
(9.5)
since the rst two terms in
(9.5) sum up to a point on the boundary of the right hand set in (9.4),
and
a⊥
points inside this set. It follows that, upon choosing
may assume that
y · e2 = ỹ · e2
s
large, we
is as large as we want. Moreover, the term
r
s− cos(2ε)
sin(2ε) b⊥ + ta⊥ clearly lies within the cone of opening angle ε around e2 .
r
Since the term
cos(2ε) a is constant, we can choose s large enough that ỹ lies
within the corresponding cone with opening angle
2ε.
Thus
|y · e1 | < tan(2ε)y · e2 .
j ∈ N,
We may now, for each
satisfying
|y j | > j
pick
yj ∈ Ω
such that
(y j )∞
j=1
is a sequence
and
|y j · e1 | < 2−j y j · e2 .
y jm /|y jm | converges to some point y . Then
yj
yj
|y · e1 | = lim m · e1 < lim 2−jm m · e2 ≤ lim 2−jm = 0.
m→∞ |y
m→∞
m→∞
|
|y |
Pick a subsequence such that
jm
jm
R > 0 be given, and consider m such that jm > R. Since Ω is convex and
0 ∈ Ω we have Ry jm /|y jm | ∈ Ω, which implies that Ry ∈ cl(Ω). Since R > 0
Let
was arbitrary, we conclude that
R+ y ⊂ cl(Ω).
x ∈ Ω. We use [x, y] to denote the line joining x and
[x, Ry] are then in cl(Ω). Letting R
go to innity one easily sees that x + Sy ∈ cl(Ω) for all xed S > 0, and thus
Now consider any other
y.
Given
R > 0,
all elements on the line
x + R+ y ⊂ cl(Ω)
as well. Thus
Ω+R+ y ⊂ cl(Ω), and as Ω+R+ y
is an open set, it immediately
follows that
Ω + R+ y ⊂ Ω.
Since the reverse inequality is obvious, the proof is complete under the assumption that
ξ ∈ ∂Θ
and
ξ 6= 0.
ξ ∈ R \ cl(Θ), we have hΩ (ξ) = ∞. Let (y j )∞
j=1 be a sequence in Ω such
that y j · ξ ≥ j and pick a subsequence such that y j /|y j | converges to some
m
m
point y . The desired conclusion then follows as above.
If
d
ξ = 0, then Θ can not equal Rd ,
y j ∈ Ω with |y j | > j for all j ∈ N. Again,
Finally, if
which means that we can pick
we can construct a
desired properties by repeating the above argument.
Lemma 9.5.
and
ζ ∈ Cd
Let
Ω
y
with the
be a convex domain and dene Θ via (9.3). Let p ∈ Pol
p(x)ex·ζ ∈ L2 (Ω) if and only if Re ζ ∈ int(Θ).
be given. Then
On General Domain Truncated Correlation and Convolution.
Proof.
Ω
is bounded if and only if
Θ = Rd ,
29
in which case there is nothing to
Θ is a proper cone and in particular 0 ∈ ∂Θ. First assume
that Re ζ ∈ int(Θ). The problem is clearly invariant under rotations and
d
dilations of R , so there is no restriction to assume that ζ = (µ + iν)e1 with
µ > 0. We use x0 to denote any variable with e1 · x0 = 0. Set
n
o
B = x0 : |x0 | ≤ 1 .
(9.6)
prove. Otherwise
By Proposition 9.2 there exists an
ε>0
C∈R
and a
such that
0
hΩ (e1 + εx ) ≤ C
for all
x0 ∈ B .
(9.7)
y = ye1 + y 0 be any point in Ω. Then (9.7)
y0
(ye1 + y 0 ) · e1 + ε 0 = y + ε|y 0 | ≤ C.
|y |
Let
gives that
The translated cone
Π=
hence includes
Ω.
Let
vd
t
(C − t)e1 + B : t > 0
ε
denote the Lebesgue volume of the unit ball in
Rd .
Then
kp(x)ex·ζ k2L2 (Ω) ≤ kp(x)ex·ζ k2L2 (Π) =
Z
∞
≤
Z
e2µ(C−t)
t
εB
0
Z
e2µx1 |p(x)|2 dx ≤
Π
!
p(C − t, x0 )2 dx0
dt < ∞,
where the last inequality follows from observing that
is a polynomial in
Now, if
Re ζ 6∈
t.
R
t
εB
p(C − t, x0 )2 dx0
int(Θ), then Proposition 9.4 and a rotation implies that we
Ω + e1 R+ = Ω and µ = Re ζ · e1 ≥ 0. By a translation we
0 ∈ int(Ω). Let B be as in (9.6), and pick r > 0 such
rB ⊂ Ω. Then Proposition 9.4 implies that Π = rB + R+ e1 ⊂ Ω. By
may assume that
may also assume that
that
Fubini's theorem we get
2
p(x)ex·ζ 2 2
≥ p(x)ex·ζ L2 (Π)
L (Ω)
Z ∞
Z 2
0
=
e2µx1
p (x1 , x0 ) eζ·x x0 dx0 dx1 = ∞,
0
B
where the last equality follows by observing that the inner integral is a nonzero polynomial in
x1 .
φ ∈ ExpPolint(Θ) . As in Lemma
φ(x
+
y)
can be expanded in a nite sum of the form
P
j ψj (x)ϕj (y) where each ψj and ϕj is in ExpPolint Θ . By Lemma 9.5 we then
2
have that these functions are in L (Ω), and it is easy to see that this implies
P
2
2
that they are also in L (Υ) and L (Ξ). Since Γφ (f ) =
j ψj (x) hf, ϕj (y)i ,
we see that Γφ is a bounded operator.
Proof of Theorem 9.3. Suppose rst that
4.2, we see that
30
Fredrik Andersson and Marcus Carlsson
Now the converse. By Theorem 9.1 we know that
φ ∈ ExpPol,
so it is of the
form
φ=
X
where the sum is nite and the
ζ j 's
pj (x)ex·ζ j ,
j
are distinct. Note that
hΩ = hΥ + hΞ
which implies by (9.3) that
ΘΩ = ΘΥ ∩ ΘΞ .
Thus we are done if we show that
Re ζ j0 ,
(9.8)
where
j0
is xed but arbitrary, is
in both int(ΘΥ ) and int(ΘΞ ).
Let
N
ex·ζ j
pj 's. By Proposition 3.1, there
f ∈ L2 (Υ) which is orthogonal (in L2 (Υ)) to all functions in PolN ·
x·ζ j0
all j 6= j0 , but not orthogonal to pj0 (x)e
. It follows that
x·ζ j0
,
Γφ f (x) = q(x)e
be the maximum of the degree of the
exists an
for
q is a polynomial which is not identically equal to 0. This means that
q(x)ex·ζ j0 ∈ L2 (Ξ), and thus Re ζ j0 ∈ int(ΘΞ ) by Lemma 9.5. By Proposition
∗
2.2, the same argument applied to Γφ also gives that Re ζ j ∈ int(ΘΥ ). By
0
the remark following (9.8), the proof is complete.
where
As an illustration, let us consider small Hankel operators.
Corollary 9.6.
A small Hankel operator
Γφ,Rd+ ,Rd+
has nite rank if and only
if
φ ∈ ExpPol(−∞,0)d .
We also remark that operators of the form
Hankel operators. Since both
of
Rd \ (−∞, 0]d ,
R+ × R
d−1
Γφ,Rd+ ,Rd \(−∞,0]d
and
R
d−1
× R+
are known as big
are (convex) subsets
Theorem 9.3 applied to the corresponding two operators
easily yields there are no non-trivial nite rank big Hankel operators.
9.1. Weights
In this nal section we outline how the results are aected upon introducing
weighted spaces. Our objective is not to give an exhaustive treatment, but
rather to demonstrate how one can obtain bounded nite rank operators with
unbounded symbols in
ExpPol,
even for
w : Rd → R+ is said
α > 1 and Cα such that
A function
exists
Υ = Ξ = Rd .
to be of sup-exponential growth if there
α
Cα e|x| < w(x).
d
2
(9.9)
Υ ⊂ R dene
R L (Υ, w) as the set of Lebesgue
kf k2 = Υ |f (x)|2 w(x) dx < ∞. Let Υ, Ξ
be two such domains and let w1 , w2 be functions of sup-exponential growth.
−1
2
2
Set Ω = Υ + Ξ and let φ ∈ Lloc (Ω) be such that φ(x + ·) ∈ L (Υ, w1 ) for
−1
2
all x ∈ Ξ and such that φ(· + y) ∈ L (Ξ, w2 ) for all y ∈ Υ. We then dene
Given any connected domain
measurable functions
f
on
Υ
with
On General Domain Truncated Correlation and Convolution.
31
Γφ : L2 (Υ, w1 ) → L2 (Ξ, w2−1 ) precisely as before, e.g. using ((1.1)). Note that
the integral exists by Hölder's inequality, but Γφ may be unbounded. If all
parameters needs to be specied we use the notation Γφ = Γφ,Υ,Ξ,w1 ,w2 . The
main result of this section is the following:
Theorem 9.7.
Let w1 and w2 be continuous sup-exponential weights and let
Υ, Ξ, and Ω be as above. Then Γφ : L2 (Υ, w1 ) −→ L2 (Ξ, w2−1 ) is bounded and
has nite rank if and only if φ ∈ ExpPol. Moreover, if γ and ξ are bounded
connected sub-domains of Υ and Ξ respectively, then
Rank Γφ,Υ,Ξ,w1 ,w2 = Rank Γφ,γ,ξ,1,1 .
Γφ = Γφ,Υ,Ξ,w1 ,w2 is bounded
φ ∈ ExpPol. We now show this. Let α be such that (9.9) is satised
for both w = w1 and w = w2 , and let γ be a number between 1 and α. Clearly
|x|γ
there exists a C > 0 such that |φ(x)| < Ce
. Thus
Z
Z
γ
γ
γ
|Γφ (f )(x)| ≤ C
e|x−y| |f (y)| dy ≤ C
e|2x| e|2y| |f (y)| dy ≤
Proof. The main diculty lies in proving that
whenever
Υ
γ
≤ Ce|2x|
Z
Υ
γ
e2|2y|
α
−|y|
1/2 Z
Υ
γ
C
≤ √ e|2x|
Cα
where
D
Z
γ
e2|2y|
α
|f (y)|2 e|y| dy
dy
−|y|α
1/2
≤
Υ
1/2
γ
kf kL2 (Υ,w1 ) = De|2x| kf kL2 (Υ,w1 )
dy
Rd
is some nite constant, and so
γ
kf kL2 (Υ,w1 ) ≤
kΓφ (f )kL2 (Υ,w−1 ) ≤ D e|2x| 2
2
L (Ξ,w2−1 )
sZ
sZ
D
e2|2x|γ
≤D
dx kf kL2 (Υ,w1 ) ≤ √
e2|2x|γ −|x|α dx kf kL2 (Υ,w1 ) ,
Cα
Ξ w2 (x)
Rd
as desired.
For the remaining part of the proof, note that
only if
Γφ,γ,ξ,1,1
Γφ,γ,ξ,w1 ,w2
is bounded if and
is, since the weights are comparable on bounded domains.
Thus these two operators have the same rank. With this in mind, the rest of
the argument follows by straightforward modications of the proof of Theorem 9.1, we omit the details.
As an illustration, note that the above theorem says that there are plenty
of nite rank convolution operators and plenty of nite rank Wiener-Hopf
operators (Toeplitz operators on
Corollary 9.8.
R)
2
if we involve Gaussian weights.
2
2
2
−x
) consider Tφ : L2 (R+ , ex ) → L2 (R+ , e−x )
Rφ ∈ L (R, e
given by Tφ (f )(x) = R φ(x − y)f (y) dy . Then Tφ has nite rank if and only
+
if φ ∈ ExpPol.
Given
32
Corollary 9.9.
Fredrik Andersson and Marcus Carlsson
2
2
φ ∈ L2 (Rd , e−|x| ) and consider Cφ : L2 (Rd , e|x| ) →
R
2
d −|x|2
L (R , e
) given by Cφ (f )(x) = Rd φ(x − y)f (y) dy . Then Cφ has nite
rank if and only if φ ∈ ExpPol.
Given
Both these corollaries follow by Theorem 9.7 and the unitary equivalence of
truncated convolution and truncated correlation operators, (recall (1.2)).
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Fredrik Andersson
Centre for Mathematical Sciences, Lund University, Sweden
e-mail: [email protected]
Marcus Carlsson
Centre for Mathematical Sciences, Lund University, Sweden
e-mail: [email protected]