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Revista de la Unión Matemática Argentina
Print version ISSN 0041-6932On-line version ISSN 1669-9637
Rev. Unión Mat. Argent. vol.49 no.1 Bahía Blanca Jan./June 2008
On the Notion of Bandlimitedness and its Generalizations
Ahmed I. Zayed
Abstract. In this survey article we introduce the Paley-Wiener space of bandlimited functions and review some of its generalizations. Some of these generalizations are new and will be presented without proof because the proofs will be published somewhere else.
Guided by the role that the differentiation operator plays in some of the characterizations of the Paley-Wiener space, we construct a subspace of vectors in a Hilbert space
using a self-adjoint operator
We then show that the space
has similar properties to those of the space
The paper is concluded with an application to show how to apply the abstract results to integral transforms associated with singular Sturm-Liouville problems.
2000 Mathematics Subject Classification. Primary: 30D15, 47D03; Secondary: 44A15
Key words and phrases. Paley-Wiener space, Bandlimited Functions, Bernstein Inequality, Self-adjoint Operators, and Sturm-Liouville Operators.
The term bandlimited functions came from electrical engineering where it means that the frequency content of a signal is limited by certain bounds from below and above. More precisely, if
is a function of time, its Fourier transform
![f.](/img/revistas/ruma/v49n1/1a099x.png)
![[- σ,σ]](/img/revistas/ruma/v49n1/1a0911x.png)
![ˆf](/img/revistas/ruma/v49n1/1a0912x.png)
![[- σ, σ].](/img/revistas/ruma/v49n1/1a0913x.png)
![σ](/img/revistas/ruma/v49n1/1a0914x.png)
![2 L (IR )](/img/revistas/ruma/v49n1/1a0915x.png)
![P W σ](/img/revistas/ruma/v49n1/1a0916x.png)
![B2 . σ](/img/revistas/ruma/v49n1/1a0917x.png)
In this survey article we shall give an overview of some of the generalizations of this space, of which some are new and will be presented without proof since the proofs will be published somewhere else. For some related work, see [1, 2, 3, 4, 8, 9, 18, 19]
We begin with the following fundamental result by Paley and Wiener on band-limited functions, which gives a nice characterization of the space
Theorem 1 (Paley-Wiener,[13]). A function is band-limited to
if and only if
![g ∈ L2 (- σ, σ)](/img/revistas/ruma/v49n1/1a0922x.png)
![f](/img/revistas/ruma/v49n1/1a0923x.png)
![f](/img/revistas/ruma/v49n1/1a0924x.png)
Another important property of the space is given by the Whittaker-Shannon-Koteln'nikov (WSK) sampling theorem, which can be stated as follows [22]:
Theorem 2. If then
can be reconstructed from its samples,
where
via the formula
![]() | (1.1) |
with the series being absolutely and uniformly convergent on .
One of the earliest generalizations of the Paley-Wiener space is the Bernstein space. Let and
The Bernstein space
is a Banach space consisting of all entire functions
of exponential type with type at most
that belong to
when restricted to the real line. It is known [5, p. 98] that
if and only if
is an entire function satisfying
![x](/img/revistas/ruma/v49n1/1a0943x.png)
![y](/img/revistas/ruma/v49n1/1a0944x.png)
Unlike the spaces the spaces
are closed under differentiation and the differentiation operator plays a vital role in their characterization. The Bernstein spaces have been characterized in a number of different ways and one can prove that the following are equivalent:
- A function
belongs to
if and only if its distributional Fourier transform has support
in the sense of distributions.
-
Let
be such that
for all
and some
then
if and only if
satisfies the Bernstein's inequality [12, p. 116]
(1.2) -
Let
be such that
for all
and some
Then
if and only if
-
Let
be such that
for some
Then
if and only if it satisfies the Riesz interpolation formula
(1.3) where the series converges in
Because this characterization is not well known, we will prove it. We have
But
hence
Now by differentiating the Riesz interpolation formula once more, we obtain formally
but the series on the right-hand side converges because
Therefore, it follows that
and in addition
satisfies the Bernstein inequality; hence,
The converse is shown in [12].
The space is the Paley-Wiener space
Hence, a function
in
belongs to the Paley-Wiener space
if and only if
![f ∈ L2 (IR )](/img/revistas/ruma/v49n1/1a0991x.png)
![PW (IR) σ](/img/revistas/ruma/v49n1/1a0992x.png)
![σ](/img/revistas/ruma/v49n1/1a0993x.png)
The result is not true for For,
vanishes at all
but it is not identically zero. However, the theorem is true for
,
Now we introduce the Zakai Space of Bandlimited Functions [21].
Definition 4. A function is said to be bandlimited with bandwidth
in the sense of Zakai if it is entire of exponential type satisfying
and
![]() | (1.9) |
for some where
is the infimum of all
such that the Fourier transform of
vanishes outside
It should be noted that if is
-bandlimited in the sense of Zakai, then
Let us denote the Zakai space by
Clearly,
since if
is bounded on the real line, the integral in Eq. (1.9) is finite. Examples of functions in
are
and
which can be written as a Fourier transform of a function with compact support, namely,
since
![F(ω )](/img/revistas/ruma/v49n1/1a09123x.png)
![Lp](/img/revistas/ruma/v49n1/1a09124x.png)
![1 ≤ p,](/img/revistas/ruma/v49n1/1a09125x.png)
Another generalization of the class of bandlimited functions is the class which is defined as follows. Let
be the class of all entire functions of exponential type satisfying
![|f (z )| ≤ C (1 + |z|)k exp (σ|ℑz|).](/img/revistas/ruma/v49n1/1a09129x.png)
![f ∈ Hkσ](/img/revistas/ruma/v49n1/1a09130x.png)
-
is a temperate distribution whose Fourier transform has support in
The class is the same as
and
is the same as the Zakai class
The class
![[- σ,σ ].](/img/revistas/ruma/v49n1/1a09139x.png)
![∞ f ∈ H σ](/img/revistas/ruma/v49n1/1a09140x.png)
![k.](/img/revistas/ruma/v49n1/1a09142x.png)
Moreover, the following sampling theorem holds [10]:
2. Bandlimited Vectors in a Hilbert Space
In this section we introduce a space of Paley-Wiener vectors in a Hilbert space As can be seen from (1.2) and (1.3) the differentiation operator plays a vital role in the characterization of classical Paley-Wiener space. In our abstract setting, the differentiation operator will be replaced by a self-adjoint operator
in a Hilbert space
. Furthermore, from the abstract setting we will be able to derive a new characterization of the classical Paley-Wiener space that connects Paley-Wiener functions to analytic solutions of a Cauchy problem involving Schrödinger equation.
According to the spectral theory [6], there exist a direct integral of Hilbert spaces and a unitary operator
from
onto
, which transforms the domain
of the operator
onto
with norm
![FD (Df ) = λ(FDf ),f ∈ D1.](/img/revistas/ruma/v49n1/1a09157x.png)
Definition 6. The unitary operator will be called the Spectral Fourier transform and
will be called the Spectral Fourier transform of
.
Definition 7. We will say that a vector in
belongs to the space
if its Spectral Fourier transform
has support in
.
The next proposition is evident.
Proposition 8. The following properties hold true:
a) The linear set is dense in
.
b) The set is a linear closed subspace in
.
In the following theorems we describe some basic properties of Paley-Wiener vectors and show that they share similar properties to those of the classical Paley-Wiener functions. The next theorem, whose proof can be found in [15], shows that the space has properties (A) and (B). See also [14, 16]
Theorem 9. The following conditions are equivalent:
1);
2) belongs to the set
![k ∈ ℕ,](/img/revistas/ruma/v49n1/1a09174x.png)
![]() | (2.1) |
3) for every the scalar-valued function
of the real variable
is bounded on the real line and has an extension to the complex plane as an entire function of exponential type
;
4) the abstract-valued function is bounded on the real line and has an extension to the complex plane as an entire function of exponential type
.
To show that the space has property (C), we will need the following Lemma.
Lemma 10. Let be a self-adjoint operator in a Hilbert space
and
If for some
the upper bound
![]() | (2.2) |
is finite, then and
Definition 11. Let for some positive number
We denote by
the smallest positive number such that the interval
contains the support of the Spectral Fourier transform
It is easy to see that and that
is the smallest space to which
belongs among all the spaces
For,
Hence, by Theorem 9, Moreover, if
for some
then from Definition 7 the spectral Fourier transform of
has support in
which contradicts the definition of
The next theorem shows that the space
has property (C).
Theorem 12. Let belong to the space
for some
Then
![]() | (2.3) |
exists and is finite. Moreover, Conversely, if
and
exists and is finite, then
and
Finally, we have another characterization of the space Consider the Cauchy problem for the abstract Schrödinger equation
![]() | (2.4) |
where is an abstract function with values in
The next theorem gives another characterization of the space from which we obtain a new characterization of the space
Theorem 13. A vector belongs to
if and only if the solution
of the corresponding Cauchy problem (2.4) has the following properties:
1) as a function of it has an analytic extension
to the complex plane
as an entire function;
2) it has exponential type in the variable
, that is
3. Applications To Sturm-Liouville Operators
In this section we apply the general results obtained in previous sections to specific examples involving differential operators. We specify our characterization of Paley-Wiener functions that are defined by integral transforms other than the Fourier transform. For related material, see [20, 23].
3.1. Integral Transforms Associated with Sturm-Liouville Operators on a Half-line. Consider the singular Sturm-Liouville problem on the half line
![]() | (3.1) |
with
and is assumed to be real-valued.
Let be a solution of equation (3.1) satisfying the initial conditions
Clearly,
is a solution of (3.1) and (3.2). It is easy to see that
and
are bounded as functions of
for
[17]. It is known [17, 11] that if
, then
![]() | (3.3) |
is well-defined (in the mean) and belongs to , and
![]() | (3.4) |
with
![]() | (3.5) |
The measure is called the spectral function of the problem. In many cases of interest the support of
is
In this case the transform (3.4) takes the form
![]() | (3.6) |
and the Parseval equality (3.5) becomes Hereafter, we assume that
is real-valued, bounded and
Because we are interested in the case where the spectrum of the problem is continuous, we shall focus on the case in which the differential equation (3.1) is in the limit-point case at infinity. Restrictions on
to guarantee continuous spectra can be found in [11, 17]. The condition
will suffice. In such a case the problem (3.1) and (3.2) is self-adjoint [7, p. 158, ], i.e.,
for all
where
consists of all functions
satisfying
is differentiable and
is absolutely continuous on
for all
and
are in
Now consider the initial-boundary-value problem involving the Schrodinger equation
![]() | (3.7) |
with
![]() | (3.8) |
and
![]() | (3.9) |
where
Set
![]() | (3.10) |
Formally, if and
are in
then
and
![]() | (3.11) |
Therefore, is a solution of the initial-boundary-value problem (3.7) -(3.9), in the sense of
Definition 14. We say that is bandlimited with bandwidth
or
if its spectral Fourier transform
according to Definition 6, has support
where
is given by (3.1) and (3.2).
It follows from the definition that if is bandlimited to
, then
In order to apply Theorem 13, we have to define the domain on which all iterations of
are self-adjoint. It is easy to see that
consists of all functions
satisfying the following conditions:
is infinitely differentiable on
is in
for all
Hence, if is bandlimited according to Definition 14,
which exists for all
Thus, by Parseval's equality
That is,
![]() | (3.13) |
which is a generalization of Bernstein inequality (1.2).
Theorem 15. A function is bandlimited in the sense of Definition 14 with bandwidth
if and only if the solution
of the initial-boundary-value problem (3.7) - (3.9) with
has the following properties:
- As a function of
it has analytic extension
to the complex plane as entire function of exponential type
- It satisfies the estimate
is bounded on the real t-line.
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Ahmed I. Zayed
Department of Mathematical Sciences,
DePaul University,
Chicago, IL 60614, USA
azayed@math.depaul.edu
Recibido: 10 de abril de 2008
Aceptado: 23 de abril de 2008