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Revista de la Unión Matemática Argentina
versión Online ISSN 16699637
Rev. Unión Mat. Argent. v.49 n.1 Bahía Blanca ene./jun. 2008
On the Notion of Bandlimitedness and its Generalizations
Ahmed I. Zayed
Abstract. In this survey article we introduce the PaleyWiener 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 PaleyWiener space, we construct a subspace of vectors in a Hilbert space using a selfadjoint 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 SturmLiouville problems.
2000 Mathematics Subject Classification. Primary: 30D15, 47D03; Secondary: 44A15
Key words and phrases. PaleyWiener space, Bandlimited Functions, Bernstein Inequality, Selfadjoint Operators, and SturmLiouville 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
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 bandlimited functions, which gives a nice characterization of the space
Theorem 1 (PaleyWiener,[13]). A function is bandlimited to if and only if
Another important property of the space is given by the WhittakerShannonKoteln'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 PaleyWiener 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
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

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
but the series on the righthand side converges because
Therefore, it follows that
The space is the PaleyWiener space Hence, a function in belongs to the PaleyWiener space if and only if
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
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

 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
Moreover, the following sampling theorem holds [10]:
2. Bandlimited Vectors in a Hilbert Space
In this section we introduce a space of PaleyWiener 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 PaleyWiener space. In our abstract setting, the differentiation operator will be replaced by a selfadjoint operator in a Hilbert space . Furthermore, from the abstract setting we will be able to derive a new characterization of the classical PaleyWiener space that connects PaleyWiener 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
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 PaleyWiener vectors and show that they share similar properties to those of the classical PaleyWiener 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
 (2.1) 
3) for every the scalarvalued 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 abstractvalued 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 selfadjoint 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 SturmLiouville Operators
In this section we apply the general results obtained in previous sections to specific examples involving differential operators. We specify our characterization of PaleyWiener functions that are defined by integral transforms other than the Fourier transform. For related material, see [20, 23].
3.1. Integral Transforms Associated with SturmLiouville Operators on a Halfline. Consider the singular SturmLiouville problem on the half line
 (3.1) 
with
and is assumed to be realvalued.
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 welldefined (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 realvalued, 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 limitpoint 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 selfadjoint [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 initialboundaryvalue 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 initialboundaryvalue 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 selfadjoint. 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 initialboundaryvalue 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
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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