Revista de la Unión Matemática Argentina
versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.49 n.1 Bahía Blanca ene./jun. 2008
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
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,). A function is band-limited to if and only if
Another important property of the space is given by the Whittaker-Shannon-Koteln'nikov (WSK) sampling theorem, which can be stated as follows :
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
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]
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
where the series converges in Because this characterization is not well known, we will prove it. We have
but the series on the right-hand side converges because
Therefore, it follows that
The space is the Paley-Wiener space Hence, a function in belongs to the Paley-Wiener 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 .
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 :
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 , there exist a direct integral of Hilbert spaces and a unitary operator from onto , which transforms the domain of the operator onto with norm
The next proposition is evident.
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 , shows that the space has properties (A) and (B). See also [14, 16]
2) belongs to the set
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.
is finite, then and
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).
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
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
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].
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 . It is known [17, 11] that if , then
is well-defined (in the mean) and belongs to , and
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
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
Formally, if and are in then
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
which is a generalization of Bernstein inequality (1.2).
- 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,
Chicago, IL 60614, USA
Recibido: 10 de abril de 2008
Aceptado: 23 de abril de 2008