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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 384545, 7 pages
A Theory of Summability on a Space of Generalized Functions
1Centre of Advanced Mathematics and Physics, National University of Sciences and Technology, Islamabad 44000, Pakistan
2Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Livingston Tower, Glasgow G1 1XH, UK
Received 30 January 2013; Revised 1 July 2013; Accepted 2 July 2013
Academic Editor: Alberto Fiorenza
Copyright © 2013 Khaula Naeem Khan and Wilson Lamb. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A theory of summability of orthonormal sets is introduced in multinormed spaces. The approach which is presented caters for infinite sets , where the index set may be uncountable, and is applied to obtain convergence results in appropriate spaces of test functions and corresponding spaces of generalized functions. These spaces are constructed in a systematic manner that relies heavily on properties of orthonormal bases in Hilbert spaces. A space of almost-periodic generalized functions, in which each generalized function can be expanded in terms of an uncountable basis of exponential functions, is obtained as a special case of our theory.
In recent papers [1, 2], we investigated the behaviour of certain fractional differential and integral operators on appropriately defined spaces of generalized functions. In each paper, a procedure for producing spaces of test functions and generalized functions, pioneered by Zemanian [3, Chapter 9], played a crucial role. Central to this procedure is the theory of complete orthonormal sequences in -type spaces. Given a complete orthonormal sequence of eigenfunctions of a suitably restricted symmetric differential operator in , where is an interval in , corresponding spaces of test functions and generalized functions are obtained in a systematic and constructive manner.
In the theory developed by Zemanian, the operator is a differential expression of the form where , the are positive integers, and the are smooth functions on . Moreover, it is assumed that satisfies the (symmetry) condition and has smooth eigenfunctions that form a complete orthonormal basis for and for which the corresponding real eigenvalues are such that as . The domain of the operator is where is the identity operator on ; see [3, Chapter 9]. The space of test functions is then taken to be the vector space equipped with the topology generated by the countable multinorm , where
The resulting topological vector space can be shown to be a Fréchet space and is denoted by . Note that is a dense subspace of , since , and that each eigenfunction . The space of generalized functions is the dual of equipped with the weak* topology generated by the multinorm , where where is the number assigned to each by . Since is the dual of a Fréchet space, it is sequentially complete with respect to the weak* topology. Also, by identifying functions with regular generalized functions via it can be shown that is continuously imbedded in . Consequently, we can write where represents a continuous embedding. Results on the weak* convergence of eigenfunction expansions of generalized functions in the space are given in [3, Chapter 9], where it is shown that It should be noted that the Zemanian theory of eigenfunction expansions has been extended in a number of ways since its introduction; see, for example, [4–7]. In particular, in  the differential operator that is used to construct the Fréchet space is replaced by any self-adjoint operator defined on an arbitrary separable Hilbert space . The space of “test-functions” is then taken to be the Fréchet space in which convergence is defined via the countable multinorm , . The corresponding space of “generalized functions” is given by equipped with the weak* topology. The Zemanian results arise as the special case when and is a self-adjoint differential operator whose spectrum contains only eigenvalues; see [5, Remark 3.5]. One of the aims of the present paper is to demonstrate that a Zemanian-type theory of eigenfunction expansions, but possibly involving an uncountable orthonormal basis, can be developed in the more general setting of an arbitrary Hilbert space. This theory will be based on an extension to complete multinormed spaces of the concept of summability in Banach spaces. It is clear that a self-adjoint operator on a separable Hilbert space cannot have an uncountable basis, as all orthonormal bases in a separable Hilbert space are countable; see [8, p.255]. Consequently, in the sequel, we do not assume that that the Hilbert space is separable.
In elementary analysis, the sum of an infinite sequence of terms in some topological vector space exists if the limit of the corresponding sequence of partial sums exists. To deal with the case of summations of a family of elements over a general index set , which could be uncountable, requires the notion of summability. There is a well-established theory of summability in a Banach space setting; see, for example, [9, Chapter 6] and [10, Chapter 4]. However, as far as we are aware, a similar theory has not been presented within the framework of a complete multinormed space. Consequently, we begin in Section 2 by giving an appropriate definition of summability in this more general setting and verify that the definition satisfies all the important computational rules.
In Section 3, we establish that the definition of summability given in Section 2 can be used to extend the Zemanian theory of eigenfunction expansions of generalized functions to the case where the orthonormal basis may be uncountable. As the spaces favoured by Zemanian are separable, we work within the more general framework of a Hilbert space , which need not be separable, and use results from . As a specific example of an uncountable basis, we go on in Section 4 to examine the nonseparable Hilbert space of almost-periodic functions and show how the theory developed in the previous sections leads nicely to eigenfunction expansions for a class of almost-periodic generalized functions.
2. Infinite Sums in Multinormed Spaces
A multinormed space is a topological vector space in which the topology is defined by a collection of seminorms. Let be a vector space and let be a separating collection of seminorms on for some index set . If is countable, then the resulting topological vector space, which we will continue to denote simply by , is a countably multinormed space, which, if complete, is a Fréchet space; see [11, p.132].
Let be an indexed family in and let denote the collection of all finite subsets of . If we use the binary operation to define a partial order on , then becomes a directed set. Associated with is the indexed set of finite partial sums, given by
The mapping from , is then a net in and so we can use the theory associated with nets to provide a definition of the sum , in terms of a limit of finite partial sums, which is independent of the way in which the terms in the sum are ordered.
Definition 1. Let be a complete multinormed space with multinorm . The vectors are said to be summable to , or, equivalently, the unordered sum of the indexed set converges to , if, for each and , there exists such that, for all with ,
In this case, we write and say that the unordered sum converges unconditionally to .
In the subsequent work, the phrases “ converges to ,” “the vectors are summable to ,” and “ is summable to ” will all be used when in .
Lemma 2. If , then is unique.
Proof. Let and also let in . Then, from Definition 1, for each and , there exists such that whenever contains , we have
Similarly, for and , there exists such that whenever contains ,
Now, setting , let be any set from containing . Then contains both and . Hence, from (12) and (13), we obtain and so Since this holds for all and is a multinorm, it follows that .
Lemma 3. Let and let in . Then (i), (ii), where is any scalar.
Proof. The proof is a direct consequence of Definition 1 and is similar to the proof of the previous lemma.
The fundamental theorem of this paper is the next one which is used to obtain a number of results in Section 3.
Theorem 4. The unordered sum is convergent in if and only if, for any and , there exists such that for all that are disjoint from .
Proof. To prove this theorem, we shall use the fact that if and are disjoint, finite subsets of , then .
Sufficiency. Let be arbitrarily fixed, and, for each , choose a finite subset of such that for any that is disjoint from . Replacing by , we can assume that . Then if , Hence is a Cauchy sequence in , and as we know that is a complete multinormed space, exists. Keeping fixed, let in (16) to obtain
Now, for any , we can choose such that . Then, whenever contains , we obtain, from (16) and (17), Thus, converges to in .
Necessity. Suppose converges to in . Given and , we can choose such that, whenever contains ,
Then, for any that is disjoint from , we get and also
Now we state another important result which shows that Definition 1 is equivalent to the usual definition of a convergent series when is countable.
Lemma 5. Let be a countable index set. And let the vectors be summable to in . Then for any increasing sequence of finite subsets of such that .
Proof. Let be summable to . Then from Definition 1, for and , there exists such that, whenever contains , we have As is an increasing sequence of sets in and , there exists such that . Moreover for all , and so Hence the result follows.
The importance of Lemma 5 is that when is countable and is summable to in , then it is often possible to write down explicitly a sequence of partial sums that converge to . As a particular case, let , the set of all integers. Then, we have the following result.
Lemma 6. If in , then , where . In this case we can write
Proof. It follows from Lemma 5.
3. Orthonormal Expansions in Spaces of Test Functions and Generalized Functions
In this section, we demonstrate how the theory of summability developed in Section 2 can be used to extend the distributional eigenfunction-expansion theory of Zemanian to the case when we have an uncountable orthonormal basis. As indicated earlier, all orthonormal bases in are countable, and so, for our extension to be meaningful, we follow the more general approach introduced in  of working with a self-adjoint operator on an arbitrary Hilbert space . However, unlike , where it is assumed that is separable, we shall allow to be nonseparable. Throughout we shall assume that has an (possibly uncountable) orthonormal basis of eigenvectors of with corresponding real eigenvalues . We shall represent the inner product and norm in by and , respectively, and, as before, we shall denote the collection of all finite subsets of by .
As discussed in the introduction, is the Fréchet space: with multinorm , where see [5, Definition 2.1, Theorem 2.3]. The space of continuous linear functionals on is denoted by . Convergence in is defined by the weak* topology which is generated by the multinorm , where
The space is sequentially complete, and each generates, uniquely, a continuous linear functional via the formula see [5, Theorems 2.6, 2.7]. As a consequence, we can regard as a subspace of , and so can write
Theorem 7. Let . Then where the unordered sum is convergent in .
Proof. For any , we can use the linearity of to write Since, for each , is summable to in , it follows that, for and , there exists such that for all that contain . But then for all that contain . Hence is summable in to .
It is also possible to characterize the unconditional convergence of sums of the form , where for all , in the following way.
Theorem 8. Let be a family of scalars. Then is summable to some if and only if is summable in for each , in which case for all .
Proof. For each , let . Then for each , we obtain
Suppose that is summable in for each , and let . Then by Theorem 4, there exists such that for any that is disjoint from . But then for any such . Since this can be done for each , is summable in .
Conversely, if is summable in , then again, by Theorem 4, for any and , there exists such that for every disjoint from . Hence, from (35), for any such , which implies that is summable in for each .
The final statement in the theorem follows from the fact that, for each fixed , where and , ; see [9, Theorem 6.26].
Corresponding results on the weak* convergence of eigenvector expansions in the space can also be established. Once again, these can be regarded as natural extensions of the theory developed in [3, Chapter 9] to the case of an index set which need not be countable. First, note that can be regarded as a collection of “regular generalized eigenvectors” in . As the following theorem shows, each can be expressed as a weak*-convergent Fourier series in terms of this collection of regular generalized eigenvectors.
Theorem 9. Each can be expressed in the form where the unordered sum is convergent in .
Proof. Let and . Then, with , where , we have
(by continuity of ; see [3, Theorem 1.8-1]).
Since in and there are only a finite number of seminorms , it follows that there exists such that for all that contain . Hence
Theorem 10. Let the null space of be finite-dimensional and let . Then in if there exists such that is convergent in . Moreover, in this case for each .
Proof. Let and, for each finite subset of , let
By the convergence assumption and Theorem 4, it follows that, for , there exists a finite subset of such that for all finite subsets of disjoint from . Also, from Theorem 8,
Hence there exists a finite subset of such that for all finite subsets of disjoint from . Let . Then, for any finite subset of that is disjoint from , we have
Therefore is convergent in , which implies that is also summable in since is finite-dimensional.
To complete the proof, we use the fact that since is a continuous linear functional, we have
4. Almost-Periodic Generalized Functions
To illustrate the theory of expansions of test functions and generalized functions in terms of an uncountable basis of eigenfunctions, we examine the differential operator on the Hilbert space of almost-periodic functions. As explained in [9, Example 6.19], to obtain this space we consider first the vector space of quasiperiodic functions of the form where and are arbitrary constants. This vector space becomes a pre-Hilbert space when we define on it the inner product
The Hilbert space is then obtained as the completion of this pre-Hilbert space. Elements in are equivalence classes of functions of the form where and the sum in (52) converges with respect to the norm induced by the inner product (51). The set of functions is an uncountable orthonormal basis for , showing that is not separable. Note that each can be regarded as an eigenfunction of , defined on the domain , with corresponding eigenvalue .
To enable a Fréchet space of test functions and associated space of generalized functions to be constructed, we require a self-adjoint extension of . The fact that the set in (53) is an orthonormal basis for enables us to define in an obvious manner as Note that, for any given , only countably many terms in the expansions in (54) will be nonzero.
The general theory developed earlier now leads immediately to the following expansion result for generalized functions in .
Theorem 11. Let be the space of generalized functions constructed around the self-adjoint extension of given by (54). Then each can be expressed in the form where and the unordered sum is convergent in .
In this particular case, we can interpret as a space of almost-periodic generalized functions.
A theory of summability has been developed and applied to obtain important convergence results of an orthonormal set in suitably constructed spaces of test functions and generalized functions. As a particular case, a space of almost-periodic generalized functions has been produced, in which each generalized function can be expanded in terms of an uncountable orthonormal basis of exponential functions.
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