Abstract
Let be the space of tempered distributions of Beurling type with test function space and let be the space of ultradifferentiable functions with arbitrary support having a period . We show that is generated by . Also, we show that the mapping is linear, onto, and continuous and the mapping is linear and onto where is the subspace of having a period and is the dual space of .
1. Introduction
In his excellent book [1], Zemanian introduced a space of periodic test functions of period and a space of periodic distributions with period . He defined -convolution, a different type of classical convolution, for and discussed Fourier series of . His results have been applied in distribution theory [2–4], transform analysis [3, 5], differential equations [2, 6, 7], and signal analysis [8, 9].
Since then, developments of periodic ultradistributions and periodic ultradifferentiable functions theory have been made. Gorbačuk characterized two periodic ultradistributions and of Roumieu type and Beurling type, respectively, by the growth rate of their coefficients in [10] and Pilipović gave structure theorem for and in [11]. Also, Taguchi studied the Fourier coefficients of ultradifferentiable functions of Gevrey classes and ultradistributions in [12].
Gorbačuk, Pilipović, and Taguchi’s results have been applied in almost periodic ultradistributions of Roumieu type and Beurling type [13, 14] and signal analysis [15].
In the meantime, Smith discussed tempered distributions by focusing on distributional convolution and some of its application to signal analysis in [16] and showed the relationship between periodic tempered distributions, , and dual space of the smooth periodic functions, , in [17]. Smith also discussed the convolution for and and Fourier series of in [17].
In this paper, we will consider periodic tempered distributions of Beurling type, , which is an extension of periodic tempered distributions and periodic ultradifferentiable functions with arbitrary support, . We show that is generated by which is a test function space of tempered distributions of Beurling type, , and the mapping is linear, onto, and continuous. Also we show that the mapping is linear and onto, where is a dual space of .
Our results are indebted to Zemanian [1] and especially to Smith [17].
2. Periodic Tempered Distributions of Beurling Type and Periodic Ultradifferentiable Functions
We review tempered distributions of Beurling type and ultradifferentiable functions and related results which we need later. For more details, we can refer to [18–20].
We denote by the set of all continuous functions on satisfying the following conditions:;; for some real and positive ;, where is the increasing continuous concave function such that .
Example 1. , ().
More detailed properties and examples of can be found in [21] and represents an element in throughout this paper.
Let and if is a real number, we write which may be finite or infinite.
Definition 2. Let be the set of all such that has compact support and for all . The continuous linear functional of is denoted by whose elements are called the distribution of Beurling type.
For each , defines the seminorms in . From [19, Proposition 1.3.6], if , then , where is the set of all infinitely differentiable functions with compact support in . Also from [19, Theorem 1.3.18] and the property of , we have . A sequence is said to converge to in if as for any . By the property of , , where the elements of are the distribution in the sense of Schwarz. The family of seminorms on is equivalent to the family by [19, Corollary 1.4.3].
Example 3 (see [19, Section ] or [20, Section ]). Let be the Poisson kernel for the upper half-plane. We define and is the conjugate harmonic function of . Let For real , define For appropriately chosen , is in with support in for any .
Definition 4. is the set of all complex-valued functions in such that for each compact subset of the restrictions to of and of some agree. The continuous linear functional of is denoted by whose elements are called the distribution with compact support of Beurling type.
The topology of is given by the seminorms for any and any compact subset of . We have from [19, Proposition 1.5.2] that is the set of all complex-valued functions in such that if , then . The topology in is also given by the seminorms for any and any . A sequence is said to converge to in if as for any and any . can be identified with the set of all elements of which have compact support in .
Definition 5. One denotes by the set of all functions with the property of and and for each and one has A continuous linear functional of is denoted by whose elements are called the tempered distributions of Beurling type.
The topology of is defined by the seminorms and for each and . It can be seen that . A sequence is said to converge to in if and for each and .
Let . We recall that is said to be periodic with period if there exists a positive real number such that for all , where .
Definition 6. One denotes the space of periodic tempered distributions of Beurling type having a period by .
Example 7. Let be the Dirac delta distribution such that and let . Define . Then we have from the property of that, for , where and are constants in the property ( of . If we take such that , converges in . Since ,Hence . As stated in [17, Example 2], is an impulse train and is useful in modelling the operation of sampling a continuous signal at sampling interval and in establishing the link between analog and digital signals via the Sampling Theorem.
We introduce the space of periodic ultradifferentiable functions with compact support.
Definition 8. For , one denotes by the set of all functions in which have period . We denote by the space of all continuous, -periodic, linear mapping from into .
is a vector space over when addition of functions and scalar multiplication of functions are defined in the usual manner. The topology in is given by the seminorms for any and any . A sequence converges to in if and only if and have period and converges to in .
3. Relationship between and and between and
In this section, we show that the mapping is linear, onto, and continuous and the mapping is linear and onto. First, we show that is generated by .
Lemma 9. Let and let . For each natural number , define a mapping by Then there exists a function such that, for each integer , the sequences and converge to and uniformly on compact subsets of , respectively.
Proof. Fix . For each , where is taken such that for a constant in the property of . Fix and let be positive integer such that . Then for and , . For any , let be positive integer such that and Here is a constant in the property of . Then for and for , we have from the property of thatHence is a Cauchy sequence, uniformly in . Since is complete, for each , there exist a function such that, for each , converges to uniformly on compact subsets of . If we take , we have from basic theorem for calculus and mathematical induction that for all . By the same process of periodicity of in Example 7, we know that is periodic. Now sincefor some constant , any constant , and any , if we take sufficiently large, is uniformly bounded. Then since is bounded, . If we replace by in the proof of first part of this theorem and usewe can show from the same process of the above that there exist such that By the Lebesgue dominated convergence theorem, we have
Definition 10. Define by , where is in Lemma 9, and denote
Example 11. Choose such that By Example 3, we can find a nontrivial . Then is nontrivial and by [19, Corollary 1.3.14]; hence If we define we have from [19, Proposition 1.3.5] that Since is nonnegative and nontrivial, we have without loss of the generality by multiplying by a suitable positive constant. Let where is the characteristic function on . Then on and by the fact that Define by and define byThen and . If we define a mapping by , then by Lemma 9. Then we have from the last part of [14, Example 18] that for all .
Theorem 12. Given , there exist such that uniformly on compact subsets in .
Proof. Let be the function in Example 11 and let . Since and , and for in , we have from Example 11 and Lemma 9 thatThe convergence in (19) is uniform on compact subsets in by Lemma 9.
Now, we show the relationship between and . Since for any , is not one-to-one, but satisfies the properties of isomorphism except for one-to-one property.
Theorem 13. The mapping is linear, onto, and continuous.
Proof. is clearly linear. Let be in Example 11 and let . Since and , . Then since is onto. To show that is continuous, let and be in such that in . Let and Since and are metric spaces, we need to show that in . But and are clearly periodic by the proof of Lemma 9, it suffices to show that in . For any , there exist such that if , then and can take with such that where and are constants in the property of . Then for and ,Then converges uniformly to on and hence on by periodicity of and by Lemma 9. Hence if we apply the Lebesgue convergence theorem, we have that, for ,as . Thus converges to in .
We show the relationship between periodic tempered distributions of Beurling type, , and the dual space of .
Definition 14. One denotes by the space of all continuous, -periodic linear mappings .
In Definition 14, -periodicity of means that for all .
Example 15. Consider the function defined by for all . Clearly, is periodic and linear. Let in . Then in , hence in for . we have from [20, Proposition 3.9] that in , that is, uniformly on compact subset of , that is, . If we take with , in . Thus .
Let and let . Since and are in by [19, Theorem ], by Leibniz’s formula. Hence and by mathematical induction and Leibniz’s formula. Also clearly . Then we can define the following: let and let , , and . For all , , , , and
Now we will show the relationship between and .
Definition 16. The mapping is defined by , where is in Definition 14. Let be in Example 11. The mapping is defined by for all .
Clearly we have that has period and is linear.
Theorem 17. Let and let be the function defined in Example 11. Then the mapping defined by for all is linear and onto.
Proof. Clearly is periodic. Now we show that is continuous. Let , and let in . Since , , and , and are in fact in . Then by [20, Proposition 3.9] in for some compact subset of ; hence uniformly on as for any Then for any and , if we let , thenas , where .
Since in we have from [19, Definition 1.3.25] that, for any ,Then, for any and , if we let , then, by (26),as , where . Hence in . By the continuity of , as . Hence is continuous, that is; is in . By the same line in [17, Proof of Theorem 2], is linear andthat is, is onto.
Example 18. Let be in Example 7 and be in Example 15. If we choose , then for we have from the periodicity of that
In [17, Theorem 2], Smith showed that the mapping from periodic tempered distribution to the dual space of periodic infinitely differentiable functions is linear, onto, and one-to-one. To guarantee one-to-one property in Theorem 17, we need to show as in for . We can show the convergence in the seminorm as follows.
Let and take such that and let and be constants in the property of . Then
Here we take such that By the periodicity of in Lemma 9, we have
Nowwhere Using (32), for any ,when we take sufficiently large. Since as uniformly on compact subset on by Lemma 9,Hence as .
But we did not succeed in the convergence of as in the seminorm , so we did not show one-to-one property in Theorem 17.
Remark 19. We have from (29) that , where is the identity operator on by onto property of , but we cannot show , where is the identity operator on by the above fact about one-to-one property of .
Example 20. Let with period . Then and are in . In fact, let and and are constants in property of . Then, for all and any , Hence if and ,Then if we take sufficiently large such that ,Clearly is linear and periodic by the periodicity of . To show that is continuous, let such that in . For any , there exists an integer such that for any and . For such an , by (38), which shows that as . Thus .
By the property of Fourier transform, we have that If we replace by with sufficiently large in (38), we have that . Through the above process, we have that .
4. Application to Fourier Analysis
Let be a sequence in . is said to converge provided that exists for all . The series is said to converge provided that the sequence of partial sums, , converges. Similarly, a sequence in is said to converge provided that exists for all . The series is said to converge provided that the sequence of partial sums, , converges.
Lemma 21. The mapping defined byis linear.
Proof. Since the composition of continuous linear functions, . Since , for any ,Therefore, has period ; that is, . The linearity of is clear.
Theorem 22. and are closed sets under convergence.
Proof. Let be a sequence in and let exist for all . Since is a Frěchet space, we have from Banach-Steinhaus theorem that is linear and continuous; that is, . Since each is -periodic, for each ,Therefore, . Now let in converge and let for all . For each positive integer , let . Then for any , from Lemma 21, Therefore, and by (43). We have from the mathematical equation of in the proof of Theorem 13 that, for each , Hence ; that is, .
For any , we have
Therefore,(a)If and , then .(b)From Theorem 22, if and , then
From (a) and (b), we have that and preserve convergence.
Example 23 (Fourier series of the (distributional) Fourier transform of an impulse train). We define the Fourier transform of by
We have from an application of the Fubini theorem that (47) generalizes the classical Fourier transform.
For each integer , define by for all . Then, for ,
Therefore, the Fourier transform of an impulse train is
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by a grant from Inje University for the Research, 20170080.