# Periodic Tempered Distributions of Beurling Type and Periodic Ultradifferentiable Functions with Arbitrary Support

Authors: Byung Keun Sohn
Journal: International Journal of Mathematics and Mathematical Sciences
 (2018)

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## Abstract

LetSω′(R) be the space of tempered distributions of Beurling type with test function space Sω(R) and let Eω,p be the space of ultradifferentiable functions with arbitrary support having a period p. We show that Eω,p is generated by Sω(R). Also, we show that the mapping Sω(R)→Eω,p is linear, onto, and continuous and the mapping Sω,p′(R)→Eω,p′ is linear and onto where Sω,p′(R) is the subspace of Sω′(R) having a period p and Eω,p′ is the dual space of Eω,p.


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## Body

## 1. Introduction

In his excellent book [1], Zemanian introduced a space PT of periodic test functions of period T and a space PT′ of periodic distributions with period T. He defined T-convolution, a different type of classical convolution, for PT′ and discussed Fourier series of PT′. 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 ultradistributionsD′{mk} and D′(mk) of Roumieu type and Beurling type, respectively, by the growth rate of their coefficients in [10] and Pilipović gave structure theorem for D′{mk} and D′(mk) 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, S1′, and dual space of the smooth periodic functions, Pp∗, in [17]. Smith also discussed the convolution for S1′ and Pp∗ and Fourier series of S1′ in [17].In this paper, we will consider periodic tempered distributions of Beurling type,Sω,p′(R), which is an extension of periodic tempered distributions and periodic ultradifferentiable functions with arbitrary support, Eω,p. We show that Eω,p is generated by Sω(R) which is a test function space of tempered distributions of Beurling type, Sω′(R), and the mapping Sω(R)→Eω,p is linear, onto, and continuous. Also we show that the mapping Sω,p′(R)→Eω,p′ is linear and onto, where Eω,p′ is a dual space of Eω,p.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 byMc the set of all continuous functions ω on R satisfying the following conditions:(α)
0=ω(0)≤ω(ξ+η)≤ω(ξ)+ω(η);(β)
∫‍ω(ξ)/(1+|ξ|)2dξ<∞;(γ)
ωξ≥a+blog⁡(1+|ξ|) for some real a and positive b;(δ)
ω(ξ)=Ω(|ξ|), where Ω is the increasing continuous concave function such that max|ξ|≤rω(ξ)≤Ω(r).Example 1.
ω ( ξ ) = log ⁡ ( 1 + ξ ), ω(ξ)=ξ1/d (d>1).More detailed properties and examples ofω can be found in [21] and ω represents an element in Mc throughout this paper.Letϕ∈L1(R) and if λ is a real number, we write (1)ϕλω=∫ϕ^ξeλωξdξ,which may be finite or infinite.Definition 2.
LetDω(R) be the set of all ϕ∈L1(R) such that ϕ has compact support and ϕλ(ω)<∞ for all λ>0. The continuous linear functional of Dω(R) is denoted by Dω′(R) whose elements are called the distribution of Beurling type.For eachλ>0, ·λ(ω) defines the seminorms in Dω(R). From [19, Proposition 1.3.6], if ω(ξ)=log⁡(1+|ξ|), then Dω(R)=Cc∞(R), where Cc∞(R) is the set of all infinitely differentiable functions with compact support in R. Also from [19, Theorem 1.3.18] and the property (γ) of ω, we have Dω(R)⊂Cc∞(R). A sequence {ϕn} is said to converge to ϕ in Dω(R) if ϕn-ϕλ(ω)→0 as n→∞ for any λ. By the property (γ) of ω, Dω′(R)⊃D′(R), where the elements of D′(R) are the distribution in the sense of Schwarz. The family of seminorms {ϕ→ϕλ(ω)} on Dω(R) is equivalent to the family {ϕ→eλω(ξ)supξ|ϕ^(ξ)|} by [19, Corollary 1.4.3].Example 3 (see [19, Section 1.3] or [20, Section 2]).
LetP be the Poisson kernel for the upper half-plane. (2)Pξ,η=ηπ1ξ2+η2.
We defineu(ξ,η)=∫‍P(t-ξ,η)ω(t)dt=∫‍P(s,η)ω(s+ξ)ds and v(ξ,η) is the conjugate harmonic function of u. Let (3)Fζ=exp⁡-2uξ,η-2ivξ,η,ζ=ξ+iη,η>0.
For realx, define f(x)=(2π)-1∫‍eixξF(ξ)dξ. For appropriately chosen x0, g(x)=f(x-x0)f(-x-x0) is in Dω(R) with support in (-δ,δ) for any δ>0.Definition 4.
E ω ( R ) is the set of all complex-valued functions ϕ in R such that for each compact subset K of R the restrictions to K of ϕ and of some ψ∈Dω(R) agree. The continuous linear functional of Eω(R) is denoted by Eω′(R) whose elements are called the distribution with compact support of Beurling type.The topology ofEω(R) is given by the seminorms ϕ→infψ=ϕinKψλ(ω) for any λ>0 and any compact subset K of R. We have from [19, Proposition 1.5.2] that Eω(R) is the set of all complex-valued functions ϕ in R such that if ψ∈Dω(R), then ψϕ∈Dω(R). The topology in Eω(R) is also given by the seminorms ϕ→ψϕλ(ω) for any λ>0 and any ψ∈Dω(R). A sequence {ϕn} is said to converge to ϕ in Eω(R) if ψϕn-ψϕλ(ω)→0 as n→∞ for any λ>0 and any ψ∈Dω(R). Eω′(R) can be identified with the set of all elements of Dω′(R) which have compact support in R.Definition 5.
One denotes bySω(R) the set of all functions ϕ∈L1(R) with the property of ϕ and ϕ^∈C∞ and for each λ>0 and α∈N one has (4)Pα,λϕ=supx∈Reλωxϕαx<∞,Πα,λϕ=supξ∈Reλωξϕ^αξ<∞.
A continuous linear functional ofSω(R) is denoted by Sω′(R) whose elements are called the tempered distributions of Beurling type.The topology ofSω(R) is defined by the seminorms Pα,λ and Πα,λ for each λ>0 and α∈N. It can be seen that Dω(R)⊂Sω(R)⊂Eω(R). A sequence {ϕn} is said to converge to ϕ in Sω(R) if Pα,λ(ϕn-ϕ)→0 and Πα,λ(ϕn-ϕ)→0 for each λ>0 and α∈N.LetT∈Sω′(R). We recall that T is said to be periodic with period p if there exists a positive real number p such that T(s)=T(s-p) for all s∈Sω(R), where s-p(t)=s(t+p).Definition 6.
One denotes the space of periodic tempered distributions of Beurling type having a periodp by Sω,p′(R).Example 7.
Letδx be the Dirac delta distribution such that δx(ϕ)=ϕ(x) and let p>0. Define Sδ,p=∑n=-∞∞‍δnp. Then we have from the property (γ) of ω that, for ϕ∈Sω(R), (5)ϕnp≤P0,λϕe-λωnp≤P0,λϕe-λa1+np-λb,where a and b are constants in the property (γ) of ω. If we take λ such that λb>2, ∑n=-NN‍δnp(ϕ)=∑n=-NN‍ϕ(np) converges in C. Since limx→∞ϕ(x)=0,(6)Sδ,pϕ-p=limN→∞∑n=-NNδnpϕ-p=limN→∞∑n=-NNϕn+1p=limN→∞∑k=-N+1N-1ϕkp+limN→∞ϕNp+limN→∞ϕN+1p=Sδ,pϕ.
HenceSδ,p∈Sω,p′(R). As stated in [17, Example 2], Sδ,p=∑n=-∞∞‍δnp is an impulse train and is useful in modelling the operation of sampling a continuous signal at sampling interval P 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.
Forp>0, one denotes Eω,p by the set of all functions in Eω(R) which have period p. We denote Eω,p′ by the space of all continuous, p-periodic, linear mapping from Eω,p into C.E ω , p ′ is a vector space over C when addition of functions and scalar multiplication of functions are defined in the usual manner. The topology in Eω,p is given by the seminorms f→fϕλ(ω) for any λ>0 and any ϕ∈Dω(R). A sequence {fn}∈Eω,p converges to f in Eω,p if and only if fn and f have period and fn converges to f in Eω(R).


## 3. Relationship betweenSω(R) and Eω,p and between Sω,p′(R) and Eω,p′

In this section, we show that the mappingSω(R)→Eω,p is linear, onto, and continuous and the mapping Sω,p′(R)→Eω,p′ is linear and onto. First, we show that Eω,p is generated by Sω(R).Lemma 9.
Letp>0 and let ϕ∈Sω(R). For each natural number N, define a mapping ϕN:R→C by (7)ϕNt=∑n=-NNϕt-np.
Then there exists a functionf∈Eω,p such that, for each integer k, the sequences {ϕN(k)} and {ϕ^N(k)} converge to f(k) and f^(k) uniformly on compact subsets of R, respectively.Proof.
Fixk∈Z+. For each t∈R, (8)ϕk≤e-k′ωtPk,k′ϕ,where k′ is taken such that k′b>2 for a constant b in the property (γ) of ω. Fix l>0 and let Nl be positive integer such that Nlp>l. Then for |t|≤l and n>Nl, |np-t|≥np-l>p(n-Nl). For any ϵ>0, let Mϵ be positive integer such that Mϵ>Nl and (9)2Pk,k′ϕp-k′be-k′a∑n=Mϵ∞11+n-Nlk′b<ϵ.
Herea is a constant in the property (γ) of ω. Then for |t|≤l and for L≥K>Mϵ, we have from the property (γ) of ω that(10)ϕKkt-ϕLkt≤∑n=-L-Kϕkt-np+∑n=KLϕkt-np≤2Pk,k′ϕ∑n=KLe-k′ωt-np≤2Pk,k′ϕ∑n=Mϵ∞e-k′a+blog⁡1+pn-Na≤2Pk,k′ϕp-k′be-k′a∑n=Mϵ∞11+n-Nlk′b<ϵ.
Hence{ϕN(k)(t)} is a Cauchy sequence, uniformly in |t|≤l. Since Sω(R) is complete, for each k∈Z+, there exist a function fk:R→C such that, for each t∈R, limN→∞ϕN(k)(t) converges to fk(t) uniformly on compact subsets of R. If we take f=f0, we have from basic theorem for calculus and mathematical induction that f0(k)=fk for all k∈Z+. By the same process of periodicity of Sδ,p in Example 7, we know that f is periodic. Now since(11)ψϕNλω≤CsupξeλωξψϕN^ξ=C2π-1eλωξψ^∗ϕ^Nξ=C2π-1eλωξ∫ψ^ξ-ηϕ^Nηdη≤C2π-1∫ψ^ξ-ηeλωξ-ηeλωη·ϕ^Nηelωηe-lωηdη=C2π-1Π0,λψΠ0,lϕN∫eλ-lωηdη,for some constant C, any constant l, and any ψ∈Dω(R), if we take l sufficiently large, ψϕNλ(ω) is uniformly bounded. Then since (12)ψfλω=ψlimN→∞ϕNλω=limN→∞ψϕNλωis bounded, f∈Eω(R). If we replace Pk,k′ by Πk,k′ in the proof of first part of this theorem and use(13)ψϕ^Nλω=C2π-1eλωξψ^∗ϕ^^Nξ≤C∫ψ^ξ-ηeλωξ-ηeλωηϕˇNηelωηe-lωηdη=CΠ0,λψP0,lϕN∫eλ-lωηdη,we can show from the same process of the above that there exist g∈Eω,p such that (14)limN→∞ϕ^Nkξ=gkξ.
By the Lebesgue dominated convergence theorem, we have(15)gkξ=limN→∞Dξk∫e-itξϕNtdt=Dξk∫e-itξlimN→∞ϕNtdt=Dξk∫e-itξftdt=f^kξ.Definition 10.
DefineLω:Sω(R)→Eω,p by Lω(ϕ)=f, where f is in Lemma 9, and denote Γω,p={γ∈Sω:Lω(γ)=1}.Example 11.
Chooseϵ>0 such that [-p/4,p/4]+[0,4ϵ]⊂[-p/2,p/2]. By Example 3, we can find a nontrivial ψ∈Dω((0,ϵ)). Then ψ∗ψˇ is nontrivial and ψ∗ψˇ⊂Dω((0,2ϵ)) by [19, Corollary 1.3.14]; hence ψ∗ψˇ¯⊂Dω((0,2ϵ)). If we define (16)φ=ψ∗ψˇ2=ψ∗ψˇψ∗ψˇ¯,we have from [19, Proposition 1.3.5] that φ⊂Dω((0,2ϵ)). Since φ is nonnegative and nontrivial, we have ∫‍φ(x)dx=1 without loss of the generality by multiplying φ by a suitable positive constant. Let (17)mx=1-p/4,p/4+0,2ϵ∗φx,where 1K is the characteristic function on K. Then m(x)=1 on [-p/4,p/4] and m∈Dω([-p/2,p/2]) by the fact that suppm⊂[-p/4,p/4]+[0,4ϵ]⊂[-p/2,p/2]. Define m~:R→R by m~(x)=(m(x)+1)/2 and define χ:R→R by(18)
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Thenχ∈Dω(R)⊂Sω(R) and suppχ⊂[-p,p]. If we define a mapping χN:R→C by χN(x)=∑n=-NN‍χ(x-np), then limN→∞χ(x)=f(x)=Lω(χ) by Lemma 9. Then we have from the last part of [14, Example 18] that Lω(χ)=f(x)=1 for all t∈R.Theorem 12.
Givenf∈Eω,p, there exist ϕ∈Sω(R) such that f(x)=limN→∞ϕN(x) uniformly on compact subsets in R.Proof.
Letχ be the function in Example 11 and let ϕ=χf. Since χ∈Dω(R) and f∈Eω(R), ϕ∈Dω(R)⊂Sω(R) and for x in R, we have from Example 11 and Lemma 9 that(19)limN→∞ϕNx=limN→∞∑n=-NNχx-npfx-np=fxlimN→∞∑n=-NNχx-np=fx.The convergence in (19) is uniform on compact subsets in R by Lemma 9.Now, we show the relationship betweenSω(R) and Eω,p. Since Lω(ϕ)=Lω(ϕnp) for any n∈Z+, Lω is not one-to-one, but Lω satisfies the properties of isomorphism except for one-to-one property.Theorem 13.
The mappingLω:Sω(R)→Eω,p is linear, onto, and continuous.Proof.
L ω is clearly linear. Let χ be in Example 11 and let g∈Eω,p. Since g∈Eω(R) and χ∈Dω(R), gχ∈Dω(R)⊂Sω(R). Then since(20)Lωgχx=limN→∞∑N=-nngx-npχx-np=gxlimN→∞χNx=gx,Lω is onto. To show that Lω is continuous, let {ϕn} and ϕ be in Sω(R) such that {ϕn}→ϕ in Sω(R). Let fn=Lω(ϕn) and f=Lω(ϕ). Since Sω(R) and Eω,p are metric spaces, we need to show that fn→f in Eω,p. But fn and f are clearly periodic by the proof of Lemma 9, it suffices to show that fn→f in Eω(R). For any ϵ>0, there exist M>0 such that if n>M, then Π0,0(ϕn-ϕ)<ϵ/6 and can take Λ with Λb>2 such that (21)Π0,Λϕn-ϕe-Λa∑m>111+m-1pΛb<ϵ2,where a and b are constants in the property (γ) of ω. Then for n>M and |ξ|≤p,(22)f^nξ-f^ξ≤∑m=-∞∞ϕ^nξ-mp-ϕ^ξ-mp≤∑m=-11ϕ^nξ-mp-ϕ^ξ-mp+∑m>1ϕ^nξ-mp-ϕ^ξ-mp≤3Π0,0ϕn-ϕ+∑m>1Π0,Λϕn-ϕe-Λωξ-mp≤3Π0,0ϕn-ϕ+Π0,Λϕn-ϕe-Λa∑m>111+m-1pΛb<ϵ.
Thenf^n(ξ) converges uniformly to f^(ξ) on |ξ|<p and hence on R by periodicity of f^n and f^ by Lemma 9. Hence if we apply the Lebesgue convergence theorem, we have that, for φ∈Dω(R),(23)φfn-fλω=∫φfn-f^ξeλωξdξ=2π-1∫φ^∗f^n-f^ξeλωξdξ⟶0,as n→∞. Thus fn converges to f in Eω(R).We show the relationship between periodic tempered distributions of Beurling type,Sω,p′(R), and the dual space of Eω,p.Definition 14.
One denotesEω,p′ by the space of all continuous, p-periodic linear mappings Λω:Eω,p→C.In Definition14, p-periodicity of Λω means that Λω(f)=Λω(f-p) for all f∈Eω,p.Example 15.
Consider the functionΛω:Eω,p→C defined by Λω(f)=f(0) for all f∈Eω,p. Clearly, Λω is periodic and linear. Let fn→f in Eω,p. Then fn→f in Eω(R), hence φfn→φf in Dω(R) for φ∈Dω(R). we have from [20, Proposition 3.9] that φfn→φf in Cc∞, that is, φfn→φf uniformly on compact subset of R, that is, φ(0)fn(0)→φ(0)f(0). If we take φ∈Dω(R) with φ(0)≠0, Λω(fn)=fn(0)→Λω(f)=f(0) in C. Thus Λω∈Eω,p′.Letf,g∈Eω(R) and let φ∈Dω(R). Since (φf)′ and φ′f are in Dω(R) by [19, Theorem 1.3.27], φf′∈Dω(R) by Leibniz’s formula. Hence f′∈Eω(R) and f(k)∈Eω(R) by mathematical induction and Leibniz’s formula. Also clearly fg∈Eω(R). Then we can define the following: let Λω∈Eω,p′ and let a∈R, k∈Z+, and g∈Eω,p. For all f∈Eω,p, Λω,a(f)=Λω(f-a), Λ~ω(f)=Λω(f~), Λω(k)(f)=(-1)kΛω(f(k)), and (gΛω)(f)=Λω(gf).Now we will show the relationship betweenSω,p′(R) and Eω,p′.Definition 16.
The mappingΨω:Eω,p′→Sω′(R) is defined by Ψω(Λω)=Λω∘Lω, where Lω is in Definition 14. Let χ be in Example 11. The mapping Φω:Eω,p′→Sω′(R) is defined by Φω(T)(f)=T(fχ) for all f∈Sω(R).Clearly we have thatΨω(f) has period p and Ψω is linear.Theorem 17.
LetT∈Sω,p′(R) and let χ be the function defined in Example 11. Then the mapping Φω:Sω,p′(R)→Eω,p′ defined by (24)ΦωTf=Tfχfor all f∈Eω,p is linear and onto.Proof.
ClearlyΦω(T) is periodic. Now we show that Φω(T) is continuous. Let {fn}, f∈Eω,p and let fn→f in Eω,p. Since {fn}, f∈Eω(R), and χ∈Dω(R), fnχ and fχ  are in fact in Dω(R). Then by [20, Proposition 3.9] fn→f in Cc∞(K) for some compact subset K of R; hence (fnχ)(k)→(fχ)(k) uniformly on K as n→∞ for any k∈Z+. Then for any k∈Z+ and λ>0, if we let suppχ=K, then(25)Pλ,kfnχ-fχ=supx∈Reλωxfnχ-fχkx≤eλωcKsupx∈Kfnχ-fχkx⟶0,as n→0, where cK=maxx∈K|x|.
Sincefn→f in Dω(R) we have from [19, Definition 1.3.25] that, for any λ>0,(26)supξ∈Reλωξ(fnχ-fχ)^ξ⟶0n⟶∞.
Then, for anyk∈Z+ and λ>0, if we let suppχ=K, then, by (26),(27)Πλ,kfnχ-fχ=supξ∈Reλωξ(fnχ-fχ)^kξ=supξ∈Reλωξ∫Ke-ixξxkfnχ-fχxdx≤CK,ksupξ∈Reλωξ∫Ke-ixξfnχ-fχxdx≤CK,ksupξ∈Reλωξ(fnχ-fχ)^ξ⟶0,as n→∞, where CK,k=supx∈R|x|k. Hence fnχ→fχ in Sω(R). By the continuity of T, (28)ΦωTfn=Tfnχ⟶Tfχ=ΦωTf,as n→∞. Hence Φω(T) is continuous, that is; Φω(T) is in Eω,p′. By the same line in [17, Proof of Theorem 2], Φω(T) is linear and(29)ΦωΨωΛω=Λω;that is, Φω(T) is onto.Example 18.
LetSδ,p be in Example 7 and Λω be in Example 15. If we choose χ∈Γω,p, then for f∈Eω,p we have from the periodicity of f that (30)ΦωSδ,pf=Φωχf=f0∑n=-∞∞χnp=Λωf.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 ϕNu→fu as N→∞ in Sω(R) for ϕ,u∈Sω(R). We can show the convergence in the seminorm Pλ,k as follows.Letx∈R and take m∈Z+ such that x∈[mp,(m+1)p] and let a and b be constants in the property (γ) of ω. Then(31)ϕNkx=∑n=-NNϕkx-np≤Pλ,kϕ∑n=-∞∞e-λωx-np≤Pλ,kϕ∑n≠m,m+1e-λa·e-λblog⁡1+x-np+2≤Pλ,kϕe-λa1pλb∑n=1∞1n2+2=Aλ,k.Here we takeλ such that λb>2. By the periodicity of f(k) in Lemma 9, we have(32)ϕNkx-fk≤Ak.Now(33)Pλ,kϕN-fu=supx∈ReλωtϕN-fukx≤supx∈Reλωt∑0≤j≤kckjϕN-fk-jxujx≤cksupx>AeλωtϕN-fk-jxujx+cksupx≤AeλωtϕN-fk-jxujx=I+II,where ck=(k+1)max0≤j≤k|ckj|. Using (32), for any ϵ>0,(34)I=cksupx>AeλωtϕN-fk-jxujx≤Ak-jcksupx>Aeλωtujx≤Ak-jckPλu,jue-λuωx≤Ak-jckPλu,jue-λua11+xλub<ϵ2Ak-jck,when we take λu sufficiently large. Since ϕN→f as N→∞ uniformly on compact subset on R by Lemma 9,(35)II=supx≤AeλωtϕN-fk-jxujx≤Pλu,jusupx≤AϕN-fk-jx≤ϵ2ck.Hence Pλ,k((ϕN-f)u)→0 as N→∞.But we did not succeed in the convergence of(ϕN-f)u→0 as N→∞ in the seminorm Πλ,k, so we did not show one-to-one property in Theorem 17.Remark 19.
We have from (29) that Φω∘Ψω=I, where I is the identity operator on Eω,p′ by onto property of Φω, but we cannot show Ψω∘Φω=I, where I is the identity operator on Sω,p′(R) by the above fact about one-to-one property of Φω.Example 20.
Letf∈L1loc with period p. Then Tf and Tf^ are in Sω,p′(R). In fact, let ϕ∈Sω(R) and a and b are constants in property (γ) of ω. Then, for all x∈R and any λ1>0, (36)ϕx≤e-λ1ωxP0,λ1ϕ.
Hence ifn∈Z and x∈[np,(n+1)p],(37)ϕx≤e-λ1ωnpP0,λ1ϕn∈Z+e-λ1ωn+1pP0,λ1ϕnisnegativeinteger.
Then if we takeλ1 sufficiently large such that λ1b>2,(38)Tfϕ=∫fxϕxdx≤∑n<0∫npn+1pfxϕxdx+∑n≥0∫npn+1pfxϕxdx≤∑n<0e-λ1ωn+1pP0,λ1ϕ∫npn+1pfxdx+∑n≥0e-λ1ωnpP0,λ1ϕ∫npn+1pfxdx≤1+∑n≤-11npλ1b+1+∑n>01npλ1b×eλ1aP0,λ1ϕ∫0pfxdx=21+1pλ1b∑n=1∞1nλ1beλ1aP0,λ1ϕ∫0pfxdx<∞.
ClearlyTf is linear and periodic by the periodicity of f. To show that Tf is continuous, let {ϕn}∈Sω(R) such that ϕn→0 in Sω(R). For any ϵ>0, there exists an integer N such that P0,λ(ϕn)<ϵ for any n>N and λ1>0. For such an n>N, by (38), (39)∫fxϕnxdx≤2ϵ1+1pλ1b∑n=1∞1nλ1beλ1a∫0pfxdxwhich shows that Tf(ϕn)→0 as n→∞. Thus Tf∈Sω,p′(R).
By the property of Fourier transform, we have that(40)Tf^ϕ=∫f^xϕxdx=∫fxϕ^xdx.
If we replaceP0,λ1 by Π0,λ2 with sufficiently large λ2 in (38), we have that |Tf^(ϕ)|<∞. Through the above process, we have that Tf^∈Sω,p′(R).


## 4. Application to Fourier Analysis

Let{Tn} be a sequence in Sω,p′(R). {Tn} is said to converge provided that limn→∞Tn(ϕ) exists for all ϕ∈Sω(R). The series ∑n=-∞∞‍Tn is said to converge provided that the sequence of partial sums, {∑n=-NN‍Tn}, converges. Similarly, a sequence {Λω,n} in Eω,p′ is said to converge provided that limn→∞Λω,n(f) exists for all f∈Eω,p. The series ∑n=-∞∞‍Λω,n is said to converge provided that the sequence of partial sums, {∑n=-NN‍Λω,n}, converges.Lemma 21.
The mappingΨω:Eω,p′→Sω,p′(R) defined by(41)ΨωΛω=Λω∘Lωis linear.Proof.
Since the composition of continuous linear functions,Ψω(Λω)∈Sω′(R). Since Lω(ϕ-p)=Lω(ϕ), for any ϕ∈Sω(R),(42)Λω∘Lωpϕ=Λω∘Lωϕ-p=Λω∘Lωϕ.
Therefore,Ψω(Λω) has period P; that is, Ψω(Λω)∈Sω,p′(R). The linearity of Ψω is clear.Theorem 22.
S ω , p ′ ( R ) and Eω,p′ are closed sets under convergence.Proof.
Let{Tn} be a sequence in Sω,p′(R) and let T(ϕ)=limn→∞Tn(ϕ) exist for all ϕ∈Sω(R). Since Sω(R) is a Frěchet space, we have from Banach-Steinhaus theorem that T is linear and continuous; that is, T∈Sω′(R). Since each Tn is P-periodic, for each ϕ∈Sω(R),(43)TPϕ=Tϕ-P=limn→∞Tnϕ-P=limn→∞Tnpϕ=limn→∞Tnϕ=Tϕ.
Therefore,T∈Sω,p′(R). Now let {Λω,n} in Eω,p′ converge and let Λω(f)=limn→∞Λω,n(f) for all f∈Eω,p. For each positive integer n, let Tn=Ψω(Λω,n). Then for any ϕ∈Sω(R), from Lemma 21, (44)limn→∞Tnϕ=limn→∞ΨωΛω,nϕ=limn→∞Λω,nLωϕ=ΛωLωϕ.
Therefore,limn→∞Tn=Λω∘Lω and Λω∘Lω∈Sω,p′(R) by (43). We have from the mathematical equation of Lω in the proof of Theorem 13 that, for each f∈Eω,p, (45)ΦωΛω∘Lωf=Λω∘Lωχf=Λωf.
HenceΦω(Λω∘Lω)=Λω; that is, Λω∈Eω,p′.For anyf∈Eω,p, we have (46)ΦωTf=Tfχ=limn→∞Tnfχ=limn→∞ΦωTn.Therefore,(a)
If{Tn}∈Sω,p′(R) and T=limn→∞Tn, then Φ(T)=limn→∞Φ(Tn).(b)
From Theorem22, if {Λω}∈Eω,p′ and Λω=limn→∞Λω,n, then Ψω(Λω)=limn→∞(Λω,n).From (a) and (b), we have thatΦω and Ψω preserve convergence.Example 23 (Fourier series of the (distributional) Fourier transform of an impulse train).
We define the Fourier transform ofT∈Sω′(R) by(47)FTf=TFf,f∈SωR.We have from an application of the Fubini theorem that (47) generalizes the classical Fourier transform.For each integern, define gn:R→C by gn(ξ)=e-inPξ for all ξ∈R. Then, for ϕ∈Sω′(R),(48)FSδ,Pϕ=Sδ,PFϕ=∑-∞∞δnPFϕ=∑-∞∞FδnPϕ=∑-∞∞Tgnϕ.Therefore, the Fourier transform of an impulse train isF(Sδ,P)=∑-∞∞‍Tgn.



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