# Shifting and Variational Properties for Fourier-Feynman Transform and Convolution

Authors: Byoung Soo Kim
Journal: Journal of Function Spaces
 (2015)

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## Abstract

Shifting, scaling, modulation, and variational properties for Fourier-Feynman transform of functionals in a Banach algebraS are given. Cameron and Storvick's translation theorem can be obtained as a corollary of our result. We also study shifting, scaling, and modulation properties for the convolution product of functionals in S.


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## Body

## 1. Introduction

LetC0[0,T] denote the Wiener space, that is, the space of real valued continuous functions x on [0,T] with x(0)=0. The concept of L1 analytic Fourier-Feynman transform for functionals on Wiener space was introduced by Brue in [1]. In [2], Cameron and Storvick introduced L2 analytic Fourier-Feynman transform. In [3], Johnson and Skoug developed Lp analytic Fourier-Feynman transform for 1≤p≤2 that extended the results in [2].In [4, 5], Huffman et al. defined a convolution product for functionals on Wiener space and showed that the Fourier-Feynman transform of a convolution product is a product of Fourier-Feynman transforms. Recently Kim et al. [6] obtained change of scale formulas for Wiener integrals related to Fourier-Feynman transform and convolution. For a detailed survey of the previous work on Fourier-Feynman transform and related topics, see [7].LetM denote the class of all Wiener measurable subsets of C0[0,T] and let m denote Wiener measure. Then (C0[0,T],M,m) is a complete measure space and we denote the Wiener integral of a functional F by(1)∫C00,TFxdmx.A subsetE of C0[0,T] is said to be scale-invariant measurable [8] provided ρE is measurable for each ρ>0, and a scale-invariant measurable set N is said to be scale-invariant null provided m(ρN)=0 for each ρ>0. A property that holds except on a scale-invariant null set is said to hold scale-invariant almost everywhere (s-a.e.).LetC+ and C+~ denote the sets of complex numbers with positive real part and the complex numbers with nonnegative real part, respectively. Let F be a complex valued measurable functional on C0[0,T] such that the Wiener integral (2)JFλ=∫C00,TFλ-1/2xdmxexists as a finite number for all λ>0. If there exists a function JF∗(λ) analytic in C+ such that JF∗(λ)=JF(λ) for all λ>0, then JF∗(λ) is defined to be the analytic Wiener integral of F over C0[0,T] with parameter λ, and for λ∈C+ we write (3)∫C00,TanwλFxdmx=JF∗λ.If the following limit exists for nonzero real q, then we call it the analytic Feynman integral of F over C0[0,T] with parameter q and we write(4)∫C00,TanfqFxdmx=limλ→-iq⁡∫C00,TanwλFxdmx,where λ approaches -iq through C+.Now we will introduce the class of functionals that we work with in this paper. The Banach algebraS, which was introduced by Cameron and Storvick [9], consists of functionals expressible in the form(5)Fx=∫L20,Texp⁡iv,xdfvfor s-a.e. x in C0[0,T], where the associated measure f is a complex Borel measure on L2[0,T] and 〈v,x〉 denote the Paley-Wiener-Zygmund stochastic integral ∫0T‍v(t)dx(t).In this paper, we study shifting, scaling, modulation, and variational properties for Fourier-Feynman transform of functionals inS. Shifting properties are some of the important properties of Fourier transform. In Section 2, we develop shifting properties for Fourier-Feynman transform. For example, time shifting, frequency shifting, scaling, and modulation properties for Fourier-Feynman transform are given.In Section3, we study variational properties for Fourier-Feynman transform of functionals in S and in the last section we develop shifting, scaling, and modulation properties for convolution product of functionals in S.The Banach algebraS is a very rich class of functionals. It is known that important functionals in quantum mechanics and Feynman integration theory belong to S. For example, functionals of the form(6)Fx=exp⁡∫0T∫0Tfs,t,xs,xtdsdtwere discussed in the book by Feynman and Hibbs [10] on path integrals and in Feynman’s original paper [11]. For appropriate f:[0,T]2×R2→C, functionals of form (6) are known to belong to S [12]. Hence the results in this paper can be immediately applied to many functionals of form (6).


## 2. Shifting Properties for Fourier-Feynman Transform

In this section we develop some of the important properties relevant to shifting (translating) and computational rules for Fourier-Feynman transform of functionals in the Banach algebraS. Let us begin with the definition of the Fourier-Feynman transform of functionals on Wiener space.Let1≤p<∞ and let q be a nonzero real number throughout this paper.Definition 1.
LetF be a functional on C0[0,T]. For λ∈C+ and y∈C0[0,T], let(7)TλFy=∫C00,TanwλFx+ydmx.For 1<p<∞, we define Lp analytic Fourier-Feynman transform Tq(p)[F] of F on C0[0,T] by the formula (λ∈C+)(8)TqpFy=l.i.m.λ→-iq⁡TλFy,whenever this limit exists; that is, for each ρ>0, (9)limλ→-iq⁡∫C00,TTλFρx-TqpFρxp′dmx=0,where 1/p+1/p′=1. We define L1 analytic Fourier-Feynman transform Tq(1)[F] of F by (λ∈C+)(10)Tq1Fy=limλ→-iq⁡TλFy,for s-a.e. y∈C0[0,T], whenever this limit exists [2–5].SinceTλ is linear, obviously Tq(p) is linear; that is,(11)TqpaF+bGy=aTqpFy+bTqpGyfor all constants a, b and functionals F, G on C0[0,T], whenever each transform exists.By Definition (4) of the analytic Feynman integral and L1 analytic Fourier-Feynman transform (10), it is easy to see that(12)Tq1Fy=∫C00,TanfqFx+ydmx.In particular, if F∈S, then F is analytic Feynman integrable and(13)Tq1F0=∫C00,TanfqFxdmx.Huffman et al. established the existence of Fourier-Feynman transform onC0[0,T] for functionals in S.Theorem 2 (Theorem 3.1 of [5]).
LetF∈S be given by (5). Then for all λ∈C+(14)TλFy=∫L20,Texp⁡iv,y-12λv2dfvfor s-a.e. y∈C0[0,T]. Moreover the Fourier-Feynman transform Tq(p)[F] exists, belongs to S, and is given by(15)TqpFy=∫L20,Texp⁡iv,y-i2qv2dfvfor s-a.e. y∈C0[0,T].The Fourier transformF turns a function f into a new function F[f]. Because the transform is used in signal analysis, we usually use t for time as the variable with f and ω as the variable of the transform F[f]; that is, (16)Ffω=∫-∞∞fte-iωtdt.Engineers refer to the variable ω in the transformed function as the frequency of the signal f [13].We will use the same convention in this paper; that is, for a Fourier-Feynman transformTq(p)[F](y) of F(x), we call the variable x a time and the variable y a frequency.Our first result in this section shows that the time shifting of the Fourier-Feynman transform is equal to the frequency shifting of the Fourier-Feynman transform.Theorem 3.
LetF be a functional on C0[0,T] and let x0∈C0[0,T]. Then one has(17)TqpF·-x0y=TqpFy-x0if all sides exist.Proof.
For allλ>0 and for s-a.e. y∈C0[0,T],(18)TλF·-x0y=∫C00,TFλ-1/2x-x0+ydmx=TλFy-x0if the Wiener integral exists. Hence we have the result.The following theorem is reminiscent of the time shifting theorem for the Fourier transform. Hence we call the following theorem the time shifting formula for Fourier-Feynman transform on Wiener space. It says that if we shift backx0 and replace F(x) by F(x-x0), then the Fourier-Feynman transform of this shifted function is equal to the Fourier-Feynman transform of Fxexp⁡iqx0,x multiplied by an exponential factor.Theorem 4 (time shifting).
LetF∈S be given by (5) and let x0∈C0[0,T]. Then one has(19)TqpF·-x0y=exp⁡-iqx0,y+iq2x02TqpF·exp⁡iqx0,·yfor s-a.e. y∈C0[0,T].Proof.
LetG(x)=F(x)exp{iq〈x0,x〉}. Using (5) we write G(x) as(20)Gx=∫L20,Texp⁡iv+qx0,xdfv=∫L20,Texp⁡iw,xdgw,where g(E)=f(E-qx0) for a Borel subset E of L2[0,T]; that is, G also belongs to S. Now by Theorem 2, we have(21)TqpGy=∫L20,Texp⁡iw,y-i2qw2dgw=∫L20,Texp⁡iv+qx0,y-i2qv+qx02dfv=exp⁡iqx0,y-iq2x02∫L20,Texp⁡iv,y-x0-i2qv2dfv.Finally by Theorem 2 again we have(22)TqpGy=exp⁡iqx0,y-iq2x02TqpFy-x0and so by (17) the proof is complete.Cameron and Storvick [14] presented a new translation theorem for the analytic Feynman integral on Wiener space. Moreover Ahn et al. [15] gave a simple proof of an abstract Wiener space version of the translation theorem. Taking p=1 and y=0 in (19) and considering (13) we obtain Cameron and Storvick’s translation theorem as follows. Hence Theorem 4 can be viewed as Cameron and Storvick’s translation theorem for the Fourier-Feynman transform.Corollary 5.
LetF∈S be given by (5) and let x0∈C0[0,T]. Then one has(23)∫C00,TanfqFx-x0dmx=exp⁡iq2x02∫C00,TanfqFxexp⁡iqx0,xdmx.Next theorem is reminiscent of the frequency shifting theorem for the Fourier transform. Using Theorem3 we have the following property for the frequency shifting of the Fourier-Feynman transform.Theorem 6 (frequency shifting).
LetF∈S be given by (5) and let y0∈C0[0,T]. Then one has(24)TqpFy-y0=exp⁡-iqy0,y+iq2y02TqpF·exp⁡iqy0,·yfor s-a.e. y∈C0[0,T].The following theorem is called a scaling theorem because we want the transform not ofF(x), but of F(ax), in which a can be thought of as a scaling factor.Theorem 7 (scaling).
LetF∈S be given by (5) and let a be a nonzero real number. Then one has(25)TqpFa·y=Tq/a2pFayfor s-a.e. y∈C0[0,T].Proof.
LetG(x)=F(ax) for x∈C0[0,T]. Using (5) we can write G(x) as(26)Gx=∫L20,Texp⁡iv,axdfv=∫L20,Texp⁡iw,xdgw,where g(E)=f(E/a) for each Borel subset E of L2[0,T]. By Theorem 2, we have(27)TqpGy=∫L20,Texp⁡iw,y-i2qw2dgw=∫L20,Texp⁡iv,ay-a2i2qv2dfvfor s-a.e. y∈C0[0,T]. Finally by Theorem 2 again we obtain the result.Our next corollary follows immediately from the scaling theorem above by puttinga=-1. This result is called time reversal because we replace x by -x in F(x) to get F(-x). The transform of this new functional is obtained by simply replacing y by -y in the transform of F(x).Corollary 8 (time reversal).
LetF∈S be given by (5). Then one has(28)TqpF-·y=TqpF-yfor s-a.e. y∈C0[0,T].Our next theorem is useful in obtaining the Fourier-Feynman transforms of new functionals from the Fourier-Feynman transforms of old functionals for which we know their Fourier-Feynman transform.Theorem 9 (modulation).
LetF∈S be given by (5) and let x0∈C0[0,T]. Then one has(29)TqpF·cos⁡qx0,·y=12KFx0,y+KF-x0,y,(30)TqpF·sin⁡qx0,·y=12iKFx0,y-KF-x0,y,where(31)KFx0,y=exp⁡iqx0,y-iq2x02TqpF·-x0yfor s-a.e. y∈C0[0,T].Proof.
Putcos(q〈x0,·〉)=1/2(exp{iq〈x0,·〉}+exp{-iq〈x0,·〉}) and use the linearity of the Fourier-Feynman transform Tq(p) to get(32)TqpF·cos⁡qx0,·y=12TqpF·exp⁡iqx0,·y+TqpF·exp⁡-iqx0,·y.Finally by the time shifting theorem or frequency shifting theorem we obtain (29). Using sin(q〈x0,·〉)=1/2i(exp{iq〈x0,·〉}-exp{-iq〈x0,·〉}) the second conclusion is proved similarly.Since the Dirac measure concentrated atv=0 in L2[0,T] is a complex Borel measure, the constant function F≡1 belongs to S. Hence we have the following corollary.Corollary 10.
Letx0∈C0[0,T]. Then one has(33)Tqpcos⁡qx0,·y=cos⁡qx0,yexp⁡-iq2x02,(34)Tqpsin⁡qx0,·y=sin⁡qx0,yexp⁡-iq2x02 for s-a.e. y∈C0[0,T].Proof.
SinceTq(p)[F(·-x0)](y)=Tq(p)[F](y-x0)=1 for F≡1, by the modulation property, Theorem 9, and Euler’s formula, (33) and (34) follow immediately.


## 3. Variational Properties for Fourier-Feynman Transform

In using the Fourier transform to solve differential equations, we need an expression relating the transform off′ to that of f. In this section we develop similar relationships for Fourier-Feynman transform on Wiener space; that is, we provide variational properties for Fourier-Feynman transform of functionals in the Banach algebra S.Definition 11.
LetF be a functional on C0[0,T] and let w1,…,wn∈C0[0,T]. Then(35)δFx,w1=∂∂tFx+tw1t=0(if it exists) is called the first variation of F(x). The higher order variations of F(x) are defined inductively. For example, the 2nd order variation of F(x) is the first variation of δF(x,w1) with respect to x and is defined by(36)δ2Fx,w2;w1=∂∂tδFx+tw2,w1t=0and nth order variation is defined by(37)δnFx,wn;…;w1=∂∂tδn-1Fx+twn,wn-1;…;w1t=0for n=3,4,…. If w=w1=⋯=wn, then we denote δnF(x,w;…;w) as δnF(x,w).Theorem 12.
LetF∈S be given by (5) with ∫L2[0,T]vn·df(v)<∞. Then for s-a.e. x and w1,…,wn in C0[0,T], δnF(x,wn;…;w1) exists, is an element of S as a function of x, and is given by the formula(38)δnFx,wn;…;w1=∫L20,Texp⁡iv,xdfnv,where fn is a complex Borel measure on L2[0,T] defined by(39)fnE=in∫Ev,w1⋯v,wndfvfor each Borel subset E of C0[0,T].Proof.
We will prove that(40)δ2Fx,w2;w1=i2∫L20,Tv,w1v,w2exp⁡iv,xdfvfor s-a.e. x and w1, w2 in C0[0,T]. Using Lemma 3.1 of [16], we have(41)δ2Fx,w2;w1=∂∂tδFx+tw2,w1t=0=∂∂t∫L20,Tiv,w1exp⁡iv,x+itv,w2dfvt=0.Now the result follows if we can pass the differentiation under the integral sign. But this is done because by the Fubini theorem (42)∫C00,T2∫L20,Tv,w1v,w2dfvdm×mw1,w2=2π∫L20,Tv2dfvwhich is finite and so ∫L2[0,T]‍|〈v,w1〉〈v,w2〉|d|f|(v)<∞ for s-a.e. w1, w2 in C0[0,T]. Now by mathematical induction we obtain general result (38).Ifw=w1=⋯=wn in Theorem 12, then we have the following corollary.Corollary 13.
LetF∈S be given by (5) with ∫L2[0,T]vn·d|f|(v)<∞. Then for s-a.e. x and w in C0[0,T], δnF(x,w) exists, is an element of S as a function of x, and is given by the formula(43)δnFx,w=∫L20,Texp⁡iv,xdfnv,where fn is the complex Borel measure on L2[0,T] defined by(44)fnE=in∫Ev,wndfvfor each Borel subset E of C0[0,T].In our next theorem, for functionals inS we establish a relationship between the Fourier-Feynman transform of the variation and the variation of the Fourier-Feynman transform. Also see Corollary 4.3 of [15] for a similar result.Theorem 14.
LetF∈S be given by (5) with ∫L2[0,T]vn·d|f|(v)<∞. Then one has(45)TqpδnF·,wn;…;w1y=δnTqpFy,wn;…;w1for s-a.e. y and w1,…,wn in C0[0,T]. Also, both of the expressions in (45) are given by the expression(46)in∫L20,Tv,w1⋯v,wnexp⁡iv,y-i2qv2dfv for s-a.e. y and w1,…,wn in C0[0,T].Proof.
By (38) and Theorem 2, we have (47)TqpδnF·,wn;…;w1y=∫L20,Texp⁡iv,y-i2qv2dfnvfor s-a.e. y and w1,…,wn in C0[0,T]. Now by (39) we know that the last expression can be rewritten as (46). On the other hand, by the same method as in the proof of Theorem 12, we see that the right hand side of (45) is also expressed as (46).Lettingw=w1=⋯=wn in Theorem 14 we have the following corollary.Corollary 15.
LetF∈S be given by (5) with ∫L2[0,T]vn·d|f|(v)<∞. Then one has(48)TqpδnF·,wy=δnTqpFy,w=in∫L20,Tv,wnexp⁡iv,y-i2qv2dfvfor s-a.e. y and w in C0[0,T].The following theorem involves an iterated Fourier-Feynman transform ofnth order variation.Theorem 16.
LetF∈S be given by (5) with ∫L2[0,T]vn·d|f|(v)<∞. Then one has(49)Tqp⋯TqpδnFx,·;…;·z1⋯zn=δnFx,zn;…;z1for s-a.e. x and z1,…,zn in C0[0,T].Proof.
LetG(x,w)=∫L20,T‍v,wexpiv,xdf(v). For λ>0, we obtain(50)TλGx,·z=∫C00,TGx,λ-1/2w+zdmw=∫C00,T∫L20,Tλ-1/2v,w+v,zexp⁡iv,xdfvdmwfor s-a.e. x and z in C0[0,T]. Since ∫C0[0,T]‍〈v,w〉dm(w)=0, the Fubini theorem enables us to conclude that (51)TλGx,·z=∫L20,Tv,zexp⁡iv,xdfv.Since the right hand side of the last expression is independent of λ, we have (52)TqpGx,·z=∫L20,Tv,zexp⁡iv,xdfvfor s-a.e. x and z in C0[0,T].
Now considering Theorem12 and applying repeatedly the first part of this proof, we obtain(53)Tqp⋯TqpδnFx,·;…;·z1⋯zn=in∫L20,Tv,z1⋯v,znexp⁡iv,xdfvfor s-a.e. x and z1,…,zn in C0[0,T]. Finally by Theorem 12 again the proof is complete.


## 4. Shifting Properties for Convolution Product

We developed in Section2 some properties relevant to shifting and computational rules for the Fourier-Feynman transform of functionals in the Banach algebra S. In this section we study similar properties for the convolution product of functionals in S. Let us begin with the definition of the convolution product of functionals on Wiener space.Definition 17.
LetF and G be functionals on C0[0,T]. For λ∈C+ and y∈C0[0,T], one defines the convolution product (if it exists) by(54)F∗Gλy=∫C00,TanwλFy+x2Gy-x2dmx,(55)F∗Gqy=∫C00,TanfqFy+x2Gy-x2dmx.Obviously the convolution is bilinear in the sense that(56)F1+F2∗G1+G2qy=F1∗G1qy+F1∗G2qy+F2∗G1qy+F2∗G2qy for all functionals Fi, Gi on C0[0,T] for i=1,2, whenever each convolution exists.Huffman et al. established the existence of the convolution product onC0[0,T] for functionals in S.Theorem 18 (Theorem 3.2 of [5]).
LetF and G be elements of S with corresponding finite Borel measures f and g, respectively. Then their convolution product (F∗G)q exists and is given by the formula(57)F∗Gqy=∫L220,Texp⁡i2v+w,y-i4qv-w2dfvdgwfor s-a.e. y∈C0[0,T].Our first result in this section is a relationship between time shifting and frequency shifting of the convolution product on Wiener space.Theorem 19.
LetF and G be functionals on C0[0,T] and let x0∈C0[0,T]. Then one has(58)F·-x0∗G·-x0qy=F∗Gqy-2x0if each side exists.Proof.
For allλ>0 and for s-a.e. y∈C0[0,T], we have(59)F·-x0∗G·-x0λy=∫C00,TFy+λ-1/2x2-x0Gy-λ-1/2x2-x0dmx=F∗Gλy-2x0if the Wiener integrals exist. Hence we have the result.The following theorem is reminiscent of the time shifting theorem for Fourier-Feynman transform (Theorem4) in Section 2. But in this theorem we have to shift back x0 for F and shift front x0 for G to obtain a concrete form of a time shifting formula for the convolution product.Theorem 20 (time shifting).
LetF and G be given as in Theorem 18 and let x0∈C0[0,T]. Then one has(60)F·-x0∗G·+x0qy=exp⁡iqx02F·exp⁡iqx0,·∗G·exp⁡-iqx0,·qyfor s-a.e. y∈C0[0,T].Proof.
LetF0(x)=F(x-x0) and G0(x)=G(x+x0). By (5) we have (61)F0x=∫L20,Texp⁡iv,xdf0v,(62)G0x=∫L20,Texp⁡iw,xdg0w,where f0(E)=∫E‍exp{-i〈v,x0〉}df(v) and g0(E)=∫E‍exp{i〈w,x0〉}dg(w) for a Borel subset E of L2[0,T]. Then, by Theorem 18, the left hand side of (60) is given by (63)F0∗G0qy=∫L220,Texp⁡i2v+w,y-i4qv-w2df0vdg0w=∫L220,Texp⁡i2v+w,y-iv-w,x0-i4qv-w2dfvdgwfor s-a.e. y∈C0[0,T].
To consider the right hand side of (60), let(64)F1x=Fxexp⁡iqx0,x=∫L20,Texp⁡iv1,xdf1v1,(65)G1x=Gxexp⁡-iqx0,x=∫L20,Texp⁡iw1,xdg1w1,where f1(E)=f(E-qx0) and g1(E)=g(E+qx0) for a Borel subset E of L2[0,T]. Then, by Theorem 18, we have(66)F1∗G1qy=∫L220,Texp⁡i2v1+w1,y-i4qv1-w12df1v1dg1w1=∫L220,Texp⁡i2v+w,y-i4qv-w+2qx02dfvdgwfor s-a.e. y∈C0[0,T]. Since v-w+2qx02=v-w2+4q〈v-w,x0〉+4q2x02, we have(67)F1∗G1qy=exp⁡-iqx02∫L220,Texp⁡i2v+w,y-iv-w,x0-i4qv-w2dfvdgwfor s-a.e. y∈C0[0,T] and this completes the proof.Considering the second part of the proof of Theorem20 above, we see that, for F and G given as in Theorem 18,(68)F·exp⁡iqx0,·∗G·exp⁡iqx0,·qy=∫L220,Texp⁡i2v+w+2qx0,y-i4qv-w2dfvdgw=exp⁡i2qx0,yF∗Gqyfor s-a.e. y∈C0[0,T].The following is a scaling theorem for the convolution product.Theorem 21 (scaling).
LetF and G be given as in Theorem 18. If a is a nonzero real number, then(69)Fa·∗Ga·qy=F∗Gq/a2ayfor s-a.e. y∈C0[0,T].Proof.
By the same method as used in the proof of Theorem7 we have (70)Fa·∗Ga·qy=∫L220,Texp⁡i2v+w,ay-a2i4qv-w2dfvdgw for s-a.e. y∈C0[0,T]. Hence by Theorem 18 we obtain the result.Our next corollary follows immediately from the scaling theorem above by puttinga=-1. This result is called time reversal because we replace x by -x in F(x) and G(x) to get F(-x) and G(-x), respectively. The convolution of these new functionals is obtained by simply replacing y by -y in the convolution of F(x) and G(x).Corollary 22 (time reversal).
LetF and G be given as in Theorem 18. Then one has(71)F-·∗G-·qy=F∗Gq-yfor s-a.e. y∈C0[0,T].Our next theorem is useful to obtain the convolution product of new functionals from the convolution product of old functionals when we know their convolution product.Theorem 23 (modulation).
LetF and G be given as in Theorem 18 and let x0∈C0[0,T]. Then(72)F·cos⁡qx0,·∗G·cos⁡qx0,·qy=14KF,G1,0,0;x0,y+KF,G-1,0,0;x0,y+KF,G0,1,1;x0,y+KF,G0,1,-1;x0,y,(73)F·cos⁡qx0,·∗G·sin⁡qx0,·qy=14iKF,G1,0,0;x0,y-KF,G-1,0,0;x0,y-KF,G0,1,1;x0,y+KF,G0,1,-1;x0,y,(74)F·sin⁡qx0,·∗G·cos⁡qx0,·qy=14iKF,G1,0,0;x0,y-KF,G-1,0,0;x0,y+KF,G0,1,1;x0,y-KF,G0,1,-1;x0,y,(75)F·sin⁡qx0,·∗G·sin⁡qx0,·qy=-14KF,G1,0,0;x0,y+KF,G-1,0,0;x0,y-KF,G0,1,1;x0,y-KF,G0,1,-1;x0,y,where(76)KF,Gα,β,γ;x0,y=exp⁡i2qαx0,y-iqβx02F·-γx0∗G·+γx0qy for s-a.e. y∈C0[0,T].Proof.
Putcos(q〈x0,·〉)=1/2(exp{iq〈x0,·〉}+exp{-iq〈x0,·〉}) and use bilinearity (56) of the convolution product to get (77)F·cos⁡qx0,·∗G·cos⁡qx0,·qy=14F·exp⁡iqx0,·∗G·exp⁡iqx0,·qy+F·exp⁡-iqx0,·∗G·exp⁡-iqx0,·qy+F·exp⁡iqx0,·∗G·exp⁡-iqx0,·qy+F·exp⁡-iqx0,·∗G·exp⁡iqx0,·qy.Finally by (60) and (68) we obtain (72). Using sin(q〈x0,·〉)=1/2i(exp{iq〈x0,·〉}-exp{-iq〈x0,·〉}) the other conclusions are proved similarly.Since the Dirac measure concentrated atv=0 in L2[0,T] is a complex Borel measure, the constant function F≡1 belongs to S. Hence we have the following corollary.Corollary 24.
Letx0∈C0[0,T]. Then one has(78)cos⁡qx0,·∗cos⁡qx0,·qy=12cos⁡2qx0,y+exp⁡-iqx02,(79)cos⁡qx0,·∗sin⁡qx0,·qy=sin⁡qx0,·∗cos⁡qx0,·qy=12sin⁡2qx0,y,(80)sin⁡qx0,·∗sin⁡qx0,·qy=-12cos⁡2qx0,y-exp⁡-iqx02for s-a.e. y∈C0[0,T].Proof.
Since[F(·-γx0)∗G(·+γx0)]q(y)≡1 for F≡G≡1, by the modulation property, Theorem 23, and Euler’s formula, the results follow immediately.



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*Source: 580653-2015-09-01.xml*