The main goal of this paper is to study properties of the fractional Fourier transform on Schwartz type space $mathscr{S}_{theta}$. Symbol class $S_{rho,sigma}^{m,theta}$ is introduced. The fractional pseudo-differential operators (f.p.d.o.) associated with the symbol $a(x,xi)$ is a continuous linear mapping of $mathscr{S}_{theta}$ into itself. Kernel and integral representations of f.p.d.o are obtained. Boundedness property of f.p.d.o. is studied. Application of the fractional Fourier transform in solving generalized Fredholm integral equation is also given

Akhilesh Prasad

Praveen Kumar

English

The main goal of this paper is to study properties of the fractional Fourier transform on Schwartz type space $\mathscr{S}_{\theta}$. Symbol class $S_{\rho,\sigma}^{m,\theta}$ is introduced. The fractional pseudo-differential operators (f.p.d.o.) associated with the symbol $a(x,\xi)$ is a continuous linear mapping of $\mathscr{S}_{\theta}$ into itself. Kernel and integral representations of f.p.d.o are obtained. Boundedness property of f.p.d.o. is studied. Application of the fractional Fourier transform in solving generalized Fredholm integral equation is also given

Prasad, Akhilesh

Kumar, Praveen

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MATHEMATICAL COMMUNICATIONS 115Math. Commun. 21(2016), 115–126Pseudo-differential operator associated with the fractionalFourier transformAkhilesh Prasad1,∗and Praveen Kumar11 Department of Applied Mathematics, Indian School of Mines, Dhanbad-826 004, IndiaReceived May 31, 2015; accepted December 15, 2015Abstract. The main goal of this paper is to study properties of the fractional Fouriertransform on Schwartz type space Sθ. Symbol class Sm,θρ,σ is introduced. The fractionalpseudo-differential operators (f.p.d.o.) associated with the symbol a(x, ξ) are a continuouslinear mapping of S into Sθ. Kernel and integral representations of f.p.d.o are obtained.The boundedness property of f.p.d.o. is studied. Application of the fractional Fouriertransform in solving a generalized Fredholm integral equation is also given.AMS subject classifications: 46F12, 35S05, 42A85, 46A16Key words: Sobolev space, pseudo-differential operator, Schwartz space, fractional Fouriertransform, generalized Fredholm integral equation1. IntroductionThe motive of a generalization of the Fourier transform (FT) into the fractionalFourier transform (FrFT) in the mathematical literature was formally first intro-duced by Victor Namias in 1980 [4]. As a generalization of the FT, the FrFT is alinear operator in the time-frequency plane. It depends on an angle θ and can beinterpreted as a rotation by θ in the time-frequency plane. It has many applicationsin solving differential equations, quantum mechanics, signal processing, image pro-cessing and other fields. The FrFT [1–6,12] with an angle θ of φ(x) ∈ L2(R) denotedby (Fθφ)(ω) = φ̂θ(ω) is given by(Fθφ)(ω) = φ̂θ(ω) =∫RKθ(x, ω)φ(x)dx, (1)where the kernel isKθ(x, ω) =Cθ ei(x2+ω2) cot θ2−ixω csc θ, θ 6= nπ,1√2πe−ixω, θ = π2 ,δ(x− ω), θ = 2nπ,δ(x+ ω), θ = (2n− 1)π, n ∈ Z,and Cθ = (2πi sin θ)−1/2eiθ/2 =√1−i cot θ2π .∗Corresponding author. Email addresses: apr bhu@yahoo.com (A.Prasad),praveen502mzn@gmail.com (P.Kumar)http://www.mathos.hr/mc ©2016 Department of Mathematics, University of Osijek116 A.Prasad and P.KumarIf φ̂θ(ω) ∈ L2(R), the corresponding inversion formula is given byφ(x) =∫RKθ(x, ω) φ̂θ(ω)dω, (2)whereKθ(x, ω) = Cθe−i(x2+ω2) cot θ2+ixω csc θ = K−θ(x, ω),Cθ =(2πi sin θ)1/2e−iθ/22π sin θ=√1 + i cot θ2π= C−θ.From this, we conclude that the inverse of a FrFT with an angle θ is the FrFT withthe angle −θ.Definition 1. The Schwartz space S is a set of rapidly decreasing complex-valuedinfinitely differentiable functions φ on R such that for every choice of α and β ofnon-negative integers it satisfiesγα,β(φ) = supx∈R∣∣xα Dβφ(x)∣∣ <∞. (3)If f is of polynomial growth and a locally integrable function on R, then f generatesa distribution in S ′ as follows:〈f, φ〉 =∫Rf(x)φ(x)dx, φ ∈ S .The space S (R) is equipped with the topology generated by the collection of semi-norms {γα,β}; it is a Fréchet space. The dual of S is denoted by S′; its elementsare called tempered distributions.Definition 2. The Schwartz type space Sθ is defined as follows: φ is a member ofSθ iff it is a complex valued C∞- function on R and for every choice of α and β ofnon-negative integers it satisfiesΓθα,β(φ) = supx∈R∣∣xα (∆∗x)βφ(x)∣∣ <∞, (4)where ∆∗x = −(ddx + ix cot θ).Definition 3. The continuous fractional convolution of two continuous functionsφ(x), ψ(x) ∈ L2(R) is defined as(φ ⋆θ ψ)(z) =∫ ∞−∞φ(x)ψ(z − x)e−i2(z2−x2) cot θdx, (5)where ⋆θ denotes the fractional convolution operator.Lemma 1. Let φ(x), ψ(x) ∈ L2(R) be two continuous functions; then the fractionalFourier transform of their convolution operator is given byFθ(φ ⋆θ ψ)(ω) =1Cθe−i2ω2 cot θφ̂θ(ω)Fθ(e−i2(.)2 cot θψ)(ω). (6)P.d.o. associated with the fractional Fourier transform 117Proof. Using equations (1) and (5), we haveFθ(φ ⋆θ ψ)(ω) =∫ ∞−∞Kθ(z, ω)(φ ⋆θ ψ)(z)dz=∫ ∞−∞∫ ∞−∞Kθ(z, ω)φ(x)ψ(z − x)e−i2(z2−x2) cot θdx dz=∫ ∞−∞∫ ∞−∞Kθ(x+ y, ω)φ(x)ψ(y)e−i2[(x+y)2−x2) cot θdx dy=1Cθe−i2ω2 cot θ∫ ∞−∞Kθ(x, ω)φ(x)dx∫ ∞−∞Kθ(y, ω)ψ(y)e−i2y2 cot θdy=1Cθe−i2ω2 cot θφ̂θ(ω) Fθ(e−i2(.)2 cot θψ)(ω).This completes the proof of the lemma.Lemma 2. Let φ, ψ ∈ Sθ ⊂ S and Kθ(x, ω) be the kernel of the fractional Fouriertransform. Then(i)∫ ∞−∞(∆∗x)rφ(x)ψ(x)dx =∫ ∞−∞φ(x)(∆x)rψ(x)dx,(ii)(∆∗x)rKθ(x, ω) = (−iω csc θ)rKθ(x, ω),(iii)(∆∗x)rKθ(x, ω) = (iω csc θ)rKθ(x, ω),(iv)(Fθ(∆x)rφ(x))(ω) = (iω csc θ)rφ̂θ(ω), ∀r ∈ N0,where ∆∗x is defined as above and ∆x =(ddx − ix cot θ).Proof. The proof is similar to [7, pp. 357-358].Lemma 3 (Peetre). For any real number s and ∀, ξ, η ∈ R, the upper boundary(1 + |ξ|21 + |η|2)s≤ 2|s|(1 + |ξ − η|2)|s|is satisfied.Proof. See [11, p. 97].Lemma 4. If f(x), φ(x) ∈ Sθ(R). Then(∆∗x)k[f(x) φ(x)] =k∑r=0Ak,r (∆∗x)rφ(x) Dk−rx f(x), k ∈ N0,where ∆∗x is the same as defined above and Ak,r are constants.Proof. See [8].Theorem 1. The fractional Fourier transform φ̂θ is a continuous linear map of Sinto Sθ.Proof. See a similar proof of Proposition 2.2 (iii) of [6, p. 242].118 A.Prasad and P.Kumar2. Fractional pseudo-differential operatorsIn this section, we discuss a special class Sm,θρ,σ , which is a generalization of Hörmanderclass Smρ,σ [10]. For θ = π/2, the class Sm,θρ,σ reduces to the Hörmander class.Definition 4. Let a(x, ξ) be a complex valued smooth function on C∞(R×R) → C.It belongs to symbol class Sm,θρ,σ ,m ∈ R, 0 ≤ σ, ρ ≤ 1 iff ∀q, α, β ∈ N0, there existAα,β,m,q,θ > 0 such that(1 + |x csc θ|)q∣∣Dβx(∆∗ξ)αa(x, ξ)∣∣ ≤ Aα,β,m,q,θ(1 + |ξ csc θ|)m−ρ|α|+σ|β|, (7)where ∆∗ξ is defined as above.Definition 5. Let the symbol a(x, ξ) ∈ Sm,θρ,σ . Then a fractional pseudo-differentialoperator T θa associated with a(x, ξ) is defined by(T θaφ)(x) =∫ ∞−∞Kθ(x, ξ)a(x, ξ)φ̂θ(ξ)dξ, φ ∈ S θ. (8)Theorem 2. Let the symbol a(x, ξ) ∈ Sm,θρ,σ . Then f.p.d.o. Tθa is a continuous linearmapping from S into Sθ.Proof. Let φ ∈ Sθ and k, q ∈ N0. Then using definitions (7)-(8), Lemma 4, Theo-rem 1 and applying the technique of [6, p. 247] and [11, p. 66], we have∣∣∣xq(∆∗x)β(T θaφ)(x)∣∣∣=∣∣∣xq(∆∗x)β∫ ∞−∞Kθ(x, ξ)a(x, ξ)φ̂θ(ξ) dξ∣∣∣=∣∣∣xq∫ ∞−∞(∆∗x)β [Kθ(x, ξ)a(x, ξ)]φ̂θ(ξ) dξ∣∣∣=∣∣∣xq∫ ∞−∞β∑α=0Bβ,α(∆∗x)αKθ(x, ξ)Dβ−αx a(x, ξ)φ̂θ(ξ) dξ∣∣∣=∣∣∣xq∫ ∞−∞β∑α=0Bβ,α(−iξ csc θ)αKθ(x, ξ)Dβ−αx a(x, ξ)φ̂θ(ξ) dξ∣∣∣≤β∑α=0Bβ,α|Cθ|∫ ∞−∞(1 + |x csc θ|)q |Dβ−αx a(x, ξ)| |ξ csc θ|α|φ̂θ(ξ)| dξ≤β∑α=0Bβ,α|Cθ| supξ∈R|φ̂θ(ξ)|∫ ∞−∞Aβ−α,m,q,θ(1 + |ξ csc θ|)m+σβ+α(1−σ) dξ≤ A′β,m,q,θ∫ ∞−∞(1 + |ξ csc θ|)m+β dξ.The integral is convergent for m+ β + 1 < 0. HenceΓθq,β(T θaφ)<∞.This completes the proof of the theorem.P.d.o. associated with the fractional Fourier transform 1193. Kernel and integral representationsIn this section, we can write fractional pseudo-differential operators (T θaφ) in differentways: For all φ ∈ Sθ, we have(T θaφ)(x) =∫ ∞−∞Kθ(x, ξ)a(x, ξ)φ̂θ(ξ)dξ=∫ ∞−∞Kθ(x, ξ)a(x, ξ)(∫ ∞−∞Kθ(y, ξ)φ(y)dy)dξ=∫ ∞−∞∫ ∞−∞e−i2(x2−y2) cot θ+i(x−y)ξ csc θa(x, ξ)φ(y) dy dξ.Let x− y = z; then(T θaφ)(x) =∫ ∞−∞∫ ∞−∞ei2(z2−2xz) cot θ+izξ csc θa(x, ξ)φ(x − z) dz dξ=∫ ∞−∞kθ(x, z)φ(x − z) dz,wherekθ(x, z) =∫ ∞−∞ei2(z2−2xz) cot θ+izξ csc θa(x, ξ)dξ.Thus(T θaφ)(x) =∫ ∞−∞Kθ(x, y)φ(y) dy,with the kernel Kθ(x, y) = kθ(x, x− y).The function aξ,θ(η) associated with the symbol a(x, ξ) and defined byaξ,θ(η) = Fθ[Kθ(x, ξ)a(x, ξ)](η), (9)will play a fundamental role in our investigation. An upper boundary for aξ,θ(η) isgiven by the following lemma.Lemma 5. Let the symbol a(x, ξ) ∈ Sm,θρ,σ . Then the function aξ,θ(η) defined by (9)satisfies the inequality∣∣aξ,θ(η)∣∣ ≤ Bt,m,q,θ(1 + |η csc θ|)−t(1 + |ξ csc θ|)m+t, ∀ t > 1 ∈ N. (10)120 A.Prasad and P.KumarProof. For r ∈ N0, using Lemma 2 (ii) and definition (7), we haveaξ,θ(η) =∫ ∞−∞Kθ(x, η)(1−∆∗x)t(1 + |η csc θ|)t[Kθ(x, ξ)a(x, ξ)]dx=∫ ∞−∞Kθ(x, η)t∑r=0(−1)r(tr)(∆∗x)r(1 + |η csc θ|)t[Kθ(x, ξ)a(x, ξ)]dx=t∑r=0(−1)r(tr)1(1 + |η csc θ|)t∫ ∞−∞Kθ(x, η)r∑s=0Br,s(∆∗x)sKθ(x, ξ)×Dr−sx a(x, ξ)dx=t∑r=0(−1)r(tr)1(1 + |η csc θ|)t∫ ∞−∞Kθ(x, η)r∑s=0Br,s(−iξ csc θ)sKθ(x, ξ)×Dr−sx a(x, ξ)dx.Therefore,|aξ,θ(η)| ≤t∑r=0r∑s=0(tr)Br,s(1 + |η csc θ|)−t∫ ∞−∞|ξ csc θ|s|Dr−sx a(x, ξ)|dx≤t∑r=0r∑s=0(tr)Br,s(1 + |η csc θ|)−t∫ ∞−∞Ar−s,m,q,θ(1 + |x csc θ|)−q×(1 + |ξ csc θ|)m+σ(r−s)+sdx≤ Bt,m,q,θ(1 + |η csc θ|)−t(1 + |ξ csc θ|)m+t∫ ∞−∞(1 + |x csc θ|)−qdx.The integral is convergent for q > 1. Hence we have∣∣aξ,θ(η)∣∣ ≤ Bt,m,q,θ(1 + |η csc θ|)−t(1 + |ξ csc θ|)m+t.This completes the proof of the lemma.Theorem 3. For any symbol a(x, ξ) ∈ Sm,θρ,σ , the associated operator Tθa can berepresented by(T θaφ)(x) =∫ ∞−∞∫ ∞−∞Kθ(x, η)aξ,θ(η)φ̂θ(ξ)dη dξ, φ ∈ Sθ,where all involved integrals are convergent.Proof. Sinceaξ,θ(η) =∫ ∞−∞Kθ(x, η)[Kθ(x, ξ)a(x, ξ)]dx,by the inversion formula, we haveKθ(x, ξ)a(x, ξ) =∫ ∞−∞Kθ(x, η)aξ,θ(η)dη.P.d.o. associated with the fractional Fourier transform 121Therefore,(T θaφ)(x) =∫ ∞−∞Kθ(x, ξ)a(x, ξ)φ̂θ(ξ)dξ=∫ ∞−∞∫ ∞−∞Kθ(x, η)aξ,θ(η)φ̂θ(ξ)dη dξ.Since φ̂θ(ξ) ∈ S , we have|φ̂θ(ξ)| ≤ Eθ (1 + |ξ csc θ|)−q , ∀ q > 0.Now using the above upper boundary and Lemma 5, we have|(T θaφ)(x)|≤ |Cθ|∫ ∞−∞∫ ∞−∞|aξ,θ(η)| |φ̂θ(ξ)|dη dξ≤∫ ∞−∞∫ ∞−∞Bt,m,q,θ(1 + |η csc θ|)−t(1 + |ξ csc θ|)m+tEθ (1 + |ξ csc θ|)−qdη dξ≤ B′t,m,q,θ∫ ∞−∞(1 + |ξ csc θ|)m+t−qdξ∫ ∞−∞(1 + |η csc θ|)−tdη,for choosing q sufficiently large such that q > m+ t+ 1 and t > 1, we can make theintegral convergent.4. Boundedness property of fractional pseudo-differential op-eratorsDefinition 6 (see Prasad et al. [9]). A tempered distribution φ ∈ S′θ (R) is saidto belong to the generalized Sobolev spaces Hs,θ(R), s ∈ R, if its fractional Fouriertransform φ̂θ satisfies||φ||Hs,θ(R) =(∫ ∞−∞∣∣(1 + |ξ csc θ|2)s/2φ̂θ(ξ)∣∣2dξ)1/2<∞. (11)Lemma 6. Let the symbol a(x, ξ) ∈ Sm,θρ,σ . Then we have(i) âθ(ξ, η) =1(1 + iξ csc θ)t∫ ∞−∞Kθ(x, ξ)(1 + ∆̄x)ta(x, η)dx,(ii) |âθ(ξ, η)| ≤ Ct,m,q,θ(1 + |η csc θ|)m+tσ(1 + ξ2 csc2 θ)−t/2.Proof. (i) Using Lemma 2 (i) and (iii) and applying the technique of [6, p. 247]122 A.Prasad and P.Kumarand [11, p. 66], we have1(1 + iξ csc θ)t∫ ∞−∞Kθ(x, ξ)(1 + ∆̄x)ta(x, η)dx=1(1 + iξ csc θ)t∫ ∞−∞Kθ(x, ξ)t∑r=0(tr)(∆̄x)ra(x, η)dx=t∑r=0(tr)1(1 + iξ csc θ)t∫ ∞−∞(∆∗x)rKθ(x, ξ) a(x, η)dx=t∑r=0(tr)1(1 + iξ csc θ)t∫ ∞−∞(iξ csc θ)rKθ(x, ξ) a(x, η)dx=1(1 + iξ csc θ)t∫ ∞−∞Kθ(x, ξ)t∑r=0(tr)(iξ csc θ)r a(x, η)dx=1(1 + iξ csc θ)t∫ ∞−∞Kθ(x, ξ)(1 + iξ csc θ)t a(x, η)dx= âθ(ξ, η).(ii) Using definition (7) and the technique of [7, p. 359], we have|âθ(ξ, η)| =∣∣∣1(1 + iξ csc θ)t∫ ∞−∞Kθ(x, ξ)t∑r=0(tr)(∆̄x)ra(x, η)dx∣∣∣=∣∣∣1(1 + iξ csc θ)t∫ ∞−∞Kθ(x, ξ)t∑r=0(tr)(−1)r(∆∗x)ra(x, η)dx∣∣∣≤t∑r=0(tr)|Cθ|(1 + ξ2 csc2 θ)−t/2∫ ∞−∞|(∆∗x)ra(x, η)|dx≤t∑r=0(tr)|Cθ|(1 + ξ2 csc2 θ)−t/2∫ ∞−∞r∑s=0s∑l=0|dl| |x|l|Dsxa(x, η)|dx≤t∑r=0(tr) r∑s=0s∑l=0|dl| |Cθ|(1 + ξ2 csc2 θ)−t/2×As,m,q,θ(1 + |η csc θ|)m+sσ∫ ∞−∞(1 + |x csc θ|)−q+tdx≤ A′t,m,q,l,θ(1 + ξ2 csc2 θ)−t/2(1 + |η csc θ|)m+tσ×∫ ∞−∞(1 + |x csc θ|)−q+tdx.The integral is convergent for q > t+1. Hence there exists a constant Ct,m,q,l,θ > 0such that|âθ(ξ, η)| ≤ Ct,m,q,l,θ(1 + ξ2 csc2 θ)−t/2(1 + |η csc θ|)m+tσ .This completes the proof of the lemma.P.d.o. associated with the fractional Fourier transform 123Theorem 4. Let the symbol a(x, ξ) ∈ Sm,θρ,σ . Then the fractional Fourier transformof T θaφ can be represented as:Fθ[ei2(.)2 cot θ(T θaφ)](ξ) = Cθ gθ(ξ), φ ∈ Sθ(R),wheregθ(ξ) =∫ ∞−∞φ̂θ(η)eiη(ξ−η) cot θâθ(ξ − η, η)dη.Proof. Using definitions (5) and (8), we have(T θaφ)(x)=∫ ∞−∞Kθ(x, η)a(x, η)φ̂θ(η)dη= Cθe−i2x2 cot θ∫ ∞−∞e−i2η2 cot θ+ixη csc θ(∫ ∞−∞Kθ(x, ζ)âθ(ζ, η)dζ)φ̂θ(η)dη= Cθe−i2x2 cot θ∫ ∞−∞Kθ(x, ξ − η)ei2(η2−2ξη) cot θ+ixη csc θgθ(ξ)dξ= Cθe−i2x2 cot θ∫ ∞−∞Kθ(x, ξ) gθ(ξ)dξ= Cθe−i2x2 cot θF−θ[gθ](x).ThereforeFθ[ei2(.)2 cot θ(T θaφ)](ξ) = Cθ gθ(ξ).This completes the proof of the theorem.Theorem 5. Let the symbol a(x, ξ) ∈ Sm,θρ,σ and Tθaφ be associated with f.p.d.o. Thenthere exists Mθm,q,s,t,σ > 0 s.t.||T θaφ||Hs,θ(R) ≤Mθm,q,s,t,σ||φ||Hm+s+tσ,θ(R), φ ∈ Sθ(R),∀s ∈ R and m, q, t, σ as above.Proof. From Theorem 4, we have(1 + |ξ csc θ|2)s/2Fθ[ei2(.)2 cot θ(T θaφ)](ξ)= Cθ(1 + |ξ csc θ|2)s/2gθ(ξ)= Cθ(1 + |ξ csc θ|2)s/2∫ ∞−∞φ̂θ(η)eiη(ξ−η) cot θâθ(ξ − η, η)dη= Cθ∫ ∞−∞(1 + |ξ csc θ|2)s/2(1 + |η csc θ|2)s/2(1 + |η csc θ|2)s/2φ̂θ(η)eiη(ξ−η) cot θâθ(ξ − η, η)dη.Using Lemma 3, we have|(1 + |ξ csc θ|2)s/2Fθ[ei2(.)2 cot θ(T θaφ)](ξ)|≤ |Cθ|2|s|/2∫ ∞−∞(1 + |(ξ − η) csc θ|2)s/2(1 + |η csc θ|2)s/2|φ̂θ(η)| |âθ(ξ − η, η)|dη.124 A.Prasad and P.KumarWe note that(1 + |ξ csc θ|)m+tσ ≤ Lm,t,σ,θ(1 + |ξ csc θ|2)(m+tσ)/2, ξ ∈ R, (12)where Lm,t,σ,θ = max[1, 2m+tσ/2].Moreover, from (12) and Lemma 6 (ii), we get|âθ(ξ− η, η)| ≤ Lm,t,σ,θCt,m,q,θ(1 + |η csc θ|2)(m+tσ)/2(1 + |(ξ− η) csc θ|2)−t/2. (13)Therefore|(1 + |ξ csc θ|2)s/2Fθ[ei2(.)2 cot θ(T θaφ)](ξ)|≤ Lm,t,σ,θCt,m,q,θ|Cθ|2|s|/2∫ ∞−∞(1 + |η csc θ|2)(m+s+tσ)/2|φ̂θ(η)|×(1 + |(ξ − η) csc θ|2)(s−t)/2dη=∫ ∞−∞fθ(η)gθ(ξ − η)dη (say)= (fθ ⋆ gθ)(ξ).Now if t > s + 2, gθ = (1 + |(ξ − η) csc θ|2)(s−t)/2 ∈ L1(R). Also, since φ̂θ ∈ S ,fθ = (1 + |η csc θ|2)(m+s+tσ)/2|φ̂θ(η)| ∈ L2(R). Then (fθ ⋆ gθ)(ξ), ∀ξ ∈ R exists andbelongs to L2(R). Hence||(1 + |ξ csc θ|2)s/2Fθ[ei2(.)2 cot θ(T θaφ)](ξ)||L2(R) ≤Mθm,q,s,t,σ||(fθ ⋆ gθ)||L2(R).Therefore||ei2(.)2 cot θ(T θaφ)(ξ)||Hs,θ (R) ≤Mθm,q,s,t,σ||fθ||L2(R)||gθ||L1(R).This implies that||(T θaφ)(ξ)||Hs,θ (R) ≤ M′θm,q,s,t,σ||φ||Hm+s+tσ,θ(R).This completes the proof of the theorem.5. Application of the fractional Fourier transform to a gener-alized Fredholm integral equationFredholm integral equations of the second kind occur in a natural way while solvinga large class of boundary value problems of mathematical physics. It is very rarethat exact solutions of such integral equations can be determined completely. Beforedescribing the generalization, we need to note that it has recently been recognizedthat Fredholm’s solution can afford a tool for obtaining analytic solutions.In this section, we discuss the method of the fractional Fourier transform that canbe used to solve a generalized Fredholm integral equation of the second kind infractional convolution form.P.d.o. associated with the fractional Fourier transform 125We consider the generalized Fredholm integral equation of the second kind with thefractional convolution kernel in the form∫ ∞−∞Kθ(x, t)f(t)dt + λf(x) = u(x), x ∈ R, (14)where kernel Kθ(x, t) = g(x− t)e− i2(x2−t2) cot θ is in L2(R2), while a known functionu(x) and an unknown function f(x) are in L2(R) and λ is a known parameter.Equation (14) can be re-written in the convolution form as(f ⋆θ g)(x) + λf(x) = u(x). (15)Applying the FrFT to both sides of equation (15) and using (6), we haveFθ(f ⋆θ g)(ω) + λf̂θ(ω) = ûθ(ω)1Cθe−i2ω2 cot θf̂θ(ω)Fθ(e−i2(.)2 cot θg)(ω) + λf̂θ(ω) = ûθ(ω).Thusf̂θ(ω) =ûθ(ω)1Cθ e− i2ω2 cot θFθ(e−i2(.)2 cot θg)(ω) + λ.The inverse fractional Fourier transform leads to a formal solutionf(x) =∫ ∞−∞Kθ(x, ω)ûθ(ω)dω1Cθe−i2ω2 cot θFθ(e−i2(.)2 cot θg)(ω) + λ. (16)In particular, if λ = 1 and g(x) = 12(x|x|), so thatFθ(e−i2(.)2 cot θg)(ω) =Cθiω csc θei2ω2 cot θ,then equation (16) reduces to the formf(x) =∫ ∞−∞Kθ(x, ω) iω csc θ ûθ(ω)dω1 + iω csc θ. (17)NowFθ(e−i2(.)2 cot θe−(.))(ω) =11 + iω csc θCθei2ω2 cot θ. (18)Using Lemma 2 (iv) and (18),f(x) =∫ ∞−∞Kθ(x, ω)1Cθe−i2ω2 cot θFθ(e−i2(.)2 cot θe−(.))(ω) Fθ[∆̄u](ω)dω=∫ ∞−∞Kθ(x, ω)Fθ[∆̄u ⋆θ e− i2(.)2 cot θe−(.)](ω)dω=[∆̄u ⋆θ e− i2(.)2 cot θe−(.)](x)=∫ ∞−∞∆̄u(y) e−[(x−y)+i2(x−y)2 cot θ] e−i2(x2−y2) cot θdy=∫ ∞−∞∆̄u(y) exp[(y − x)(1 + ix cot θ)]dy.126 A.Prasad and P.KumarAcknowledgementThis work was supported by NBHM, Govt. of India, under grant No. 2/48(14)/2011/-R&D II/3501.References[1] L.B.Almeida, The fractional Fourier transform and time-frequency representations,IEEE Trans. Signal Process. 42(1994), 3084–3091.[2] B.N.Bhosale, M. S.Chaudhary, Fractional Fourier transform of distribution ofcompact support, Bull. Calcutta. Math. Soc. 94(2002), 349–358.[3] F.H.Kerr, Namias’ fractional Fourier transform on L2 and applications to differen-tial equations, J. Math. Anal. Appl. 136(1988), 404–418.[4] V.Namias, The fractional order Fourier transform and its applications to quantummechanics, J. Inst. Math. 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Pseudo-differential operator associated with the fractional Fourier transform

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