# Asymptotic Series of General Symbol of Pseudo-Differential Operator Involving Fractional Fourier Transform

Authors: S. K. Upadhyay; Anuj Kumar; Jitendra Kumar Dubey
Journal: ISRN Mathematical Analysis
 (2013)

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## Abstract

An asymptotic series of general symbol of pseudo-differential operator is obtained by using the theory of fractional Fourier transform.


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## Body

## 1. Introduction

Namias [1] introduced fractional Fourier transform which is a generalization of Fourier transform. Fractional Fourier transform is the most important tool, which is frequently used in signal processing and other branches of mathematical sciences and engineering. The fractional Fourier transform can be considered as a rotation by an angle α in time-frequency plane and is also called rotational Fourier transform or angular Fourier transform. The fractional Fourier transform [2, 3], with angle α of a function u(x), is defined by (1)u^α(x)=(Fαu)(ξ)=∫ℝ‍Kα(x,ξ)u(x)dx,
where
(2)Kα(x,ξ)={Cαei(x2+ξ2)cotα/2-ixξcosecαifα≠nπ12πe-ixξifα=π2,∀n∈𝕀,Cα=(2πisinα)-1/2eiα/2=1-icotα2π.
The corresponding inversion formula is given by
(3)u(x)=12π∫ℝ‍Kα(x,ξ)¯u^α(ξ)dξ,
where the kernel
(4)Kα(x,ξ)¯={Cα′e-i(x2+ξ2)cotα/2+ixξcosecαifα≠nπ12πeixξifα=π2,∀n∈𝕀,Cα′=(2πisinα)1/2sinαe-iα/2=2π(1+icotα).
Zayed [3] and Bhosale and Chaudhary [4] studied fractional Fourier transform of distributions with compact support. Pathak and others [5] defined the pseudo-differential operator involving fractional Fourier transform on Schwartz space S(ℝ) and studied many properties.Our main aim in this paper is to generalize the results of Zaidman [6] and to find an asymptotic series of general symbol of pseudo-differential operator involving fractional Fourier transform.Now we are giving some definitions and properties which are useful for our further investigations.Linearity of fractional Fourier transform is given as(5)Fα[c1u1(x)+c2u2(x)]=c1Fα(u1(x))+c2Fα(u2(x)),
where c1 and c2 are constants and u1(x) and  u2(x) are two input functions.LetLp(ℝ) denote the class of measurable functions f defined on ℝ such that
(6)∥f∥Lp(ℝ)=[∫ℝ‍|f(x)|pdx]1/p<∞,
where 1≤p<∞.From [5], generalized Sobolev space 𝒢αs,p(ℝ) involving fractional Fourier transform is defined by
(7)∥u∥s,p=[∫ℝ‍(1+|ξ|2)sp/2|u^α(ξ)|pdξ]1/p,s∈ℝ and u∈S′(ℝ).The convolution of two functionsf∈L1(ℝ) and g∈L1(ℝ) is defined [5, 7] as
(8)(f*g)(x)=∫ℝ‍f(y)g(x-y)dy,
provided that the integral exists.Let𝒢 be a class of all measurable complex-valued functions a(x,ξ) which are defined on ℝ×ℝ-{0}. Then, we assume the following properties.(i)
limx→∞a(x,ξ)=a(∞,ξ) exists for all ξ∈ℝ-{0} and is bounded to mesaurable function.(ii)
We definea′(x,ξ)=a(x,ξ)-a(∞,ξ), then
(9)a′(x,ξ)=Cα′2π∫ℝ‍e-i(x2+ξ2)cotα/2+ixξcosecα(Fαa′)(η,ξ)dη,η∀(x,ξ)∈ℝ×ℝ-{0},where (Fαa′)(η,ξ) is complex-valued function defined on ℝ×ℝ-{0}, which is measurable in η and ξ for all (η,ξ)∈ℝ×ℝ-{0} and satisfies the estimate:
(10)|(Fαa′)(η,ξ)|≤K(η),∀η∈ℝ,
where (1+|η|2)lK(η)∈L1(ℝ)  ∀l=0,1,2,3,….Let(rj)0∞ be a strictly decreasing sequence; that is, r0>r1>r2>⋯>rj→-∞ as j→∞ and ψ∈C∞(ℝ) such that 0≤ψ(ξ)≤∞ for all ξ∈ℝ-{0},
(11)ψ(ξ)={0if0<ξ≤12,1ifξ≥1.
Let {aj(x,ξ)}0∞ be an infinite sequence of function defined on ℝ×ℝ-{0}.Then, we define a function(12)a(x,ξ)=∑j=0∞‍ψ(ξtj)|ξ|rjaj(x,ξ),
where (tj)0∞ is a sequence of positive real numbers such that tj→∞ as j→∞.From (12), it is clear that a(x,ξ)=0, for |ξ|≤1/2, x∈ℝ, and t∈[1,∞).The global estimate of the above defined functiona(x,ξ) and of remainders of order N is given as
(13)bN(x,ξ)=a(x,ξ)-∑j=0N-1‍ψ(ξtj)|ξ|rjaj(x,ξ)=∑j=N∞‍ψ(ξtj)|ξ|rjaj(x,ξ),∀ξ∈ℝ-{0}.Theorem 1.
Let{tj}0∞ be a sequence of positive real numbers such that the following inequalities:
(14)|a(x,ξ)|≤C|ξ|r0,|bN(x,ξ)|≤C|ξ|rN,∀(x,ξ)∈ℝ×ℝ-{0},
are satisfied for N=1,2,3…. In particular the estimates are as follows:
(15)|a(∞,ξ)|≤C|ξ|r0,|bN(∞,ξ)|≤C|ξ|rN,∀ξ∈ℝ-{0}.Proof.
The proof of the above theorem is obvious from [6, pages 233-234].Theorem 2.
Let{tj}0∞ be a sequence of positive real numbers such that the following estimates:
(16)|(Fαa′)(x,ξ)|≤K(λ)|ξ|r0,|(FαbN′)(x,ξ)|≤K(λ)|ξ|rN,∀(λ,ξ)∈ℝ×ℝ-{0},
are satisfied for N=1,2,3…. In particular the estimates are as follows:
(17)|(Fαa′)(∞,ξ)|≤C|ξ|r0,|(FαbN′)(∞,ξ)|≤C|ξ|rN,∀ξ∈ℝ-{0},
where (1+|λ|2)pK(λ)∈L1(ℝ), (1+|λ|2)pKN(λ)∈L1(ℝ), ∀p=0,1,2,3…, and N=1,2,3,….Proof.
The proof of above theorem is also obvious by using the same arguments from [6, pages 133–135].


## 2. Asymptotic Expansion of Pseudo-Differential Operator Associated with General Symbol

Definition 3.
Leta(x,ξ) be a general symbol belonging to 𝒢. Then pseudo-differential operator a(x,D)=Aa,α associated with symbol a(x,ξ) is defined by
(18)(a(x,ξ)u)(x)=12π∫ℝ‍Kα(x,ξ)¯a(x,ξ)u^α(ξ)dξ,
where u^α(ξ) is defined in (3) ∀(x,ξ)∈ℝ×ℝ-{0}.Definition 4.
An infinitely differentiable complex-valued functionu is member of S(ℝ) if and only if for every choice of μ and ν of non-negative integers, it satisfies
(19)γμ,ν(u)=supx∈ℝ|xμDνu(x)|<∞.Lemma 5.
A functionu∈C∞(ℝ) satisfies (19) if and only if  (20)τμ,ν(u)=supx∈ℝ|(1+|x|2)m/2Dβu(x)|<∞,∀m,β∈N0.Lemma 6 (Peetre).
For any real number t and for allξ,η∈ℝ, the estimate
(21)((1+|ξ|2)(1+|η|2))t≤2|t|(1+|ξ-η|)|t|
is satisfied.Theorem 7.
Leta′(x,ξ)∈𝒢; then one has the following relation:
(22)Fα[eix2cotα/2a′(x,D)u(x)](ξ)=Cα′2π∫ℝ‍e-i(η2-ξη)cotαa^α′(ξ-η,ξ)u^α(η)dη,
where u∈S(ℝ), x∈ℝ.Proof.
By the definition of fractional Fourier transform (1), we have
(23)Fα[eix2cotα/2a′(x,D)u(x)](ξ)=Cα∫ℝei(x2+ξ2)cotα/2-ixξcosecα××a′(x,ξ)eix2cotα/2u(x)dx=Cα∫ℝ‍ei(x2+ξ2)cotα/2-ixξcosecαa′(x,ξ)eix2cotα/2×Cα′2π×{Cα′2π∫ℝ‍e-i(x2+η2)cotα/2+ixηcosecαu^α(η)dη}dx=CαCα′2π∬ℝei(x2+ξ2-x2-η2+x2)cotα/2-ix(ξ-η)cosecα××a×a′(x,ξ)u^α(η)dηdx=CαCα′2π∬ℝei(x2+ξ2-η2)cotα/2-ix(ξ-η)cosecα××a×a′(x,ξ)u^α(η)dηdx=CαCα′2π∬ℝei(x2+ξ2+η2-2ξη)cotα/2-ix(ξ-η)cosecαa′(x,ξ)×××ei(ξη-η2)cotαu^α(η)dηdx=CαCα′2π∬ℝei(x2+ξ2+η2-2ξη)cotα/2-ix(ξ-η)cosecαa′(x,ξ)×××ei(ξη-η2)cotαu^α(η)dηdx=Cα′2π∫ℝ‍e-i(η2-ξη)cotα××{Cα∫ℝ‍ei[x2+(ξ-η)2]cotα/2-ix(ξ-η)cosecα××a×a′(x,ξ)dx∫ℝ‍}u^α(η)dη=Cα′2π∫ℝ‍e-i(η2-ξη)cotαa^α′(ξ-η,ξ)u^α(η)dη.Theorem 8.
Ifa(x,ξ)∈𝒢  is a symbol and a(x,D)=a(∞,D)+a′(x,D) is the associated operator, then, one has the following relation: (24)Fα[eix2cotα/2a(x,D)u(x)](ξ)=a(∞,ξ)Fα[eix2cotα/2u(x)](ξ)+Cα′2π∫ℝ‍e-i(η2-ξη)cotαa^α′(ξ-η,ξ)u^α(η)dη,
where u∈S(ℝ), x∈ℝ.Proof.
We have(25)Fα[eix2cotα/2a(x,D)u(x)](ξ)=Fα[eix2cotα/2{a(∞,D)+a′(x,D)}u(x)](ξ).
By linearity of fractional Fourier transform (5) we get
(26)Fα[eix2cotα/2a(x,D)u(x)](ξ)=Fα[eix2cotα/2a(∞,D)u(x)](ξ)+Fα[eix2cotα/2a′(x,D)u(x)](ξ).
Now using (1) and (22), we get the required result.Theorem 9.
Leta(x,ξ)∈𝒢 be a symbol and a(x,D) the associated operator; then one has the following relation:
(27)∥(a(x,D)u)∥s,p≤Cs,p∥u∥s+r0,p,
where u∈S(ℝ), s∈ℝ.Proof.
Firstly from (7) we have(28)∥eix2cotα/2(a(∞,D)u)∥s,p=[∫ℝ‍(1+|ξ|2)sp/2|Fα[eix2cotα/2a(∞,D)u(x)](ξ)|pdξ]1/p=[∫ℝ‍(1+|ξ|2)sp/2|a(∞,ξ)|p|Fα[eix2cotα/2u(x)](ξ)|pdξ]1/p.Now the argument of (15) yields
(29)∥eix2cotα/2(a(∞,D)u)∥s,p≤C[∫ℝ‍(1+|ξ|2)sp/2(1+|ξ|2)r0p/2××|Fα[eix2cotα/2u(x)](ξ)|pdξ∫ℝ‍]1/p=C[∫ℝ‍(1+|ξ|2)(s+r0)p/2×|Fα[eix2cotα/2u(x)](ξ)|pdξ∫ℝ‍]1/p=C∥eix2cotα/2u∥s+r0,p.
Thus we have
(30)∥eix2cotα/2(a(∞,D)u)∥s,p≤C∥eix2cotα/2u∥s+r0,p,∥eix2cotα/2(a(∞,D)u)∥s,p≤C∥u∥s+r0,p,
for every u∈S(ℝ), s∈ℝ.
Now we consider the functionUs,α(ξ)=(1+|ξ|2)s/2Gα(ξ), where Us,α(ξ)=0 for |ξ|≤1/2 and for |ξ|≥1/2 and Gα(ξ) is defined as
(31)Gs(ξ)=Cα′2π∫ℝ‍e-i(η2-ξη)cotαa^α′(ξ-η,ξ)u^α(η)dη.
Therefore,
(32)Us,α(ξ)=Cα′2π(1+|ξ|2)s/2∫ℝ‍e-i(η2-ξη)cotαa^α′(ξ-η,ξ)u^α(η)dη=Cα′2π∫ℝ‍(1+|ξ|2)s/2(1+|η|2)s/2e-i(η2-ξη)cotα××(1+|η|2)s/2a^α′(ξ-η,ξ)u^α(η)dη.
Using (21) we get
(33)|Us,α(ξ)|≤2|s|/2Cα′2π∫ℝ‍(1+|ξ-η|2)|s|/2(1+|η|2)s/2×a^α′ξ××|a^α′(ξ-η,ξ)||u^α(η)|dη.
From Theorem 2 we get
(34)|Us,α(ξ)|≤2|s|/2Cα′2π∫ℝ‍(1+|ξ-η|2)|s|/2(1+|η|2)s/2×K(ξ-η)×K(ξ-η)(1+|ξ|2)r0/2|u^α(η)|dη,=2|s|/2Cα′2π∫ℝ‍(1+|ξ-η|2)|s|/2(1+|η|2)(s+r0)/2×K(ξ-η)×K(ξ-η)(1+|ξ|2)r0/2(1+|η|2)r0/2|u^α(η)|dη,=2(|s|+|r0|)/2Cα′2π∫ℝ‍(1+|ξ-η|2)|s|/2(1+|ξ-η|2)|r0|/2×(1+|η|2)(s+r0)×(1+|η|2)(s+r0)/2K(ξ-η)|u^α(η)|dη,=2(|s|+|r0|)/2-1Cα′π×∫ℝ‍(1+|ξ-η|2)(|s|+|r0|)/2(1+|η|2)(s+r0)/2××()×K(ξ-η)|u^α(η)|dη,=∫ℝ‍Φ(ξ-η)hα(η)dη=(Φ*hα)(ξ).
Here Φ(ξ-η)∈L1(ℝ) by (10) for all s∈ℝ and since u^α(η)∈S(ℝ), therefore hα(η)∈Lp(ℝ).
Thus, we have(Φ*hα)(ξ)∈Lp(ℝ) and the inequality
(35)∥Φ*hα∥Lp(ℝ)≤∥Φ∥L1(ℝ)∥hα∥Lp(ℝ).
This implies that
(36)∥Us,α(ξ)∥p≤Cα∥u∥s+r0,p,∀u∈S(ℝ),s∈ℝ.
So that
(37)∥eix2cotα/2(a′(x,D)u)∥s,p=∥(1+|ξ|2)s/2Fα[eix2cotα/2(a′(x,D)u)(x)]∥p=∥Us,α(ξ)∥p.
Using (36) we get
(38)∥eix2cotα/2(a′(x,D)u)∥s,p≤Cα∥u∥s+r0,p.
Therefore,
(39)∥eix2cotα/2(a(x,D)u)∥s,p=∥eix2cotα/2{a(∞,D)u+a′(x,D)u}∥s,p≤∥eix2cotα/2(a(∞,D)u)∥s,p+∥eix2cotα/2(a′(x,D)u)∥s,p.
Now using (30) and (38) we get
(40)∥eix2cotα/2(a(x,D)u)∥s,p≤(C+Cα)∥u∥s+r0,p.
This implies that
(41)∥eix2cotα/2(a(x,D)u)∥s,p≤Cs,p,α∥u∥s+r0,p.
This implies the required result (27).Theorem 10.
One has the following estimates:(42)∥ψr(1tD)u∥s,p≤C∥u∥s+r,p,∀u∈S(ℝ),s,r∈ℝ.Proof.
Letψ∈C∞(ℝ), then by using (11) and (12) we find that ψr(ξ/t)=ψ(ξ/t)|ξ|r for |ξ|≥t/2. Therefore, we have
(43)|ψr(ξt)|≤C(1+|ξ|2)r/2,∀ξ∈ℝ.
Now we define the operator ψr((1/t)D) given on S(ℝ) by
(44)ψr(1tD)u=Fα-1(ψr(ξt)u^α(ξ)).
Hence,
(45)∥ψr(1tD)u∥s,p=(∫ℝ‍(1+|ξ|2)sp/2|ψr(ξt)|p|u^α(ξ)|pdξ)1/p≤C(∫ℝ‍(1+|ξ|2)sp/2|u^α(ξ)|p(1+|ξ|2)rp/2dξ)1/p=C(∫ℝ‍(1+|ξ|2)(s+r)p/2|u^α(ξ)|pdξ)1/p.
Thus,
(46)∥ψr(1tD)u∥s,p≤C∥u∥s+r,p.
Since 𝒢α∞=∩s∈ℝ𝒢αs,p(ℝ),  ψr(((1/t)D)u) is a linear operator 𝒢α∞ into itself.Definition 11.
A linear operatorL with r∈ℝ, and ∀s∈ℝ, there exists a constant Cs>0 such that
(47)∥Lu∥≤Cs∥u∥s+r,p,∀u∈𝒢α∞.
The infimum of all orders r of L is called true order of L.Definition 12.
Letψr(((1/t)D)a(x,D)) be a linear operator from ∀u∈𝒢α∞ into itself and satisfy the following inequality:
(48)∥ψr(1tD)a(x,D)∥s,p≤C∥u∥s+r,p,∀s∈ℝ,u∈𝒢α∞(ℝ).
Then ψr((1/t)D)a(x,D) is said to be a canonical operator of degree r, where r∈ℝ.Definition 13.
Let(rj)0∞ be a strictly decreasing sequence of real numbers and {ψrj((1/t)D)aj(x,D)} a sequence of canonical operators of degree rj. Then, corresponding a sequence of positive real numbers (tj)0∞ and {aj(x,ξ)}0∞⊂𝒢, a linear operator M:𝒢α∞→𝒢α∞ is asymptotically expanded into the series ∑j=0∞‍ψrj((1/t)D)aj(x,D) if it satisfies the following inequality: (49)t.o[M-∑j=0N‍ψrj(1tjD)aj(x,D)]<rN.Theorem 14.
Let{aj(x,ξ)}0∞ be a sequence of symbols belonging to  𝒢 and (rj)0∞  a strictly decreasing sequence of real numbers that tends to -∞. Then, there exists a sequence of canonical operators of degree rj,Kj,α and a linear operator Aa,α in 𝒢α∞ such that(i)
t.o(Aa,α)≤r0;(ii)
Aa,α~∑j=o∞‍Kj,α, that is, t.o  [Aa,α-∑j=oN‍Kj,α]<rN.Proof.
Note that (i) is obviously true by using Theorem9 and for (ii) by using the arguments of [6, pages 241-242]. We can define the canonical operator by the following way:
(50)(Kj,αu)(x)=e-ix2cotα/2Fα-1×[ψrj(ξtj){aj(∞,ξ)Fα(eix2cotα/2u(x))(ξ)+Cα′2π∫ℝ‍e-i(η2-ξη)cotα×a^j,α′(ξ-η,ξ)u^α(η)dη(eix2cotα/2u(x))}(ξtj)],
where a(x,ξ)=∑j=o∞‍ψrj(ξ/t)[aj(∞,ξ)+aj′(x,ξ)] belongs to class of symbol 𝒢. Also we have
(51)(Aa,αu)(x)=e-ix2cotα/2Fα-1×[a(∞,ξ)Fα{eix2cotα/2u(x)}(ξ)+Cα′2π∫ℝ‍e-i(η2-ξη)cotα××a^α′(ξ-η,ξ)u^α(η)dη{eix2cotα/2u(x)}].
Therefore,
(52)Fα[eix2cotα/2(Aa,αu-∑j=0∞‍Kj,αu)](ξ)=[{a(∞,ξ)-∑j=0N‍ψrj(ξtj)aj(∞,ξ)}×Fα{eix2cotα/2u(x)}(ξ)+Cα′2π∫ℝ‍e-i(η2-ξη)cotα×{a^α′(ξ-η,ξ)-∑j=0N‍ψrj(ξtj)a^j,α′(ξ-η,ξ)}×u^α(η)dη{a(∞,ξ)-∑j=0N‍ψrj(ξtj)aj(∞,ξ)}].
Using the previous definitions (12) and (13), we have
(53)a(∞,ξ)-∑0N‍ψrj(ξtj)aj(∞,ξ)=bN+1(∞,ξ).
From (15) and (16), we get |bN+1(∞,ξ)|≤C(1+|ξ|2)rN+1/2, where ξ∈ℝ-{0} and a^α′(ξ-η,ξ)-∑j=0N‍ψrj(ξ/tj)a^α,j(ξ-η,ξ)=b^α,N+1′(ξ-η,ξ).
Also|b^α′(ξ-η,ξ)|≤CKN+1(ξ-η)(1+|ξ|2)(rN+1/2). Now using these arguments in (52) we get
(54)Fα[eix2cotα/2(Aa,α-∑j=0N‍Kj,αu)](ξ)=bN+1(∞,ξ)Fα{eix2cotα/2u(x)}(ξ)+Cα′2π∫ℝ‍e-i(η2-ξη)cotαb^α′(ξ-η,ξ)u^α(η)dη.
Using Theorem 9, we get
(55)∥Aa,αu-∑j=0N‍Kju∥s,p≤C∥u∥s+rN+1,p,∀u∈S(ℝ),s∈ℝ.
Hence, we get
(56)t.o[Aa,α-∑j=0N‍Kj,α]≤rN+1<rN.



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