THE PSEUDO DIFFERENTIAL-TYPE OPERATOR ( x 1 D) (α β) ASSOCIATED WITH THE FOURIER-BESSEL TYPE SERIES REPRESENTATION

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1 Palestine Journal of Mathematics Vol. 5(2) (216), Palestine Polytechnic University-PPU 216 THE PSEUDO DIFFERENTIAL-TYPE OPERATOR ( x 1 D) (α β) ASSOCIATED WITH THE FOURIER-BESSEL TYPE SERIES REPRESENTATION B.B.Waphare and S.G.Gajbhiv Communicated by Ayman Badawi MSC 21 Classifications: Primary 47 G 3, 26 A 33; Secondary 46 F 12. Keywords and phrases: Pseudo differential-type operator, Hankel-type transform, Open mapping theorem, almost inverse. Abstract. For certain Frechet space F consisting of complex valued C functions defined on I = (, ) and characterized by their asymptotic behavior near the boundaries, we show that : (i) The Pseudo differential operator ( x 1 D) (α β), (α β) R, D = d dx, is an automorphism (in the topological sense) on F. (ii) ( x 1 D) (α β) is almost an inverse of the Hankel type transform h α,β in the sense that h α,β (x 1 D) (α β) (ϕ) = h (ϕ),for all ϕ F and (α β) R (iii) ( x 1 D) (α β) has a Fourier-Bessel type series representation on a subspace F b F and also on its dual F b. 1 Introduction The theory of pseudo differential operators has been developed by many researchers in India and abroad. In recent years pseudo differential operators involving Hankel transform, Hankel convolution, Bessel operators etc. has been studied by many mathematicians. It is the purpose of this paper to give the Fourier-Bessel type series representation of the pseudo differential type operator ( x 1 D) (α β). We denote by F the space of all C - complex valued functions ϕ(x) defined on I = (, ), such that ϕ(x) = k a i x 2i + O(x 2k ) (1.1) i= near the origin and is rapidly decreasing as x. For (α β) > 1/2, we define a (α β) th order Hankel-type transform h α,β on F by where Φ(y) = [h α,β ϕ(x)](y) = dm(x) = m (x)dx = [ 2 α β Γ(3α + β) ] 1 x 4α dx, ϕ(x)j α,β (xy)dm(x) (1.2) J α,β (x) = 2 α β Γ(3α + β)x (α β) J α β (x),

2 176 B.B.Waphare and S.G.Gajbhiv and J α β (x) is the Bessel-type function of order (α β). The inversion formula for (1.2) is given by [1, 3, 4], ϕ(x) = In the present paper we will show that for every real (α β): Φ(y)J α,β (xy)dm(y). (1.3) (i) The pseudo differentialâăştype operator ( x 1 D) (α β) is a topological automorphism on F. (ii) The Hankel-type transform h α,β is also an automorphism on F. (iii) On F,( x 1 D) (α β) is almost an inverse of h α,β in the sense that [h α,β ( x 1 D) (α β) ](ϕ) = h (ϕ), ϕ F. (iv) On a certain subspace F b F and on its dual F b, ( x 1 D) (α β) has Fourier-Bessel type series representation. All automorphisms are topological automorphisms in the sequel. 2 Notations and Terminology For any real number (α β) 1/2, F α,β is the space of all C - complex valued functions ϕ(x) defined on I such that m,k (ϕ) = Sup x I x m k α,β,x ϕ(x) < (2.1) For each m, k =, 1, 2... where α,β,x = D 2 + x 1 (4α)D. We can easily note that F α,β { is a } Frechet space. Its topology is generated by the countable family of separating seminorms m,k m,k=,1,2. By Lee [3, Theorem 2.1(i), page 429], we have F α,β = F a,b = F (as a set) for each (α β), (a b)( 1/2) R. Thus for each (α β)( 1/2), we have a topology T α β on F generated by the countable family of seminorms m,k. Hence (F, T α β) is a Frechet space. When (α β) = 1/2, F 1/2 F, since the factor x 1 (4α)D in α,β,x responsible for the even nature of ϕ(x) F α,β (x) near the origin, vanishes. For example e x F 1/2 but e x / F α,β. Following Zemanian [7, 8] we define Hankel-type transform h α,β with (α β) 1/2 by Ψ(y) = [ h α,β ψ(x) ] (y) = ψ(x)(xy) α+β J α β (xy)dx (2.2) It can be easily proved that h α,β is an automorphism on the space H α,β that consists of complex valued C functions defined on I and satisfies the relation m,k (ψ) = Sup x I x m (x 1 D) k [ x 2β 1 ψ(x) ] < (2.3) for each m, k =, 1, 2... where D = d/dx. Now we will need following theorem which is an important result for the development of our theory. Theorem 2.1. Let (α β), (a b) 1/2; a, b are real numbers. Then (i) The operation ϕ x 2α ϕ is an homeomorphism from F onto H α,β (ii) ( x 1 D ) n : F F is an automorphism of F. (iii) (F, T a b ) and (F, T α β ) are equivalent topological spaces. (iv) h α,β (ϕ) = ( 1) [ ( n h α,β,n x 1 D ) n] (ϕ), ϕ F, (α β) 1/2, n =, 1, 2....

3 THE PSEUDO DIFFERENTIAL-TYPE OPERATOR ( x 1 D) (α β) 177 Notation: In view of Theorem 2.1((i),(ii)), we will always write x 2α ϕ = ϕ(x) H α,β for ϕ F and drop the suffix, (α β) from T α β. So henceforth the topological linear space (F, T ) will be denoted by F. Proof of Theorem 2.1:(i)We use induction on n and noting that It can be proved that α,β,x = x 2 (x 1 D) 2 + 2(3α + β)(x 1 D) (2.4) n α,β,x= x 2n (x 1 D) 2n + a 1 x 2(n 1) (x 1 D) 2n a n (x 1 D) n (2.5) where a i are the constants depending on α β. Now ϕ F if and only if ϕ H α,β (see remark II of Lee [3]) and taking a = 1, it follows from (2.5) that m,k (ϕ) 2k i=k a 2k i (ϕ) (2.6) m+2(i k),i Proving the continuity of the inverse operation ϕ x 2β 1 ϕ. By using open mapping theorem [6, page.172], F being Frechet space, we complete the proof. (ii) Letϕ j be a sequence tending to zero in F. Then ϕ j in H α,β for arbitrary (α β) 1/2. Hence [ m,k (x 1 D) n ϕ j (x) ] 2k i=k a 2k i m+2(i k),i+n ϕ j, as j (by(2.6)) Now it remains to be shown that (x 1 D) n is bijective. It is enough to prove this for n = 1. So, let x 1 Dϕ 1 (x) = x 1 Dϕ 2 (x) for ϕ 1, ϕ 2 F. Hence ϕ 1 (x) ϕ 2 (x) = constant. But ϕ 1 (x)and ϕ 2 (x) are of rapid descent as x ϕ 1 = ϕ 2 (x). Now let ψ F, then ϕ(x) = x tψ(t)dt, defined uniquely (since ψ is of rapid descent as x ) in F, is such that x 1 Dϕ(x) = ψ(x). Thus we see that (x 1 D) n is a continuous bijection on F. The space F being a Frechet space, the Open Mapping Theorem shows that(x 1 D) n is a bicontinuous bijection on (F, T α β ) for each (α β) R {1/2}. (iii)let α β = a b + d, d R and ϕ n be a sequence tending to zero in (F, T a b ). Then x m,k (ϕ m n) = Sup [ x I a,b,x +2d(x 1 D) ] k ϕn (x) < [ Sup x I x m k k i a,b,x (2dx 1 D) i ϕ n (x) + i= k (2dx 1 D) k i i a,b,x ϕ n (x) i= + terms of type i 1 a,b,x (2dx 1 D) i 2 i 3 a,b,x... ϕ n(x) ] and (2dx 1 D) j 1 j 2 a,b,x (2dx 1 D) j 3... ϕ n (x) (where i 1 + i 2 + i 3 + = j 1 + j 2 + j 3 + = k) as n, for each m, k =, 1, 2,... Since i a,b,x and (x 1 D) i are continuous on (F, T a b ). This follows from integration by parts and induction on n. Remark 2.2. It can be shown that on F k α,β,x (x 1 D) n = (x 1 D) n k α,β n,x This proof follows by induction on k.

4 178 B.B.Waphare and S.G.Gajbhiv Definition 2.3. In view of Theorem 2.1(iv), we define the Hankel-type transform h α,β formally for any (α β) R as h α,β (ϕ) = h α,β,n (x 1 D) n ϕ, ϕ F (2.7) where n is chosen such that α β + n > 1/2. This is well defined definition as (x 1 D) n is an automorphism. Definition 2.4. Let F be the dual space of F.Then for f F define the generalized Hankel-type transform h α,β f = f ˆ of f by h α,β f, h α,β ϕ = f, ϕ, ϕ F, (α β) R Theorem 2.5. For (α β) R, h α,β is an automorphism on F and hence on F. Proof:Let ϕ(x) F. Then h α,β (ϕ) = Φ(y) = (x 1 D) 2n ϕ(x)j α,β,2n (xy)dm(x) = y 2β 1 h a,b ( ψ(x) ) (y), (2.8) where a b = α β + 2n > 1/2 and ψ(x) = x 2a ψ(x) = x 2a (x 1 D) 2n ϕ(x). Let ϕ n (x) in F ψ m (x) in H a,b h a,b (ψ m ) in H a,b h a,b (ϕ m ) in F. Now h a,b,the Hankel-type transform being bijective, (2.7) shows that h α,β is a bijection. Finally by making use of the Open Mapping Theorem we can complete the proof. Writing α = β =, in (2.7) and h, = h in definition(2.1) we get h (ϕ) = h n ( x 1 D) n ϕ(x), ϕ F. The above equation motivates us to propose the following definition. Definition 2.6. For (α β) R, define ( x 1 D) α β by ( x 1 D) α β (ϕ) = h 1 α,β h (ϕ), (2.9) Then ( x 1 D) α β is clearly an automorphism on F for each real (α β). From equation (2.9) we get ( x 1 D) α β ϕ(x) = dm(y)j α,β (xy) For distribution f F, define ( x 1 D) α β by ( x 1 D) α β f, ϕ = f, ( x 1 D) α β ϕ, ϕ F Now we modify theorem 2.1 (ii) to give our main result. dmϕ(x)j (xy) (2.1) Theorem 2.7. The pseudo differential operator ( x 1 D) α β is an automorphism on F and hence on F for each (α β) R. 3 The Fourier-Bessel type Series expansion of ( x 1 D) α β Equation (2.1) gives the integral representation of the operator ( x 1 D) α β. To get the Fourier-Bessel type series expansion, we modify our leading function space F suitably as follows (similar to the ones as in Zemanian [7, 9]). For b >, define F b = {ϕ F ϕ = for x > b} (3.1) The topology of F b is generated by a countable family of seminorms k (ϕ) = Sup <x<b k α,β,x ϕ(x) < k =, 1, 2... (3.2) Clearly all the topologies obtained by choosing different (α β) s are equivalent.

5 THE PSEUDO DIFFERENTIAL-TYPE OPERATOR ( x 1 D) (α β) 179 Remark 3.1. Without loss of generality, we may take (α β) > 1/2. Definition 3.2. We define finite Hankel type transform h α,β by Φ(z) = [h α,β ϕ] (z) = b ϕ(x)j α,β (xz)dm(x) (3.3) Then Φ(z) is an even entire function by Griffith Theorem [2, 9]. Let z = y + iw and G b = {Φ(z) Φ(z)is an even entire function satisfying (3.4)}. ηb k e (Φ) = Sup b w z C z 2k Φ(z) < (3.4) for k =, 1, Then G b is a linear topological space with ηb k as seminorms. Both spaces F b and G b are Hausdorff locally convex topological linear spaces satisfying the axiom of first countability. They are sequentially complete spaces. Theorem 3.3. h α,β is an homeomorphism from F b onto G b. Proof:Let ϕ F b. Then Hence z 2m Φ(z) = Φ(z) = h α,β,2m [ ( x 1 D) 2m ϕ(x) ] m N b From the asymptotic formula J α β) (z) 2/πz cos x 4α+2m [ ( x 1 D) 2m ϕ(x) ] (xz) (α β) J α β+2m (xz)dz ( (α β)π z π ) 2 4 z, argz < π and from the fact that z (α β) J α β+m (z) is an entire function, it follows that for all x, z, e b w (xz) (α β) J α β+2m (xz) < C m(α β) (a constant) Thus η m b (Φ) C m(α β) b 2(m+3α+β) [ (x 1 D) 2m ϕ(x) ] < (3.5) (x 1 D) 2m being an automorphism (also on F b ), (3.5) implies the continuity of h α,β. h α,β is clearly injective. For any Φ(z) G b, take ϕ(x) = Φ(y)J α,β (xz)dm(y) Then it follows from Griffith Theorem [2] that ϕ is zero almost everywhere for x > b. Also k (ϕ) = Sup <x<b k α,β,x Φ(y)J α,β (xz)dm(y) = Sup <x<b Φ(y)( 1) k y 4α+2k (xy) (α β) J α β (xy)dy <, [ Since k α,β,x (xy) (α β) J α β (xy) ] = ( 1) k y 2k (xy) (α β) J α β (xy), Φ(y) is of rapid descent as y, and [ (xy) (α β) J α β (xy) ] is bounded for < y <. Therefore, ϕ F b. Hence h α,β is surjective. Now the Open Mapping Theorem completes the proof. Theorem 3.4. Let ϕ F b. Then 2 ϕ(x) = lim ϵ + b 2 ( ) α β λn J α β (xλ n ) λ ϵ (x) x J3α+β 2 (bλ n) Φ(λ n) (3.6) n=1

6 18 B.B.Waphare and S.G.Gajbhiv where the λ n s are the positive roots of J α β (bz) = arranged in the ascending order and for < ϵ < b/4, E ( ) x 2ϵ < x < 2ϵ 1 2ϵ x b 2ϵ λ ϵ (x) = 1 E ( ) x b+2ϵ 2ϵ b 2ϵ < x < b x b and E(u) = u e 1 x(x 1) dx 1 e 1 x(x 1) dx Proof: By following [5] proof can be completed. Theorem 3.4 gives the required Fourier-Bessel type series expansion for the pseudo differential type operator ( x 1 D) α β, which we obtain in the following. Theorem 3.5 (The Fourier-Bessel type series). For ϕ F b, we have where [ ( x 1 D) α β] 2 ϕ(x) = lim ϵ + b 2 ( ) α β λn J α β (xλ n ) λ ϵ (x) x J3α+β 2 (bλ n) Φ (λ n ) (3.7) n=1 Φ (y) = h [ϕ(x)](y) Proof: Equation (2.9) along with Theorem 3.4 gives the required proof. Note that, λ 2α n Φ (λ n ) A k(α β) λ 2α 2k n J α β (xλ n ) A k(α β) constant and x α β λ 1/2 n J3α+β 2 (bλ n) is smooth and bounded on < x < b, < λ n <. Hence the truncation error 2 ( ) α β λn J α β (xλ n ) E N = lim ϵ + b 2 λ ϵ (x) x J3α+β 2 (bλ n) Φ (λ n ) n=n+1 has exponential decay for large N. This completes the proof. Theorem 3.5 gives the Fourier-Bessel type series representation of the operator ( x 1 D) α β on the testing function space F b. We wish to investigate the nature of the Fourier-Bessel series for the pseudo-differential type operator ( x 1 D) α β on the distribution space F b. The spaces F b and G b are dual spaces of F b and G b respectively. They are assigned the weak topologies generated by the seminorms and P ϕ (f) = f, ϕ, ϕ F b, f F b P ϕ (h α,β f) = h α,β f, h α,β ϕ, h α,β ϕ G b, h α,β f G b respectively.both the spaces are sequentially complete. Definition 3.6. For f F b, ϕ F b, we define the generalized finite hankel-type transform h α,β f by h α,β f, h α,β ϕ = f, ϕ (3.8) Theorem 3.7. For (α β) R, h α,β is an homeomorphism from F b onto G b. Theorem 3.8. For every ϵ (, b/4) and each f F b, the function ˆ f ϵ (y) = f(x), y 2β 1 λ ϵ (x)m (y)j α,β (xy) (3.9) where λ ϵ (x) is defined as in Theorem 3.4 is a smooth function of slow growth and defined a regular generalized function in G b

7 THE PSEUDO DIFFERENTIAL-TYPE OPERATOR ( x 1 D) (α β) 181 Proof: We note that (x 1 D) k λ ϵ (x) is bounded on < x < b for each k.using (2.6), it is simple to see that y 2β 1 λ ϵ (x)m (y)j α,β (xy) F b. Hence (3.9) is well defined. The result of the proof is similar to that of Zemanian [ Lemma 12]H. Theorem 3.9. The finite Hankel-type transform h α,β f of a generalized function f F b distributional limit, as ϵ + of the family fϵ ˆ (z) defined by (3.9). is the Proof: Proof is simple and hence omitted. Theorem 3.1. Let f F b and f ˆ = hα,β f. Then in the sense of convergence in F b, we have 2 f(x) = lim N b 2 N n=1 x 3α+β J α β (xλ n ) λn J3α+β 2 (bλ f(λ n) ˆ n ) (3.1) Proof: The proof follows easily from Theorem 3.4 and 3.9. Remark 3:For f F b, such that either f is regular or suppf [, b], the limit of fϵ ˆ (z) as ϵ + exists as an ordinary function and is equivalent to the finite Hankel-type transform of f [5]. A consequence of the above theorem is the following: Theorem Let f, g F b. If (h α,βf)(λ n ) = (h α,β g)(λ n ), for n = 1, 2, 3... then f = g and h α,β f = h α,β g. Definition For f F b, define ( x 1 D) α β f by ( x 1 D) α β f, ϕ = f, ( x 1 D) α β ϕ, ϕ F b, (α β) R (3.11) From equations (2.9),(3.8) and (3.11), it follows that ( x 1 D) α β f, ϕ = f, ( x 1 D) α β ϕ = h 1 h α,βf, ϕ, f F b, ϕ F b Hence Applying Theorem 3.1 to equation (3.12) we get ( x 1 D) α β f = h 1 h α,βf on F b (3.12) Theorem 3.13 (The Fourier-Bessel type series). Let f F b and f ˆ = hα,β f. Then in the sense of convergence in F b, we have References ( x 1 D) α β 2 f(x) = lim N b 2 N x J (xλ n ) λn J1 2(bλ f(λ n) ˆ n ). (3.13) [1] L.S.Dubey and J.N.Pandey, On the Hankel transformation of distributions, Tohoku Math.J.27(1975), [2] J.L.Griffith, Hankel transforms of functions zero outside a finite interval, J.Proc.Roy.Soc.New South Wales 89(1955) [3] W.Y.Lee, On Schwartz Hankel transformation of certain space of distributions, SIAM J.Math.Anal. 6(1975), [4] A.L.Schwartz, An inversion theorem for Hankel transforms, Proc.Amer.Math.Soc. 22(1969), [5] O.P.Singh, On distributional finite Hankel transform, Appl.Anal. 21(1986), [6] F.Traves,Topological vector Spaces,Distributions and Kernels, Academic Press, New york and London, [7] A.H.Zemanian, Generalized integral transformations, Interscience, New York [8] A.H. Zemanian, A Distributional Hankel transform, J.SIAM.Appl.Math. 14(1966), [9] A.H. Zemanian, The Hankel transformations of certain distributions of rapid growth, J.SIAM.Appl.Math. 14(1966), n=1

8 182 B.B.Waphare and S.G.Gajbhiv Author information B.B.Waphare, MIT ACSC, Alandi, Pune, Maharashtra, India. S.G.Gajbhiv, Research Scholar, Savitribai Phule Pune University and Assistant Professor MIT Academy of Engineering, Alandi, Pune, Maharashtra, India. Received: May 22, 215. Accepted: October 23, 215.