Fourier-Bessel Transform for Tempered Boehmians

Transcription

1 Int. Journal of Math. Analysis, Vol. 4, 1, no. 45, Fourier-Bessel Transform for Tempered Boehmians Abhishek Singh, Deshna Loonker P. K. Banerji Department of Mathematics Faculty of Science, J. N. V. University Jodhpur , India banerjipk@yahoo.com Abstract In the present paper the tempered Boehmian is introduced as an extension of tempered distribution, properties of which are studied using relation between the Fourier the Hankel transform. Mathematics Subject Classification: 46F1, 44A5, 44A4, 46F99 Keywords: tempered distributions, Boehmians, Fourier transform, Hankel transform, Bessel function 1. INTRODUCTION The Hankel transform is also called the Fourier-Bessel transform, relation of which with the Fourier transform is discussed in the sequel that follows. The Fourier transform of a function with respect to each of the n-independent variable is defined [6, p.76] as F (1) (ξ 1,x,...,x n )=F (1) [f (x 1,x,...,x n ); x 1 ξ 1 ] (1) the double transform is F () (ξ 1,ξ,x 3,...,x n )=F () [f (x 1,x,...,x n ); x 1 ξ 1,x ξ ], () which in succession gives F (n) (ξ 1,ξ,...,ξ n )=F (n) [f (x 1,x,...,x n ); x 1 ξ 1,...,x n ξ n ]. (3)

2 A. Singh, D. Loonker P. K. Banerji Considering X ξ by n - vectors (x 1,x,...,x n ) (ξ 1,ξ,...,ξ n ),which has the inner product x 1 ξ 1 +x ξ +...+x n ξ n =X ξ, the n - dimensional Fourier transform of f (X) is defined by F (n) (ξ) =F (n) [f (X); x ξ] (4) F (n) (ξ) =(π) n E n f (X)e i(ξ X) dx, (5) where E n is the n-dimensional Euclidean space. The inversion formula for the Fourier transform of f (X) the convolution of that of for f (X) g(x) of the vector X are defined, respectively, by f (X) = (π) n E n F (n) (ξ)e i(ξ X) dξ, (6) F (n) [f g (X); X ξ] =F (n) (ξ)g (n) (ξ). (7) For two dimensional Fourier transform in plane polar coordinates (r, θ) in x 1 x - plane (ρ, ϕ) inξ 1 ξ -plane, we have x 1 = r cos θ, x = r sin θ,ξ 1 = ρ cos ϕ, ξ = ρ sin ϕ, which yields (X ξ) =rρ cos(θ ϕ),from which we obtain We also have [7] F () [f(r),x 1 ξ 1,x ξ ] = 1 π rdr f (r)e irρ cos(θ ϕ) dθ. (8) π J (rρ) = 1 π π e irρ cos(θ ϕ) dθ (9)

3 Fourier-Bessel transform for tempered Boehmians 1 such that (8) is written as in other words, F () [f(r),x 1 ξ 1,x ξ ]= rf(r)j (rρ)dr, (1) F () [f(r),x 1 ξ 1,x ξ ]=H [f(r); ρ], (11) H is the Hankel transform of order zero, properties of which can be proved by using relation between the Fourier transform those given by (8) (11), respectively. Howell [, Art..4.3]defined the transform of circularly system function the Hankel transform by replacing (x, y) (ω, υ) with polar equivalents in F (ω, υ) =F[f(x, y)] ω,υ = f(x, y)e i(ωx+υy) dxdy (1) such that (x, y) =(r cos θ, r sin θ) (ω, υ) =(ρ cos ϕ, ρ sin ϕ), which reduce (1) as F (ρ cos ϕ, ρ sin ϕ) = π f(r cos θ, r sin θ) e irρ cos(θ ϕ) rdθdr. (13) The two - dimensional inverse Fourier transform in polar form is f(r cos θ, r sin θ) = 1 F (ρ cos ϕ, ρ sin ϕ) 4π π e irρ cos(θ ϕ) ρdϕdρ. (14) If f(r cos θ, r sin θ) is separable with respect to r, θ, i.e., then (13) reduces to f(r cos θ, r sin θ) =f r (r)f θ (θ), (15) F (ρ cos ϕ, ρ sin ϕ) = rf r (r)k (rρ, ϕ)dϕdr, (16)

4 A. Singh, D. Loonker P. K. Banerji where K (z, ϕ) = i.e. π f θ (θ)e iz cos(θ ϕ) dθ K (z, ϕ) = π f θ (θ + ϕ) =e iz cos(θ ) dθ, θ =(θ ϕ). (17) Similarly, if F (ρ cos ϕ, ρ sin ϕ) is separable with respect to ρ, ϕ, i.e., then (14) gives F (ρ cos ϕ, ρ sin ϕ) =F ρ (ρ)f ϕ (ϕ), (18) where f(r cos θ, r sin θ) = 1 ρf 4π ρ (ρ)k + (rρ, θ)dρ, (19) K + (z, θ) = π F (θ + θ)e iz cos ϕ dϕ () Therefore, it follows from (16) through (19) that if either f(x, y) orf (ω, υ) is circularly symmetric, then f(r cos θ, r sin θ) =f r (r) (1) F (ρ cos ϕ, ρ sin ϕ) =F ρ (ρ) () The Bessel function identity [7] is πj (z) = cos(z cos ω)dω π

5 Fourier-Bessel transform for tempered Boehmians 3 K ± (rρ, ω) = e ±irρ cos ωdω π = cos(rρcos ω)dω =πj (rρ), (3) π where J is the Bessel function of first kind of order zero. Further, due to above relations, (16) (19) reduce to F ρ (ρ) =π f r (r)j (rρ)rdr (4) f r (r) = 1 π F ρ (ρ)j (rρ)ρdρ. (5) The Hankel transform of order zero for the function g(r) [] is g(ρ) = g(r)j (rρ)rdr (6) by which we express (4) (5) in terms of the Hankel transform of order zero as F ρ (ρ) =π f r (r) (7) f r (r) = 1 F ρ (ρ) (8) π This is an evidence to view the Hankel transform of order zero as two dimensional Fourier transform of circularly symmetric functions. Since the Fourier transform for the tempered distribution tempered Boehmian is

6 4 A. Singh, D. Loonker P. K. Banerji established [cf. [4, 5]], we, therefore, use relation between two-dimensional Fourier transform the Hankel transform to establish Boehmians of slow growth (the tempered Boehmians), an illustration follows. The distribution space of slow growth S, for the Fourier transform is defined in terms of the Parseval equation, by [8] f, ϕ =π f,ϕ (9) f, ϕ =π f,ϕ, (3) respectively. Similarly, using relation between two - dimensional Fourier transform the Hankel transform, the tempered distribution space S for the Hankel transform is given by in other words, g(ρ), ϕ(ρ) = g(r),ϕ(r), (31) i.e. F (f), F (ϕ) = f,ϕ H (f), H (ϕ) = f,ϕ. (3) The pair of sequence (f n,ϕ n ) is called a quo- f n /ϕ n, whose numerator belongs to some set Tempered Boehmians : tient of sequence, denoted by G the denominator is a delta sequence such that f n ϕ m = f m ϕ n, n, m N. (33) Two quotients of sequence f n /ϕ n g n /ψ n are said to be equivalent if f n ψ n = g n ϕ n, n N. (34) The equivalence classes are called the Boehmians, for details, see [1]. The space of Boehmians is denoted by B, an element of which is written as

7 Fourier-Bessel transform for tempered Boehmians 5 x = f n /ϕ n. Application of construction of Boehmians to function spaces with the convolution product yields various spaces of generalized functions. The spaces, so obtained, contain the stard spaces of generalized functions defined as dual spaces. For example, if G =C(R N ) a delta sequence defined as sequence of functions ϕ n Dsuch that (i) ϕ n dx =1, a N (ii) ϕ n dx C, for some constant C n N, (iii) supp ϕ n (x), as n, then the space of Boehmian that is obtained, contains properly the space of Schwartz distributions. Similarly, this space of Boehmians also contains properly the space of tempered distributions S, when G is the space of slowly increasing functions with delta sequence. The Hankel transform of tempered Boehmian form a proper subspace of Schwartz distribution D. Boehmian space have two types of convergence, namely, the δ- Δ- convergences, which are stated as: (i) A sequence of Boehmians (x n ) in the Boehmian space B is said to be δ δ- convergent to a Boehmian x in B, which is denoted by x n x if there exists a delta sequence (δ n ) such that (x n δ n ), (x δ n ) G, n N (x n δ k ) (x δ k )asin G, k N. (ii) A sequence of Boehmians (x n )inb is said to be Δ- convergent to a δ Boehmian x in B, denoted by x n x if there exists a delta sequence (δ n ) Δ such that (x n x) δ n G, n N (x n x) δ n asin G. For details of the properties convergence of Boehmians one can refer to [3]. We have employed following notations definitions. A complex valued infinitely differentiable function f, defined on R N,is called rapidly decreasing, if ( sup sup 1+x 1 + x + + ) m x N D α f(x) <, x R N α m for every non-negative integer m. Here α = α α N, D α = α x = x α. 1 1 xα N N The space of all rapidly decreasing functions on R N is denoted by S. The delta sequence, i.e., sequence of real valued functions ϕ 1,ϕ,... S, is such that α (i) ϕ n dx =1, a N (ii) ϕ n dx C, for some constant C n N,

8 6 A. Singh, D. Loonker P. K. Banerji (iii) lim x ɛ x k ϕ n dx =, for every k N,ɛ>. If ϕ S ϕ = 1, then the sequence of functions ϕ n is a delta sequence. A complex-valued function f on R N is called slowly increasing if there exists a polynomial p on R N such that f(x)/p(x) is bounded. The space of all increasing continuous functions on R N is denoted by I. If f n I, {ϕ n } is a delta sequence under usual notion, then the space of equivalence classes of quotients of sequence will be denoted by B I, elements of which will be called tempered Boehmians. For F =[f n /ϕ n ] B I, define D α F =[(f n D α ϕ n )/(ϕ n ϕ n )]. If F is a Boehmian corresponding to differentiable function, then D α F B I. If F =[f n /ϕ n ] B I f n S, for all n N, then F is called a rapidly decreasing Boehmian. The space of all rapidly decreasing Boehmian is denoted by B S. If F =[f n /ϕ n ] B I G =[g n /ψ n ] B S,then the convolution is F G =[(f n g n )/(ϕ n ψ n )] B I. The convolution quotient is denoted by f/ϕ f denotes a usual quotient. ϕ Let f I. Then the Hankel transformation of f, denoted as H (f), is defined for distribution spaces (given as in Equation (31)) of slowly increasing function f in the following forms H (f),ϕ = f,h (ϕ), ϕ S H (f) =F ρ (ρ) =π f r (r)j (rρ)rdr.. HANKEL TRANSFORM FOR TEMPERED BOEHMIAN Theorem 1 : If [f n /ϕ n ] B I, then the sequence H {f n }, n =1,,..., converges in D. Moreover, if [f n /ϕ n ]=[g n /ψ n ],then H {f n } H {g n } converges to the same limit for the Hankel transformation of tempered Boehmians. Proof. Let ϕ D(testing function space) k N be such that H ϕ k > on the support of ϕ. Then we write H { f n },ϕ n = H {f n },ϕ H{ϕ k } H {ϕ k } ϕ = H {f n } H {ϕ k }, H {ϕ k }

9 Fourier-Bessel transform for tempered Boehmians 7 Since the sequence ϕ = H {f k } H {ϕ n }, H {ϕ k } = { } ϕ H ϕ n H ϕ n H {f k }, ϕ H {ϕ n } H {ϕ k } converges to ϕ H ϕ k. in D. Indeed, the sequence {H (f n ),ϕ} converges.this proves that the sequence H {f n } converges in D (dual of space D). Now, assume that [f n /ϕ n ]=[g n /ψ n ] B I. Let us define h n = f n+1 g n ϕ n ψ n+1, if n is odd, if n is even. δ n = ϕ n+1 ϕ n ψ n+1 ψ n, if n is odd, if n is even. Then [h n /δ n ]=[f n /ϕ n ]=[g n /ψ n ] the sequence H {h n } converges in D. Moreover, lim H {h n 1 }(ϕ n ) = lim H {f n ψ n } (ϕ) = lim H {f n ψ n } (ϕ) = lim H {f n }{H (ψ n ) (ϕ)} = lim H {f n }(ϕ). Thus, sequence H {f n } H {h n } have the same limit, which implies that H {h n } H {g n } will also have the same limit.this completes the proof of the theorem. Definition 1 : Let F =[f n /ϕ n ] B I the limit of sequence H {f n } is in D. The Hankel transform of H {F } of F, thus, has a mapping from B I into D, which is linear. Theorem : Let F =[f n /ϕ n ] B I G =[g n /ψ n ] B S. Then (i) H (G) is an infinitely differentiable function (ii) H [F G] =H [F ]H [G] (iii) H (F ) H (ϕ n )=H (f n ), n N Proof. (i) Let G =[g n /ψ n ] B S U be a bounded open subset of R N (the n- dimensional Euclidean space). H {ψ m } > onu, also Then there exists m N such that

10 8 A. Singh, D. Loonker P. K. Banerji H (G) = lim H {g n } = lim H {g n }H {ψ m } H {ψ m } = lim H {g n }H {ψ m } H {ψ m } = H {g m } H {ψ m } lim H {ψ n } = H {g m } H {ψ m } on U. Since H {g m }, H {ψ m } S H {ψ m } > onu, thush {G} is an infinitely differentiable function on U. (ii) If ϕ D, then there exists m N such that H {ψ m } > on the support of ϕ. Thus, we have H {F G}{ϕ} = lim H {f n g n }{ϕ} = lim H {f n g n }(ϕ) = lim H {f n }{H g n (ϕ)} { = lim H {f n } H g n ϕ H } ψ m H ψ m, m N { } H g m H ψ = lim H {f n } n ϕ H ψ m { } H g m = lim H {f n } ϕh (ψ H ψ n ) m = lim H {f n }{H (G)ϕH (ψ n )} = H {G} lim {H (f n )H (ψ n )}(ϕ)

11 Fourier-Bessel transform for tempered Boehmians 9 = H {G} lim H (f n ψ n )(ϕ) (iii) Let ϕ D. Then = H {F }H {G}{ϕ} = H {F } H {G}{ϕ} (35) {H F H ϕ m }{ϕ} = {H F }{H (ϕ m )ϕ}, m N = lim {H f n }{H (ϕ m )ϕ} = lim {H (f n ) H (ϕ m )}{ϕ} = lim {H (f m ) H (ϕ n )}{ϕ} = lim H {f m }{H (ϕ n ) ϕ} = H {f m }{ϕ} = H {F m }{ϕ} The theorem is, therefore, completely proved. Theorem 3 : A distribution f is the Hankel transform of tempered Boehmian if only if there exists a delta sequence {δ n } such that {fh (δ n )} is a tempered distribution for every n N. Proof. Let F =[f n /ϕ n ] B I f = H {F }.Then fh {ϕ n } = H {F }H {ϕ n }. Thus, fh (ϕ n ) is a tempered distribution. Now let f D, (δ n )bea delta sequence such that fh (δ n ) is tempered distribution for every n N. We define [ ] {fh (δ n )} δ n F = (36) (δ n δ n ) where {fh (δ n )} is the inverse Hankel transform of {fh (δ n )}. Since {fh (δ n )} is a tempered distribution, therefore, {fh (δ n )} is also a tempered distribution. Acknowledgement This work is supported by the the JNV University Research Scholarship No. 899, the DST (SERC) Young Scientist Scheme, Sanction No. SR/FTP/MS- /7, the Emeritus Fellowship (UGC, India) Sanction No. F.6-6/3/(SA- II), sanctioned to the first author (AS), the second author (DL) the third author (PKB), respectively.

12 1 A. Singh, D. Loonker P. K. Banerji References [1] T. K. Boehme, The support of Mikusinski Operators, Trans. Amer. Math. Soc. 176 (1973), [] K. B. Howell, The Hankel Transform, in The Transforms Applications Hbook (Poularikas, Alexar D.(Ed.)) Second Edition, CRC Press LLC, Boca, Raton,. [3] P. Mikusiński, Convergence of Boehmians, Japan. J. Math. 9 (1) (1983), [4] P. Mikusiński, The Fourier Transform of Tempered Boehmians, Fourier Analysis, Lecture Notes in Pure & Applied Mathematics, Marcel Dekker, New York (1994), [5] P. Mikusiński, Tempered Boehmians ultradistributions, Proc. Amer. Math. Soc. 13 (3) (1995), [6] I. N. Sneddon, The Use of Integral Transforms, Tata McGraw-Hill Publ. Co. Ltd., New Dehli,1974. [7] G. N. Watson, The Theory of Bessel Functions, nd Edition, Cambridge Univ. Press, London,1994. [8] A. H. Zemanian, Distriution Theory Transform Analysis, McGraw Hill, New York,1965. Received: May, 1