Cauchy Representation of Fractional Fourier Transform for Boehmians

Abhishek Singh and P.K. Banerji abstract: Results relating to fractional Fourier transform and their properties in the Lizorkin space are employed in this paper to investigate the Cauchy representation of fractional Fourier transform for integrable Boehmians. An inversion formula for the fractional Fourier transform is addressed. The conclusion remark of the paper spells the initiation for the present investigation.

Let S be the space of rapidly decreasing test functions namely, the space of infinitely differentiable functions: v(x) on R satisfying the relation Denoted by V (R), is the set of functions v ∈ S, satisfying The Lizorkin space [7] Φ(R) is introduced as the Fourier pre-image of the space V (R) in the space S, According to the definition of the Lizorkin space, any function ϕ ∈ Φ(R) satisfies the orthogonality condition The space Φ(R) is also invariant with respect to the Fourier transform (1.1) and its inverse (1.2), and simultaneously, these transforms are inverse of each other (1.9) Definition 1.1.Let u be a function belonging to Φ(R).Then the fractional Fourier transform of order α, 0 < α ≤ 1, is defined by whereas the corresponding inverse fractional Fourier transform of order α is given by Now, considering L 1 as the space of complex valued Lebesgue integrable functions on real line R, the norm of the function is defined by and the convolution product is an element of L 1 , and f * g ≤ f g .
Lemma 1.2.Let u be a function belonging to Φ(R).Then the fractional Fourier transform of order α, 0 < α ≤ 1, is (1.12) Lemma 1.3.Let f and g are functions belonging to Φ(R).Then The proof of the Lemma may be referred to [14].

Cauchy representation and the fractional Fourier transform
Let us enumerate the basic properties and some definitions of the Cauchy representation of the fractional Fourier transform by following [3], and [6].
, and f (z), z = x + iy, be the Cauchy representation of f .Then Definition 2.2.A function f is called a locally integrable function if f is a continuous function and for some α, (2.3) Definition 2.3.The generalized fractional Fourier transform and the generalized inverse fractional Fourier transform of a tempered function f are defined and denoted, respectively, by and where z = x + iy.
Following identities hold true between Fα (f, z) and the fractional Fourier transform: where H(t) is a positive function, which is convenient to consider in the proof of inversion theorem and has a positive fractional Fourier transform whose integral is easily calculated.Therefore, we have Proposition 2.5.Let f is a locally integrable function, then the generalized fractional Fourier transform of f has the property for all ϕ ∈ S and ǫ > 0.
Proposition 2.6.For a given locally integrable function f , the inversion formula for the generalized fractional Fourier transform Fα (f, z) has the property and Proposition 2.7.Let T is a functional for the m th derivative of the locally integrable function defined in L 1 , then the generalized fractional Fourier transform of T is defined as Let T ∈ S ′ and Fα (T, z) be a generalized fractional Fourier transform of T .Then Fα (t, z) is an analytic (Cauchy) representation of Fα (T ) in the sense, that where S ′ is the space of linear functional on S or the space of tempered distributions.
Definition 2.8.The generalized fraction Fourier transform F α (f, z) for a locally integrable function f , for multi-variables is defined by and where z j = x j + iy j , j = 1, 2, 3, .... Proposition 2.9.Suppose f ∈ L 1 , F α (f ) ∈ L 1 and F = F α (f ), then the Cauchy representation of F (z) by means of the Cauchy kernel is given by Definition 2.10.For any tempered distribution space S, the space of linear functional S ′ on S, the space D of all infinitely differentiable functions on R n with compact support (A set K ⊂ X, a topological space, is called compact if every open cover of K contains a finite subcover) and its dual D ′ if S ∈ D ′ and T ∈ E ′ , then the convolution of the distributions is defined by (2.16) The space E (a, b) is the space of smooth functions on (a, b) and E ′ (a, b), or simply E ′ , is the dual of the space E .

Cauchy Representation for Integrable Boehmians
The general construction of Boehmians is given in [9,10] which when applied to various function spaces, various Boehmian spaces result.The term Boehmian is used for all objects by an abstract algebraic construction, similar to that of the field of quotients.Let G be an additive commutative semigroup and S ⊆ G, is a sub-semigroup, which has a mapping * from G × S to G such that (i) if δ, η ∈ S, then (δ * η) ∈ S and δ * η = η * δ The member of the class ∆ of sequence from S are called the delta sequence, which satisfies the following: We consider a quotient of the sequence f n /ϕ n , numerator of which belongs to G and the denominator is a delta sequence, and Two quotients of sequences f n /ϕ n and g n /ψ n are said to be equivalent if The equivalence classes, thus obtained, are called Boehmians, the space of all of which bears the notation B and an element of which is written as x = f n /ϕ n .If we consider G to be the set of all locally integrable functions on R, then the Boehmian space B is called the space of locally integrable Boehmian B L1 , which has properties of addition, scalar by multiplication, convolution in a convolution algebra [11].
A sequence of Boehmians F n is ∆-convergent to a Boehmian F if there exists a delta sequence (δ n ) such that (F n − F ) * δ n ∈ L 1 , for every n ∈ N and (F n − F ) * δ n → 0, as n → ∞.
A sequence of Boehmians F n is δ-convergent to a Boehmian F if there exists a delta sequence (δ n ) such that F n * δ k ∈ L 1 and F * δ k ∈ L 1 , for every n, k ∈ N and (F n − F ) * δ k → 0, for each k ∈ N.For convergence of Boehmians, see [9] and refer to [10] for the properties of integrable Boehmians.If where ∆ is a class of sequence (δ n ) (n = 1, 2, . ..).For (δ n ), the quotient [δ n /δ n ] corresponds to Dirac delta function δ, all the derivatives of δ are also integrable Boehmians.
The integral of a Boehmian as follows from the property, that, if For a function from L 1 , the definition, given by (3.4), happens to be the analogous to the definition of the Lebesgue integral.However, there are functions which are integrable in the sense of Boehmians, but not so as an ordinary function.For instance, a continuously differentiable function on L 1 is such that its derivative does not exist in L 1 .
Lemma 3.1.Let f ∈ L 1 and z = x + iy, from Properties 2.3, 2.4 and 2.5, we have Then the sequence converges uniformly on each compact set in R.
Proof: If (δ n ) is a delta sequence, then F α (δ n ) converges uniformly on each compact set to a constant function 1.Therefore, for each compact set K, F α (δ k ) > 0 on K, and for almost all k ∈ K, we have In view of Lemma 3.1, the fractional Fourier transform of an integrable Boehmian F = [f n /δ n ] can be defined as the limit of F α (f n ) in the space of continuous functions on R. Hence, this proves that the fractional Fourier transform of an integrable Boehmian is a continuous function and, thereby, the lemma is proved.

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Cauchy Representation of Fractional Fourier Transform for Boehmians 63 Proof: By virtue of the properties of fractional Fourier transform [8], indeed, the proofs of (i) and (ii) are obvious.By the definition of the convolution transform [5, p. 785] and by Lemma 1.2, the property (iii) can easily be proved.The proofs of properties (iv) and (v) are same as in [5, pp.785-786], we obtain these results after some simplifications.Proof of the property (vi) is as follows : We have uniformly on each compact set.Let (δ n ) be a delta sequence such that where k is well defined.Then This, explicitly, proves the property (vi).The theorem is thus, completely proved.✷

Conclusion
This paper can be considered a fair and reasonable generalization of the work given in the citation [6] in this paper.The concerned paper investigates the Cauchy representation of integrable and tempered Boehmians, which justifies the natural generalization of the tempered distributions (in the sense of Schwartz).As a result and for natural consequence, the present article becomes more of general nature by addressing the extension with the involvement of the fractional integral operator, the reference [14] of this paper highlights the concept.