KAMAL TRANSFORM FOR INTEGRABLE BOEHMIANS

In this paper, basic properties of Kamal transform for Integrable Boehmians are demonstrated. Also inversion theorem for Kamal transform is proved.


INTRODUCTION
The study of generalized functions has become a major area of research for last five decades. The  [10]. Similar to Mikusinski operators, a very general new class of generalized functions, called as Boehmians was constructed by Mikusinski P. [12]. Several properties of Kamal transform are investigated with applications in [1][2][3]. The Various integral transforms on Boehmians spaces are developed and studied in [4][5][6][7][8][9][13][14].
In this paper we mainly deal with a case of Kamal transform and developed Boehmian space for it.

GENERALIZED FUNCTIONS [15-17]:
The straight forward approach for generalized function is given by Temple G.
[16] with the fact that different sequences may have the same generalized functions.
Hence it is a need to define an equivalence relation between sequences that represent the same

INTEGRABLE BOEHMIANS
A general construction of Boehmians was studied in [11]. The space of Boehmians with two notions of convergence was well defined in [12]. The integral transforms have been extended to the context of Boehmian spaces. Fourier [5,6,7,13,14], Hilbert [8,9] are some of them.
If ( ) ( ) are delta sequences then ( * ) is also a delta sequence.
The delta sequences have approximate identities or summability kernels as other notions.
A pair of sequences ( , ) is called quotient of sequences; denoted by / , if ∈ 1 ( ) is a delta sequence and * = * , for all , ∈ ℕ.
The equivalence class of quotient of sequences is called as an Integrable Boehmian.
The space of all these Integrable Boehmians is denoted by 1 .
The relation between these two types of convergence is given in [12] as an equivalence. Hence, we can define the integral of a Boehmian as if The integral is same as Lebesgue integral for the function in 1 . However, continuously differentiable functions in 1 whose derivatives are not in 1 are Integrable as Boehmians but not Integrable as functions.

THE KAMAL TRANSFORM
The Kamal transform of the function ( ) defined in [1] is Here the constant M should be finite number, may be infinite and v is a variable of transform.
Off course these conditions are sufficient for existence of Kamal transform of ( ).    (ii) (v) If ∆ − lim = ℎ ̃→̃ uniformly on each compact set.
Proof: To prove (v) it is sufficient if we show that − lim = which implies that ̃→̃, converges uniformly on each compact set.
The case (i) to (iv) can be proved easily.  Hence we can also have ∆ − lim = . Hence, we can have the following characterization- exists and equal to 0 at each point then Γ(t) is continuous function.

CONCLUSION
The space 1 contains some elements which are not Schwartz distributions. The Kamal transform for Integrable Boehmians is obtained with some basic properties. An inversion theorem for Kamal transform is also discussed.