Modified Fourier transform and its properties

In this manuscript we recommend a new description of the modified Fourier transform for a function which is absolutely integrable, having finite number of maxima and minima and finite number of discontinuities which further takes the form of simple Fourier transform for substituting α = e where α > 0 and α ≠ 1. Moreover we prove various results of the modified Fourier transform and also we show that the set that consists of whole modified Fourier transformable functions under the convolution operation is a commutative semi group as well as form an abelian group under the operation of addition