Fractional Fourier Integral Theorem and Fractional Fourier Sine and Cosine Transform

FRACTIONAL FOURIER INTEGRAL THEOREM AND FRACTIONAL FOURIER SINE AND COSINE TRANSFORM Saleem Iqbal, S.M. Raza, * LalaRukh Kamal and Farhana Sarwar Department of Mathematics/Physics, University of Balochistan, Quetta, Pakistan e-mail: fs1005,saleemiqbal81,[email protected]. ABSTRACT: The fractional Fourier transform (FRFT) is a generalization of the ordinary Fourier transform (FT). Recently several properties of FRFT have been developed by generalizing the properties of the ordinary FT. In this paper we show that a fractional Fourier integral theorem (FRFIT) can be derived by using the inversion formula of the FRFT in a similar way as for the ordinary FT. We also derive the Fractional Fourier cosine and sine transforms by using the FRFIT in a similar way as one can find the Fourier cosine and sine transform from Fourier integral theorem of the ordinary FT. We also show that fractional Fourier cosine and sine transforms are the generalized form of Fourier cosine transform (FCT) and Fourier sine transform (FST) and possesses all the properties of the FRFT.