On the generalized fourier sine- and cosine-transforms

Some results concerning generalized Fourier sine- and cosine- transforms are discussed. The well known sine (or cosine) Fourier transform is an isometric mapping of L2 (0,∞) on itself (see [1]). It is interesting to consider the expression of a function with respect to sin(αξ + ϕ), where ϕ is a constant. Such an approach can be found in the papers of G.H. Hardy [2], R.G. Cooke [3], in [4] (the formula 7.10), and in a recent paper by A. Zilberglat and N. Lebedev [5]. In these works it was shown that an integrable function on (0, ∞) of a bounded variation over (0, ∞) can be repersented in the form of an integral of a hypergeometric function. In th Paper we consider generalized Fourier sine- and cosine- transforms of functions belonging to the space L2 (0, ∞). We shown the uniqueness and continuity of such an representation. Moreover, we obtain relations between formulae from the papers [2], [3] and [5].