please answer this question it is about fourier analysis

The purpose of this assignment is to study an application of the Fourier Transform to medical imaging (see Ch.6, Section 5 of Stein-Shakarchi for context). Given t E R and 0 E (-7, 7], consider the line Lto = {(x, y) ER2 : x = tcos0 + ssin0, y = tsin0 - scos0, s ER). (t cos 0, t sin 0) & V Lt,e (1) For f E S(R2), define the X-ray transform of f to be the function xf (t, 0) := / f = / f(tcos0 + ssin0,tsin0 - scos0) ds, (t,0) ERx (-7, 7). It.0 Compute the X-ray Transform of the 2-D Gaussian, G(x, y) = e-7(22+y?). (2) For a fixed 0 E (-7, 7], let Xf(,0) be the 1-dimensional Fourier Transform of the function Xf(t, 0) in the t variable. Prove the following identity: xf(,0) = f(E cos0, { sin0) for all & E R and 0 E (-7, 7] Note that the Fourier Transform on the right-hand side is the 2-dimensional Fourier Transform of the function f(x, y). HINT: Note that if 7 := (cos 0, sin 0) and 71 := (sin 0, - cos 0), then Et = $7 . (ty+ syl) for all &, t, s E R. Use this observation to convert the 1-dimensional Fourier Transform to a 2-dimensional one. (3) Using the identity in (2), prove that if f E S(IR2) satisfies X f (t, 0) = 0 for all (t, 0) E Rx (-7, "], then f is identically zero on R2. This shows the map f - Xf is injective