how to solve first two questions and how to start it?

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). Ty (t cos 0, t sin 0) Lt,0 (1) For f E S(R2), define the X-ray transform of f to be the function xf (t, 0 ) := / f = / f(tcos0 + ssino, tsino-scos0) ds, (t, 0) ER x (-7, TT]. Lt , 0 Compute the X-ray Transform of the 2-D Gaussian, G(x, y) = e-T(x2ty?). (2) For a fixed 0 E (-7, 7], let Xf($, 0) be the 1-dimensional Fourier Transform of the function X f(t, 0) in the t variable. Prove the following identity: x f (5,0) = f(5 cos 0, { sin 0) for all & E R and 0 E (-7, T]. 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 @) and 71 := (sin 0, - cos 0), then Et = $7 . (ty + syl) for all , t, s ER. 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(R2) satisfies X f (t, 0) = 0 for all (t, 0) ERx (-7, T], then f is identically zero on R2. This shows the map f - Xf is injective