On Invertibility of the Radon Transform and Compressive Sensing

This thesis contains three articles. The first two concern inversion and local injectivity of the weighted Radon transform in the plane. The third paper concerns two of the key results from compressive sensing. In Paper A we prove an identity involving three singular double integrals. This is then used to prove an inversion formula for the weighted Radon transform, allowing all weight functions that have been considered previously. Paper B is devoted to stability estimates of the standard and weighted local Radon transform. The estimates will hold for functions that satisfy an a priori bound. When weights are involved they must solve a certain differential equation and fulfill some regularity assumptions. In Paper C we present some new constant bounds. Firstly we present a version of the theorem of uniform recovery of random sampling matrices, where explicit constants have not been presented before. Secondly we improve the condition when the so-called restricted isometry property implies the null space property.