Range Theorems and Reconstruction Algorithms
A.G. Ramm, A.I. Katsevich · 2020
This chapter discusses the classical range theorem on smooth functions. It also discusses range theorem on the Sobolev spaces. The range theorem is reviewed in terms of the Fourier coefficients. A linear combination of exponentials is derived based on the Fourier coefficients. A complete linear metrizable topological vector space of functions is illustrated based on the linear combination of exponentials. Range theorem for X-ray transform is deduced based on smooth functions. Numerical solution of a specific equation with noisy data is provided. A positive elliptic pseudo-differential operator is utilized for determining the numerical solution of the equation having the noisy data.