Reed, M. and Simon. B. (1980) Methods of Modern Mathematical Physics: Functional Analysis I. Revised and Enlarged Edition, Academic Press, San Diego.
has been cited by the following article:
TITLE: Construction of Equivalent Functions in Anisotropic Radon Tomography
AUTHORS: William Menke
KEYWORDS: Geotomography, Radon Transform, Acoustic Anisotropy, Non-Uniqueness, Null Space
JOURNAL NAME: Applied Mathematics, Vol.10 No.1, January 25, 2019
ABSTRACT: We consider a real-valued function on a plane of the form m(x,y,θ)=A(x,y)+Bc(x,y)cos(2θ)+Bs(x,y)sin(2θ)+Cc(x,y)cos(4θ)Cs(x,y)sin(4θ) that models anisotropic acoustic slowness (reciprocal velocity) perturbations. This “slowness function” depends on Cartesian coordinates and polar angle θ. The five anisotropic “component functions” A (x,y), Bc(x,y), Bs(x,y), Cc(x,y) and Cs(x,y) are assumed to be real-valued Schwartz functions. The “travel time” function d(u, θ) models the travel time perturbations on an indefinitely long straight-line observation path, where the line is parameterized by perpendicular distance u from the origin and polar angle θ; it is the Radon transform of m ( x, y, θ). We show that: 1) an A can always be found with the same d(u, θ) as an arbitrary (Bc,Bs) and/or an arbitrary (Cc,Cs) ; 2) a (Bc,Bs) can always be found with the same d(u, θ) as an arbitrary A, and furthermore, infinite families of them exist; 3) a (Cc,Cs) can always be found with the same d(u, θ) as an arbitrary A, and furthermore, infinite families of them exist; 4) a (Bc,Bs) can always be found with the same d(u, θ) as an arbitrary (Cc,Cs) , and vice versa; and furthermore, infinite families of them exist; and 5) given an arbitrary isotropic reference slowness function m0(x,y), “null coefficients” (Bc,Bs) can be constructed for which d(u, θ) is identically zero (and similarly for Cc,Cs ). We provide explicit methods of constructing each of these “equivalent functions”.