Finite section convolution integral operatiors-structure of resolvent and solution of first kind equations

Abstract

This thesis investigates the structure of the solutions of Fredholm integral equations of the first and second kinds with convolution kernels on finite intervals and some of its applications. The thesis is divided into three chapters. In Chapter 1, we consider the second-kind equation: f(x)???0tk(x?y)f(y)?dy=g(x)(0.1)f(x) - \lambda \int_0^t k(x-y) f(y) \, dy = g(x) \tag{0.1}f(x)???0t?k(x?y)f(y)dy=g(x)(0.1) where t>0t > 0t>0, k(x)?L2[0,t]k(x) \in L^2[0,t]k(x)?L2[0,t], g(x)?L2[0,t]g(x) \in L^2[0,t]g(x)?L2[0,t], and f(x)?L2[0,t]f(x) \in L^2[0,t]f(x)?L2[0,t], fff being the unknown function. Let KKK denote the integral operator on L2[0,t]L^2[0,t]L2[0,t] defined as: (Kf)(x)=?0tk(x?y)f(y)?dy(0.2)(Kf)(x) = \int_0^t k(x-y) f(y) \, dy \tag{0.2}(Kf)(x)=?0t?k(x?y)f(y)dy(0.2) Then, the above equation (0.1) may be written as: (I??K)f=g.(I - \lambda K) f = g.(I??K)f=g. If ?\lambda? is not in the spectrum of the operator KKK, the resolvent (I??K)?1=R?(I - \lambda K)^{-1} = R_\lambda(I??K)?1=R?? exists as a well-defined, bounded linear operator on L2[0,t]L^2[0,t]L2[0,t]. We show that R?R_\lambdaR?? is determined by its action on the constant function and one more specially chosen function, y(x)y(x)y(x). In obtaining the representation for R?R_\lambdaR??, essential use is made of the following theorem: Theorem: Let TTT be any bounded linear operator on L2[0,t]L^2[0,t]L2[0,t]. Then there exists a kernel r(x,y)r(x,y)r(x,y) such that: (Tf)(x)=?0tr(x,y)f(y)?dy?f?L2[0,t].(Tf)(x) = \int_0^t r(x,y) f(y) \, dy \quad \forall f \in L^2[0,t].(Tf)(x)=?0t?r(x,y)f(y)dy?f?L2[0,t]. (i) For every fixed x?[0,t]x \in [0,t]x?[0,t], r(x,y)?L2[0,t]r(x,y) \in L^2[0,t]r(x,y)?L2[0,t] as a function of yyy. It is also shown that the representation for R?R_\lambdaR?? takes a simpler form in the case of self-adjoint operators; and that in the case of Volterra-type operators, it is determined by its action on the constant function alone. In Chapter 2, we apply the above results to obtain a generalization of the well-known Chandrasekhar-Ambartsumyan XXX and YYY functions occurring in radiative transfer ([1]; [2]), for a class of bounded linear operators on L2[0,t]L^2[0,t]L2[0,t]. We also obtain a generalization and representation for the scattering function S(z,u)S(z,u)S(z,u) in terms of the XXX and YYY functions, analogous to that in radiative transfer [2]. In Chapter 3, we get a representation for the solutions of the Fredholm integral equation of the first kind with convolution kernel on a finite interval, i.e., an integral equation of the form: ?0tk(x?y)f(y)?dy=g(x).\int_0^t k(x-y) f(y) \, dy = g(x).?0t?k(x?y)f(y)dy=g(x). Let W2={g?L2[0,t]:g?,g???L2[0,t]}W^2 = \{ g \in L^2[0,t] : g', g'' \in L^2[0,t] \}W2={g?L2[0,t]:g?,g???L2[0,t]}. Whenever g?W2g \in W^2g?W2, it is shown that the solution can be expressed in terms of the solutions for special right sides, namely g=1g = 1g=1 and g=xg = xg=x. The above representation is illustrated with an example of a classical kernel arising from applications, namely log??x?y?\log|x - y|log?x?y?.