Convergence Rates for Regularized Solutions of Integral Equations from Discrete Noisy Data

Given data yᵢ = (Kg) (uᵢ) + ᵢ where the 's are random errors, the u's are known, g is an unknown function in a reproducing kernel space with kernel r and K is a known integral operator, it is shown how to calculate convergence rates for the regularized solution of the equation as the evaluation points \uᵢ\ become dense in the interval of interest. These rates are shown to depend on the eigenvalue asymptotics of KRK^, where R is the integral operator with kernel r. The theory is applied to Abel's equation and the estimation of particle size densities in stereology. Rates of convergence of regularized histogram estimates of the particle size density are given.