A Data-Driven Approach to Solving First-Kind Fredholm Integral Equations and Their Convergence Analysis

We investigate the statistical recovery of solutions to first-kind Fredholm integral equations with discrete, scattered, and noisy pointwise measurements. Assuming the forward operator's range belongs to the Sobolev space of order $m$, which implies algebraic singular-value decay $s_j\le Cj^{-m}$, we derive optimal upper bounds for the reconstruction error in the weak topology under an a priori choice of the regularization parameter. For bounded-variance noise, we establish mean-square error rates that explicitly quantify the dependence on sample size $n$, noise level $\sigma$, and smoothness index $m$; under sub-Gaussian noise, we strengthen these to exponential concentration bounds. The analysis yields an explicit a priori and a posteriori rule for the regularization parameter. Numerical experiments validate the theoretical results and demonstrate the efficiency of our practical parameter choice.