In this paper, the best approximate solution of Fredholm integral equations of the first kind with some scattered observations is studied. An explicit approximate solution has been obtained by our proposed method and converges to the exact solution with minimum norm of the integral equations with probability 1, which is identical with the solution by means of the regularization method. In addition, based on the H-$H_k$ formulation, the gap between the range space and its closure is fully characterized, in which the infinite reproducing kernel Hilbert spaces are strictly embedded. We prove that if the right-hand term of the integral equation lies in this gap, the $H$-norm of the best approximate solution will diverge to infinity, as the number of observations increases. Finally, an integral equation confirms our conclusions.

本文研究了第一类Fredholm积分方程在一些分散观测下的最佳近似解。通过我们提出的方法得到了一个显式的近似解，并以概率1收敛到积分方程组的最小范数的精确解，这与正则化方法的解相同。此外，基于H -$H_{ķ}$公式中，范围空间及其闭包之间的差距被充分表征，其中无限再现核希尔伯特空间被严格嵌入。我们证明，如果积分方程的右手项位于这个间隙中，则$H$随着观察次数的增加，最佳近似解的范数将发散到无穷大。最后，一个积分方程证实了我们的结论。