The Fredholm integral equations of the first kind is a typical ill-posed problem, so that it is usually difficult to obtain its analytical minimal-norm solution. This paper gives a closed-form minimal-norm solution for the degenerate kernel equations based on the H-HK formulation. Furthermore, it has been shown that the structure of solutions to degenerate kernel equations and matrix equations are consistent. Subsequently, the obtained results are extended to non-degenerate integral equations. Finally, the validity and applicability of the proposed method are demonstrated by some examples.

翻译：第一类Fredholm积分方程是典型的不适定问题，通常难以获得其解析的极小范数解。本文基于H-HK公式给出了退化核方程的闭式极小范数解。进一步证明了退化核方程与矩阵方程的解结构具有一致性。随后，将所得结果推广至非退化积分方程。最后通过若干算例验证了所提方法的有效性与适用性。