The paper by Erhard Schmidt, titled "Development of Arbitrary Functions into Systems of Prescribed Functions," is part of a series on the theory of linear and nonlinear integral equations. The introduction reviews Fredholm's solution to the inhomogeneous linear integral equation and Hilbert's extension of this to the homogeneous equation, introducing the concept of eigenfunctions and eigenvalues. Hilbert's work on the existence of normal functions and the development of arbitrary functions into them is highlighted, along with the introduction of the Green's function. The paper then delves into theorems related to the existence of eigenfunctions for symmetric kernels and the development of arbitrary functions into them. It also discusses the approximation of functions using eigenfunctions and the conditions for the approximation to be optimal. The paper concludes with a discussion on the extension of these results to nonlinear integral equations and their applications in partial and ordinary differential equations.The paper by Erhard Schmidt, titled "Development of Arbitrary Functions into Systems of Prescribed Functions," is part of a series on the theory of linear and nonlinear integral equations. The introduction reviews Fredholm's solution to the inhomogeneous linear integral equation and Hilbert's extension of this to the homogeneous equation, introducing the concept of eigenfunctions and eigenvalues. Hilbert's work on the existence of normal functions and the development of arbitrary functions into them is highlighted, along with the introduction of the Green's function. The paper then delves into theorems related to the existence of eigenfunctions for symmetric kernels and the development of arbitrary functions into them. It also discusses the approximation of functions using eigenfunctions and the conditions for the approximation to be optimal. The paper concludes with a discussion on the extension of these results to nonlinear integral equations and their applications in partial and ordinary differential equations.