The kernel function plays an important role in the 1-D problem because of the spectral representation of the electric potential for a stratified model with a point source. Functional analysis establishes the equivalence between the differential equation (which governs the kernel function) and a boundary integral equation, called a jump summation equation for the 1-D case. In this equation, the jumps of the weighted Wronskian of two distinct models are summed over all the singular points. Numerous applications of this general equation demonstrate its flexibility. An appropriate choice of models and of the weight function leads to two splitting theorems and two imbedding theorems. The basic idea is to split the stratification into two models for the splitting theorems and into three models for the imbedding theorems. An application of these theorems concerns the handling of underground and underwater sounding measurements. Three possible configurations are examined and their performances are compared. With these examples, a simple method of layer stripping is introduced in the kernel space. These theorems are also used to establish the shift properties for a given set of layers surrounded by two homogeneous half-spaces. The consequences of these shift properties especially concern electrical tomography, where a case of equivalence is shown. The general character of these theorems may generate other applications.