New classes of linear many-dimensional Volterra-type equations of the first kind (Q1921702)

The author studies systems that can be put in the Volterra series form \[ y(t) = \sum^\infty_{m = 1} \int^t_0 \cdots \int^t_0 K_m (t,s_1, \dots, s_m) \prod^m_{i = 1} x(s_i) ds_i, \quad t \in [0,T], \] where \(x(t)\) is the input and \(y(t)\) is the output. The problem considered is how one can find the functions \(K_m\), in particular what test functions one should take as input and how one can then derive the kernels from the output. Both the time invariant case (that leads to convolution equations) and the nonstationary case are considered. The equations used to solve these problems can be considered as special cases of equations of the form \(V \varphi = f\) studied by the author.