Form preserving linear integral operators
The present paper is concerned with integral operators of the following form \[ (Tf)(t)=\int^1_0K(t,s)f(s)ds,\quad t\in[0,1], \] where the kernel function \(K(t,s)\) is continuous on \([0,1] \times [0,1]\) and input functions are continuous on \([0,1]\). The authors are interested in characterizations of the kernel \(K(t,s)\) such that the corresponding operator \(T\) leaves various families of input functions invariant. Here several of these families including symmetric, skew symmetric, balanced, absolutely balanced functions, and functions with nonnegative symmetric sums are considered and characterizations of the kernel functions are given. In the last section several applications of the obtained results are mentioned. In addition, the authors point out that some results can be extended to square integrable kernel functions and integrable input functions.
form preserving linear integral operators
scientific article; zbMATH DE number 915118