The exact solution of system of linear operator equations in reproducing kernel spaces

In this paper, we discuss the problem of how to solve the system of linear operator equations in Hilbert spaces and applied the result to reproducing kernel spaces. Under the assumption that the operator equations has a unique solution, its formal solution is given. As a result, when H, H₁ are both reproducing kernel spaces, the formal solution turned into analytic solution. Truncating the series (where formal solution and the analytic solution are denoted by a series), the approximate solution of the operator equations is obtained. When increasing the number of the nodes, the error of the approximate solution is monotone decreasing in the sense of the norm of the Hilbert space. The final numerical experiment shows the efficiency of our method.