求解无穷线性方程组

利用再生核空间讨论了无穷线性方程组的求解,给出了无穷线性方程组Ay=b精确解的表达式.假定A是l2→l2的有界线性算子,建立l2和再生核空间的1-1映射,将方程Ay=b转化为再生核空间中的方程Ku=f,给出Ku=f的精确解u的表达式;最后给出无穷线性方程组的精确解.实际数值计算中,因为方程Ku=f的精确解是以级数形式给出的,级数截断得到近似解,从而得到无穷线性方程组Ay=b的近似解.还给出了无穷线性方程组有解的充分必要条件.

Solving the infinite system of linear equations

It is discussed how to solve the infinite system of linear equations by using methods of reproducing kernel space.A representation of the exact solution for the infinite system of linear equations Ay=b is presented.For a bounded linear operator A from l~2 onto l~2,a one-to-one mapping from l~2 to a reproducing kernel space is established.This map transforms Ay=b into Ku=f which is an operator equation on a reproducing kernel space.Give the representation of the exact solution for equation Ku=f.Finally,the representation of the exact solution for the infinite system of linear equations is given.In practical numerical calculation,since exact solution of Ku=f is given in the form of series,we obtain an approximate solution of Ku=f by truncating the series,and thereby we can get an approximate solution of Ay=b.Moreover,a sufficient and necessary condition for the existence the infinite system of linear equations is given.