Hermite插值算子及其误差的Hilbert变换表示

A Hilbert transform representation of Hermite interpolation and its error

插值算子与被插函数的误差有着诸多表示,如积分,差分表示等.论文将Hermite插值算子的误差表示与Hilbert变换结合起来,给出了该插值算子及其误差的Hilbert变换的表示.并由此误差表示,证明了某一类函数,即:在区间(-a,a)上解析,但在任一以-a,a为焦点的椭圆内不解析的函数,其Hermite插值算子的收敛性.

Error between function and its interpolation operator has many form, for example, the integral form, divided difference form, etc. In this paper, a new representation for the error of Hermite interpolation involving the Hilbert transforms is presented, and the convergence of Hermite interpolation to functions which are analytic in （-a, a） but not analytic in an ellipse with foci at -a, a is proved.