基于正交函数的加权Landaus型不等式

The weighted Landaus type inequality based on the orthogonal function

Landau s型不等式在函数逼近论和不等式理论具有一定意义.Landau首先提出函数及其导数的范数之间的Landau s型不等式.Varma和Bojanov研究了基于Hermite函数的代数多项式的Landau s型不等式.对于一类经典的正交多项式Qn(x):这个多项式可以是Hermite函数Hn(x),也可以是广义的Laguerre函数L(ns)(x)(s>-1),或者是Jacobi多项式P(nα,)β(x)(α,β>-1),给出了统一的加权Landau s型不等式;利用Qn(x)正交性,建立了代数多项式pn(x)的加权Landau s型不等式,并且指出其不等式的系数在某种意义上是最好的.

Landau＇s type inequality has some contribution to approximation theory of function and theory of inequality. Landau is the first person who raised Landau＇s type inequality between norm for function and norm for derivative of function. Varma and Bojanov studied Landau＇s type inequality of algebraic polynomials based on Hermite function. In this paper, we propose a united weighted Landau＇s type inequality for classical orthogonal polynomials Qn （x）, Qn （x） can be Hermite function H. （x）, generalized Laguerre function Ln^（s） （x）（s〉-1）, or Jacobi function Pn^（α,β） （x） （α,β〉-1）. Making the best use of orthogonality of the Qn（x）, we establish weighted Landau＇s type inequality of algebraic polynomials pn（x）. We think that the coefficients in our inequality are the best in some sense.