how to estimate maximum of Lebesgue function of arbitrary nodes?











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Denote $S_n=left{x_{n}=(x_{n0},x_{n1},...,x_{nn})|aleq x_{n0}<x_{n1}<cdots<x_{nn}leq bright}$ with $-infty<a<b<+infty$ and $ngeq 1$. For any $x_nin S_n$, define the Lebesgue function $L_{x_n}(x)$ by
$$
L_{x_n}(x):=sumlimits_{i=0}^{n}left|frac{prodlimits_{0leq jleq n,~jneq i}(x-x_{nj})}{prodlimits_{0leq jleq n,~jneq i}(x_{ni}-x_{nj})}right|.
$$

How to show that $inflimits_{x_nin S_n}maxlimits_{xin[a,b]}L_{x_n}(x)geq Clog n$ for some constant $C>0$ ?










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    down vote

    favorite












    Denote $S_n=left{x_{n}=(x_{n0},x_{n1},...,x_{nn})|aleq x_{n0}<x_{n1}<cdots<x_{nn}leq bright}$ with $-infty<a<b<+infty$ and $ngeq 1$. For any $x_nin S_n$, define the Lebesgue function $L_{x_n}(x)$ by
    $$
    L_{x_n}(x):=sumlimits_{i=0}^{n}left|frac{prodlimits_{0leq jleq n,~jneq i}(x-x_{nj})}{prodlimits_{0leq jleq n,~jneq i}(x_{ni}-x_{nj})}right|.
    $$

    How to show that $inflimits_{x_nin S_n}maxlimits_{xin[a,b]}L_{x_n}(x)geq Clog n$ for some constant $C>0$ ?










    share|cite|improve this question


























      up vote
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      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Denote $S_n=left{x_{n}=(x_{n0},x_{n1},...,x_{nn})|aleq x_{n0}<x_{n1}<cdots<x_{nn}leq bright}$ with $-infty<a<b<+infty$ and $ngeq 1$. For any $x_nin S_n$, define the Lebesgue function $L_{x_n}(x)$ by
      $$
      L_{x_n}(x):=sumlimits_{i=0}^{n}left|frac{prodlimits_{0leq jleq n,~jneq i}(x-x_{nj})}{prodlimits_{0leq jleq n,~jneq i}(x_{ni}-x_{nj})}right|.
      $$

      How to show that $inflimits_{x_nin S_n}maxlimits_{xin[a,b]}L_{x_n}(x)geq Clog n$ for some constant $C>0$ ?










      share|cite|improve this question















      Denote $S_n=left{x_{n}=(x_{n0},x_{n1},...,x_{nn})|aleq x_{n0}<x_{n1}<cdots<x_{nn}leq bright}$ with $-infty<a<b<+infty$ and $ngeq 1$. For any $x_nin S_n$, define the Lebesgue function $L_{x_n}(x)$ by
      $$
      L_{x_n}(x):=sumlimits_{i=0}^{n}left|frac{prodlimits_{0leq jleq n,~jneq i}(x-x_{nj})}{prodlimits_{0leq jleq n,~jneq i}(x_{ni}-x_{nj})}right|.
      $$

      How to show that $inflimits_{x_nin S_n}maxlimits_{xin[a,b]}L_{x_n}(x)geq Clog n$ for some constant $C>0$ ?







      approximation-theory interpolation-theory






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      edited 6 hours ago

























      asked yesterday









      Lin Xuelei

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