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 $C>0$ ?
approximation-theory interpolation-theory
asked yesterday
Lin Xuelei
10310
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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 $C>0$ ?
approximation-theory interpolation-theory
asked yesterday
Lin Xuelei
10310
add a comment |
up vote
0
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 $C>0$ ?
approximation-theory interpolation-theory
asked yesterday
Lin Xuelei
10310
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
approximation-theory interpolation-theory
asked yesterday
Lin Xuelei
10310
asked yesterday
Lin Xuelei
10310
edited 6 hours ago
asked yesterday
Lin Xuelei
10310
asked yesterday
Lin Xuelei
10310
asked yesterday
Lin Xuelei
10310
10310
add a comment |
add a comment |
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