how to estimate maximum of Lebesgue function of arbitrary nodes?
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 Nov 13 at 5:29
Lin Xuelei
10810
add a comment |
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 Nov 13 at 5:29
Lin Xuelei
10810
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 Nov 13 at 5:29
Lin Xuelei
10810
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 Nov 13 at 5:29
Lin Xuelei
10810
asked Nov 13 at 5:29
Lin Xuelei
10810
edited Nov 14 at 12:58
asked Nov 13 at 5:29
Lin Xuelei
10810
asked Nov 13 at 5:29
Lin Xuelei
10810
asked Nov 13 at 5:29
Lin Xuelei
10810
10810
add a comment |
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2996349%2fhow-to-estimate-maximum-of-lebesgue-function-of-arbitrary-nodes%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
