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Todd Trimble
Todd Trimble
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For every table of interpolating nodes, there is a +vepositive continuous function whose interpolating polynomials are not +ve i.opositive infinitely often

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user521337
user521337
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For every table of interpolating nodes, there is a +ve continuous function whose interpolating polynomials are not +ve i.o.

Fix an interval $[a,b]$. Is it true that for every table of interpolating nodes $\{x_{0,n},x_{1,n}...,x_{n,n}\}_{n=1}^{\infty}$, there exists a continuous function $f:[a,b]\to (0,\infty)$ such that the sequence of interpolating polynomials $p_n(x)$ of $f$ at those points satisfy $(\infty,0]\cap p_n([a,b])\ne \phi$$(-\infty,0]\cap p_n([a,b])\ne \phi$ for infinitely many $n$ ?

NOTE: If this is true, then it would imply that for every table of interpolating nodes $\{x_{0,n},x_{1,n}...,x_{n,n}\}_{n=1}^{\infty}$, there is a continuous function $f:[a,b]\to (0,\infty)$ such that the sequence of interpolating polynomials $p_n(x)$ of $f$ does not converge uniformly to $f$ on $[a,b]$. i.e. a positive answer to my question implies Faber's theorem.

For every table of interpolating nodes, there is a +ve continuous function whose interpolating polynomials are not +ve i.o.

Fix an interval $[a,b]$. Is it true that for every table of interpolating nodes $\{x_{0,n},x_{1,n}...,x_{n,n}\}_{n=1}^{\infty}$, there exists a continuous function $f:[a,b]\to (0,\infty)$ such that the sequence of interpolating polynomials $p_n(x)$ of $f$ at those points satisfy $(\infty,0]\cap p_n([a,b])\ne \phi$ for infinitely many $n$ ?

NOTE: If this is true, then it would imply that for every table of interpolating nodes $\{x_{0,n},x_{1,n}...,x_{n,n}\}_{n=1}^{\infty}$, there is a continuous function $f:[a,b]\to (0,\infty)$ such that the sequence of interpolating polynomials $p_n(x)$ of $f$ does not converge uniformly to $f$ on $[a,b]$. i.e. a positive answer to my question implies Faber's theorem.

For every table of interpolating nodes, there is a +ve continuous function whose interpolating polynomials are not +ve i.o

Fix an interval $[a,b]$. Is it true that for every table of interpolating nodes $\{x_{0,n},x_{1,n}...,x_{n,n}\}_{n=1}^{\infty}$, there exists a continuous function $f:[a,b]\to (0,\infty)$ such that the sequence of interpolating polynomials $p_n(x)$ of $f$ at those points satisfy $(-\infty,0]\cap p_n([a,b])\ne \phi$ for infinitely many $n$ ?

NOTE: If this is true, then it would imply that for every table of interpolating nodes $\{x_{0,n},x_{1,n}...,x_{n,n}\}_{n=1}^{\infty}$, there is a continuous function $f:[a,b]\to (0,\infty)$ such that the sequence of interpolating polynomials $p_n(x)$ of $f$ does not converge uniformly to $f$ on $[a,b]$. i.e. a positive answer to my question implies Faber's theorem.

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user521337
user521337
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For every table of interpolating nodes, there is a +ve continuous function whose interpolating polynomials are not +ve i.o.

Fix an interval $[a,b]$. Is it true that for every table of interpolating nodes $\{x_{0,n},x_{1,n}...,x_{n,n}\}_{n=1}^{\infty}$, there exists a continuous function $f:[a,b]\to (0,\infty)$ such that the sequence of interpolating polynomials $p_n(x)$ of $f$ at those points satisfy $(\infty,0]\cap p_n([a,b])\ne \phi$ for infinitely many $n$ ?

NOTE: If this is true, then it would imply that for every table of interpolating nodes $\{x_{0,n},x_{1,n}...,x_{n,n}\}_{n=1}^{\infty}$, there is a continuous function $f:[a,b]\to (0,\infty)$ such that the sequence of interpolating polynomials $p_n(x)$ of $f$ does not converge uniformly to $f$ on $[a,b]$. i.e. a positive answer to my question implies Faber's theorem.