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Consider real polynomials on the interval $I=[-1,1]$. It is easy to see that the smallest degree for a non-negative polynomial with given zeros $x_1,\dots,x_s\in I^\circ$ is $n=2s$ (e.g. $P(x) = \prod_{i=1}^s (x-x_i)^2$ works).

My question is:

What is the smallest degree for a polynomial such that it is bounded by $\pm 1$ on $I$ and attains the value $1$ on a set $x_1^+,\dots,x_s^+$ and the value $-1$ on a set $x_1^-,\dots,x_r^-$?

Background: I know that the fact about nonnegative polynomials with presribed zeros can be generalized to "generalized polynomials" built from Tchebycheff-systems (due to a theorem by Krein). I would love to see a similar theorem on bounded generalized polynomials which attain the bounds at prescribed points.

Edit: In this question I leanerd from the answer of Gjergji Zaimi that there are bounds on the degree of increasing interpolating polyomials. How does the bounds change for monotone interpolation are described above? Are the there (algebraical or numerical) methods to calculate the polynomial?

It seems to me that monotone interpolating polynomials are not treated in the current literature and are not subject of current research. Is that right, and if so is there a special reason for that?

3 corrected title.

# Polynomials with with prescribed points to match prescribed bounds

Consider real polynomials on the interval $I=[-1,1]$. It is easy to see that the smallest degree for a non-negative polynomial with given zeros $x_1,\dots,x_s\in I^\circ$ is $n=2s$ (e.g. $P(x) = \prod_{i=1}^s (x-x_i)^2$ works).

My question is:

What is the smallest degree for a polynomial such that it is bounded by $\pm 1$ on $I$ and attains the value $1$ on a set $x_1^+,\dots,x_s^+$ and the value $-1$ on a set $x_1^-,\dots,x_r^-$?

Background: I know that the fact about nonnegative polynomials with presribed zeros can be generalized to "generalized polynomials" built from Tchebycheff-systems (due to a theorem by Krein). I would love to see a similar theorem on bounded generalized polynomials which attain the bounds at prescribed points.

Edit: In this question I leanerd from the answer of Gjergji Zaimi that there are bounds on the degree of increasing interpolating polyomials. How does the bounds change for monotone interpolation are described above? Are the (algebraical or numerical) method methods to calculate the polynomial?

It seems to me that monotone interpolating polynomials are not treated in the current literature and are not subject of current research. Is that right, and if so is there a special reason for that?

Consider real polynomials on the interval $I=[-1,1]$. It is easy to see that the smallest degree for a non-negative polynomial with given zeros $x_1,\dots,x_s\in I^\circ$ is $n=2s$ (e.g. $P(x) = \prod_{i=1}^s (x-x_i)^2$ works).

My question is:

What is the smallest degree for a polynomial such that it is bounded by $\pm 1$ on $I$ and attains the value $1$ on a set $x_1^+,\dots,x_s^+$ and the value $-1$ on a set $x_1^-,\dots,x_r^-$?

Background: I know that the fact about nonnegative polynomials with presribed zeros can be generalized to "generalized polynomials" built from Tchebycheff-systems (due to a theorem by Krein). I would love to see a similar theorem on bounded generalized polynomials which attain the bounds at prescribed points.

Edit: In this question I leanerd from the answer of Gjergji Zaimi that there are bounds on the degree of increasing interpolating polyomials. How does the bounds change for monotone interpolation are described above? Are the (algebraical or numerical) method to calculate the polynomial?

It seems to me that monotone interpolating polynomials are not treated in the current literature and are not subject of current research. Is that right, and if so is there a special reason for that?

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