Polynomials with prescribed points to match prescribed bounds - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T17:29:03Z http://mathoverflow.net/feeds/question/79921 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79921/polynomials-with-prescribed-points-to-match-prescribed-bounds Polynomials with prescribed points to match prescribed bounds Dirk 2011-11-03T12:58:14Z 2011-11-07T12:06:30Z <p>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).</p> <blockquote> <p>My question is:</p> <p>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 $x_1^-,\dots,x_r^-$?</p> </blockquote> <p>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.</p> <p><strong>Edit:</strong> In <a href="http://mathoverflow.net/questions/16673/finite-interpolation-by-a-nondecreasing-polynomial" rel="nofollow">this question</a> I leanerd from the answer of Gjergji Zaimi that there are bounds on the degree of <em>increasing</em> interpolating polyomials. How does the bounds change for monotone interpolation are described above? Are there (algebraical or numerical) methods to calculate the polynomial?</p> <p>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?</p>