Stone-Weierstrass for monotone functions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T07:08:03Z http://mathoverflow.net/feeds/question/85376 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/85376/stone-weierstrass-for-monotone-functions Stone-Weierstrass for monotone functions Ricky Demer 2012-01-10T23:46:44Z 2012-05-04T20:33:40Z <p>Let $\; f : [0,1] \to \mathbb{R} \;$ be continuous and non-decreasing. $\;\;$ Let $\epsilon$ be a real number such that $\; 0 &lt; \epsilon \;$. <br> Does it follow that that there exists a real polynomial $p$ such that $p$ is non-decreasing on</p> <ol> <li>$\;$ $[0,1]$ <br></li> <li>$\;$ all of $\mathbb{R}$</li> </ol> <p>and for all members $x$ of $[0,1]$, $\; |f(x)+(-(p(x)))| &lt; \epsilon \;\;$?</p> http://mathoverflow.net/questions/85376/stone-weierstrass-for-monotone-functions/85392#85392 Answer by Anatoly Kochubei for Stone-Weierstrass for monotone functions Anatoly Kochubei 2012-01-11T06:49:53Z 2012-01-11T06:49:53Z <p>In fact this question is a simple prototype of a serious problem of approximation maintaining additional qualitative properties of a function, with precise error estimates. See <a href="http://mathworld.wolfram.com/ComonotoneApproximation.html" rel="nofollow">http://mathworld.wolfram.com/ComonotoneApproximation.html</a> for the case of piecewise monotone functions. There are many problems and results of this kind (for example, with convex functions), which are useful in engineering applications and are far from trivial. </p> http://mathoverflow.net/questions/85376/stone-weierstrass-for-monotone-functions/96020#96020 Answer by Algernon for Stone-Weierstrass for monotone functions Algernon 2012-05-04T20:33:40Z 2012-05-04T20:33:40Z <p>In fact, the <a href="http://en.wikipedia.org/wiki/Bernstein_polynomial" rel="nofollow">Bernstein polynomials</a> approximating $f$ are non-decreasing on $[0,1]$. A cute way to see this is via coupling (I learned this from Lindvall's book <a href="http://books.google.nl/books?id=GB290HEW724C&amp;lpg=PA142&amp;hl=nl&amp;pg=PA145#v=onepage&amp;q&amp;f=false" rel="nofollow">Lectures on the Coupling Method</a>):</p> <p>The $n$th Bernstein polynomial $p_n(x)$ can be written as $\mathbf{E}\Big[f\big(\frac{\sum_{i=1}^n Z^x_i}{n}\big)\Big]$, where $Z^x_i$ are Bernoulli random variables with parameter $x$. If $0\leq x\leq y\leq 1$, then we can define the variables $Z^x_i$ and $Z^y_i$ on the same probability space, such that $Z^x_i\leq Z^y_i$, which immediately gives $p_n(x)\leq p_n(y)$.</p>