How well can you approximate a function by a band-restricted function? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T01:53:01Z http://mathoverflow.net/feeds/question/111445 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/111445/how-well-can-you-approximate-a-function-by-a-band-restricted-function How well can you approximate a function by a band-restricted function? H A Helfgott 2012-11-04T10:39:17Z 2012-11-06T14:40:37Z <p>Say I have a compactly supported $C^1$ function $f:\mathbb{R} \to \mathbb{R}$. Let $R>0$. Let $\nu$ be some reasonable measure on $\mathbb{R}$ -- take, for instance, (a) $d\nu(t)=dt$ or (b) $d\nu(t)=e^{-t}$ for $t>0$ and $d\nu(t)=0$ for $t\leq 0$.</p> <p>Let $\delta(R)$ be the minimum of <code>$|f-\widehat{g}|_2 = \left( \int_\mathbb{R} |f(t)-\widehat{g}(t)|^2 d\nu(t)\right)^{1/2}$</code> over all functions $g:\mathbb{R} \to \mathbb{C}$ supported on $\lbrack -R,R\rbrack$.</p> <p>What is $\delta(R)$? How fast does it decrease as $R\to \infty$? Given $R$, can one construct a $g$ that attains the minimum?</p> <p>(A variation on the same question: allow measures, not just functions $g$, supported on $\lbrack -R,R\rbrack$.)</p> <p>Update: for $d\nu(t) = t$ this is very easy by isometry, as mentioned below; the minimum is attained for $g$ equal to the restriction of $\widehat{f}$ to $\lbrack -R,R\rbrack$ -- and so, if $f$ is in $C^k$, $\delta(R)$ decreases at least as fast as $1/R^{k-1}$ as $R\to \infty$. I am really more interested in the answers for the measure $\nu$ given in (b) above.</p>