Trigonometrical approximation for the characteristic function of an interval - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T22:11:35Z http://mathoverflow.net/feeds/question/77336 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/77336/trigonometrical-approximation-for-the-characteristic-function-of-an-interval Trigonometrical approximation for the characteristic function of an interval Emmanuel Preissmann 2011-10-06T07:23:49Z 2011-11-22T23:00:17Z <p>Hello, </p> <p>Denoting $e(x)$ for $e^{2i\pi x}$, set</p> <p><code>$$E(R):=\left\{f\ \left|\ f(x)=\sum_{r=0}^{R-1}a_re(rx)\mbox{ where }a_r\in\mathbb{C}\ \forall r\mbox{ and }\sum_r|a_r|^2=1\right\}\right.$$</code></p> <p>$h(a,R):=\inf_{f\in E(R)}\int_0^a|f(x)|^2dx$ with $0\leq a\leq 1$ .</p> <p>I am curious to know the behavior of $h(a,R)$ particularly if $R\rightarrow\infty$ </p> <p>Thank you in advance for an idea!</p> http://mathoverflow.net/questions/77336/trigonometrical-approximation-for-the-characteristic-function-of-an-interval/81663#81663 Answer by Ben Adcock for Trigonometrical approximation for the characteristic function of an interval Ben Adcock 2011-11-22T23:00:17Z 2011-11-22T23:00:17Z <p>This is directly related to the so-called prolate matrix (see, for example, this <a href="http://www.sciencedirect.com/science/article/pii/002437959390142B" rel="nofollow">paper</a>), and specifically, its minimal eigenvalue. To see this, note that $$\int^a_0 | f(x) |^2 dx = \sum^{R-1}_{r,s=0} a_r \overline{a_s} \int^{a}_{0} e^{2 \pi i (r-s) x} d x.$$ Let $b_r = e^{\pi i r a} a_r$, and write $A = \{ A_{r-s} \}^{R-1}_{r,s=0}$ for the prolate matrix, i.e. $$A_0 = a,\qquad A_r = \frac{\sin \pi r a}{\pi r},\ r \neq 0.$$ If $b=(b_0,\ldots,b_{R-1})$ then $$h(a,R) = \min_{\sum_r |b_r|^2 = 1} b^* A b = \lambda_{\min}(A).$$ Eqn (3.2) of the above paper now gives the exact formula for $h(a,R)$.</p>