Hessians of Fourier transforms of positive radial functions - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T11:41:16Z http://mathoverflow.net/feeds/question/103457 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103457/hessians-of-fourier-transforms-of-positive-radial-functions Hessians of Fourier transforms of positive radial functions Liviu Nicolaescu 2012-07-29T16:21:44Z 2012-07-30T13:42:49Z <p>$\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\ii}{\boldsymbol{i}}$ $\newcommand{\eW}{\mathscr{W}}$ </p> <p>While investigating the distribution of critical points of random funtions on tori I was lead to the following analytical questions. Suppose that $w :\bR\to [0,\infty)$ is a Schwartz function. Define</p> <p>$$f: \bR\to \bR,\;\; f(t)=w(t^2).$$</p> <p>We can use the function $f$ to define for any positive integer $m$ a Schwartz function</p> <p>$$F_m:\bR^m\to \bR,\;\;F_m(\vec{x})=f(|\vec{x}|)=w(|\vec{x}|^2).$$</p> <p>We denote by $\widehat{F}_m(\vec{\xi})$ its Fourier transform and by $H_m(w, \vec{\xi})$ the Hessian of $\widehat{F}_m$ at $\vec{\xi}$.</p> <p><strong>Problem 1.</strong> </p> <blockquote> <p>Fix a positive integer $m$. Describe the set $\eW_m$ of weights $w$ such that</p> <p>$$H_m(w,\xi) > H_m(w, 0), \;\; \forall \vec{\xi}\neq 0.$$</p> </blockquote> <p>Above, for two symmetric matrices $A$, $B$ the inequality $A&lt; B$ signifies that $B-A$ is positive definite. Hence $w\in\eW_m$ if an only if the origin is a strict, absolute minimum for the Hessian map $$\vec{\xi}\to H_m (w,\vec{\xi}).$$</p> <p><strong>Problem 2.</strong> </p> <blockquote> <p>Describe the set</p> <p>$$\eW:=\bigcap_{m>0}\eW_m.$$</p> </blockquote> <p>Here is some information about $\eW$. </p> <p><strong>A.</strong> For any positive Schwartz function and any $m>0$ we have</p> <p>$$H_m(w,\vec{x})> H_m(w,0)$$</p> <p>if $|\vec{\xi}|$ is sufficiently small, or sufficiently large. In other words, the origin is always a local minimum of the Hessian map. Thus the problem is about what happens in between.</p> <p><strong>B.</strong> For any $c>0$ we have $w(s)=e^{-c s/2}\in\eW$. Indeed $f(t)=e^{-ct^2/2}$</p> <p>$$\widehat{F}_m(\vec{\xi})=const_m e^{-\frac{|\vec{\xi}|^2}{2c}}.$$</p> <p>E.g., for $m=1$ we have $$\frac{d^2}{dt^2} e^{-t^2/2}=(t^2-1) e^{-t^2/2} > -1 = \frac{d^2}{dt^2} e^{-t^2/2}|_{t=0}.$$</p> <p><strong>C.</strong> $\eW$ is a convex cone. In particular, any linear combination</p> <p>$$w(t)=\sum_i A_i e^{-c_it/2},\;\;A_i>0,$$</p> <p>belongs to $\eW$. This implies that if the Schwartz function $w(t)$ is the Laplace transform of a positive finite measure $\mu$ on $[0,\infty)$, then $w(t)\in\eW$. The classical <em>Hausdorff-Bernstein theorem</em> shows that this happens if and only if $w$ is completely monotone, i.e.,</p> <p>$$(-1)^k w^{(k)}(t)\geq 0,\;;\forall t>0, \;\;\forall k\in\mathbb{Z}_{\geq 0}.$$</p> <p>Thus $\eW$ contains all the completely monotone Schwartz functions.</p> <p>Here is a plausible</p> <p><strong>Conjecture</strong> (a) The weight $w$ belongs to $\eW$ if and only if $w$ is completely monotone.(Compare this with Schoenberg's theorem characterising completely monotone functions.)</p> <p>(b) $\eW_m \neq \eW$, $\forall m$.</p>