﻿ oipapio

### fa.functional analysis - Convolution in $\ell_p$ when $0<p<1$

BackgroundIt is known that given real sequences $a = (a_n)_{n \in \mathbb Z} \in \ell_p$ and $b = (b_n)_{n \in \mathbb Z} \in \ell_q$, their convolution defined as$$a * b (n) = \sum_{k \in \mathbb Z} a_{n-k} b_k$$is in $\ell_r$ if $1 \le p, q < \infty$ and $\frac 1 r = \frac 1 p + \frac 1 q -1$.QuestionWhat happens when $0 < p, q < 1$? Obviously, since $a$ and $b$ are in $\ell_1$, their convolution $a * b$ is in $\ell_1$. Can we say better, i.e. $a*b$ is in $\ell_r$ for some $r < 1$?...Read more

### fa.functional analysis - About the critical points of quasi-convex functions

What do we know about the structure of critical points of quasi-convex functions? I am looking for statements like "the critical points of a quasi-convex function are always either a global minima or a saddle point" or "for a quasi-convex function the number of negative eigenvalues of its Hessian at a critical point is constant across intervals of values of the function evaluated at the critical points" Are statements like this known?...Read more

### fa.functional analysis - solution uniqueness of non-linear Fredholm equations

the equation is $F(x)=G\left(\int k(x,y)f(y)dy\right)$ $(*)$where $f(x)=\frac{dF(x)}{dx}$ is unknown and it's required to be non-negative. With integration by parts we'll have the form of a non-linear Fredholm equation. In addition, k(x,y)>0 is square integrable, $k_x=\frac{dk(x,y)}{dx}<0$, $k_y=\frac{dk(x,y)}{dy}<0$; And $G$ is a differentiable, weakly increasing function ranging in $[0,1]$. I can prove uniqueness for the case when $k_{xy}>0$ and $G$ is convex (or conversely $k_{xy}<0$ while $G$ is concave). Yet I believe the resu...Read more