User angelo lucia - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T09:56:05Z http://mathoverflow.net/feeds/user/16988 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/126411/convolution-in-ell-p-when-0p1 Convolution in $\ell_p$ when $0<p<1$ Angelo Lucia 2013-04-03T16:31:53Z 2013-04-09T13:44:52Z <h2>Background</h2> <p><a href="http://mathoverflow.net/questions/107980/convolution-of-sequences" rel="nofollow">It is known</a> 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 $1 \le p, q &lt; \infty$ and $\frac 1 r = \frac 1 p + \frac 1 q -1$.</p> <h2>Question</h2> <p>What happens when $0 &lt; p, q &lt; 1$? 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 $r &lt; 1$?</p> http://mathoverflow.net/questions/22255/usefulness-of-frechet-versus-gateaux-differentiability-or-something-in-between/72771#72771 Answer by Angelo Lucia for Usefulness of Frechet versus Gateaux differentiability or something in between. Angelo Lucia 2011-08-12T14:06:38Z 2011-08-12T14:06:38Z <p>I know this question has been inactive for such a long time, but I will try to add something.</p> <p>Dieudonne in <em>Treatise on analysis</em> defines an "intermediate" concept of differentiability between Fréchet and Gâteaux ones called <strong>quasi-differentiability</strong>. Using your notation, <code>$V$</code> is quasi-differentiable at <code>$x$</code> if <code>$V \circ g$</code> is differentiable at 0, for all <code>$g : [0,1] \to L$</code> which are continuous, right-differentiable in <code>$0$</code>, and <code>$g(0) = x$</code>.</p> <p>(I am not sure about it, but i think you could caracterize quasi-differentiability in terms of uniform convergence on compact sets, i.e. the limit <code>$\lim_{t\to 0} \frac{V(x+tu) - V(x)}{t}$</code> is uniform on compact sets.)</p> <p>If <code>$L$</code> is finite dimensional, then quasi-differentiability implies Fréchet differentiability. If <code>$V$</code> is Lipschitz, then Gâteaux differentiability implies quasi-differentiability.</p> <p>Nice thing about quasi-differentiability is that the theorem of differentiation of composite functions holds (wich is not the case with Gâteaux differentiability). Moreover, if <code>$F$</code> is quasi-differentiable and <code>$G$</code> is Gâteaux differentiable, then <code>$F \circ G$</code> is G-differentiable.</p> <p>Unfortunately, it's possible to construct a function <code>$F : L^2 \to L^2$</code> which is everywhere quasi-differentiable, its differential is a right inverse (thus injective), but is nowhere locally injective. This is not possible if <code>$F$</code> is Fréchet differentiable.</p> http://mathoverflow.net/questions/72419/a-good-book-of-functional-analysis/72452#72452 Answer by Angelo Lucia for A good book of functional analysis Angelo Lucia 2011-08-09T07:54:57Z 2011-08-09T07:54:57Z <p>Apart from the <em>classics</em> already mentioned (Yosida, Brezis, Rudin), a good book of functional analysis that I think is suitable not only as a reference but also for self-study, is Fabian, Habala et al. <em>Functional Analysis and Infinite-Dimensional Geometry</em>. It has a lot of nice exercises, it's less abstract than the usual book and provides a lot of "concrete" theorems.</p> <p>And I'm not sure about it, but I heard there is a spanish translation (the original is of course in english).</p> http://mathoverflow.net/questions/72232/equivalent-metrics-on-frechet-spaces-and-lipschitz-maps Equivalent metrics on Fréchet spaces and Lipschitz maps Angelo Lucia 2011-08-06T10:47:08Z 2011-08-08T07:52:46Z <p>Lipschitz maps are defined over metric space as maps <code>$f:(X,d_X) \to (Y,d_Y)$</code> such that <code>$$d\left( f(x),f(x^\prime) \right)_Y \le k d(x,x^\prime)_X \ \forall x,x^\prime \in X,$$</code> where <code>$k$</code> is a positive constant. We usually say that <code>$f$</code> is a <em>contraction</em> if <code>$k&lt;1$</code>.</p> <p>It is well know that a different equivalent metric on $X$ does not preserve contractions, i.e. a map can be a contraction with respect to a metric but not with respect to an equivalent one.</p> <p>In the Banach space setting, where the spaces <code>$X$</code> and <code>$Y$</code> are endowed with a norm defining the topology, there is a somehow "canonical" distance given by <code>$$d(x,x^\prime) = \lVert x-x^\prime \rVert .$$</code> With this distance, Lipschitz maps can be characterized as maps satisfying, for some <code>$k&gt;0$</code> <code>$${\left\lVert f(x) - f(x^\prime) \right\rVert}_Y \le k {\left \lVert x-x^\prime \right \rVert}_X \ \forall x, x^\prime \in X.$$</code> It is obvious that, if <code>$f$</code> satisfies the above relation, then is a <code>$k$</code>-lipschitz map.</p> <hr> <p>In the Fréchet space setting, the topology is defined by a countable family of semi-norms <code>$({\lVert\cdot\rVert}_n)$</code>. The classical example of metric inducing the same topology is given by <code>$$d(x,x^\prime) = \sum_{n=0}^\infty {2^{-n}} \frac{{\lVert x-x^\prime\rVert}_n}{1+{\lVert x-x^\prime\rVert}_n} .$$</code></p> <p>In analogy with the Banach case, I would like to characterize (at least some) Lipschtiz maps between Fréchet spaces as maps satisfying <code>$${\left\lVert f(x) - f(x^\prime) \right\rVert}_n \le k {\left \lVert x-x^\prime \right \rVert}_n \ \forall x, x^\prime \in X,\ \forall n \in \mathbb{N}.$$</code> Again, maps satisfying the last equation <strong>are</strong> Lipschitz maps with respect to the metric defined above, but the Lipschitz constant is not <code>$k$</code> anymore, and in particular contraction with respect to the semi-norms (i.e. maps satisfying the last equation with <code>$k&lt;1$</code>) are not contraction with respect to the metric.</p> <p>Are there equivalent distances on <code>$X$</code> and <code>$Y$</code> such that <em>every</em> contraction with respect to the semi-norms is a contraction with respect with the new distance? If this is not possibile for <em>every</em> contraction, is it possible for a specific one?</p> http://mathoverflow.net/questions/126411/convolution-in-ell-p-when-0p1/126964#126964 Comment by Angelo Lucia Angelo Lucia 2013-04-09T13:36:43Z 2013-04-09T13:36:43Z Just one one: should it be &quot;Fubini&quot;, not &quot;Funini&quot;? http://mathoverflow.net/questions/126411/convolution-in-ell-p-when-0p1 Comment by Angelo Lucia Angelo Lucia 2013-04-09T10:42:51Z 2013-04-09T10:42:51Z Looks like the optimal exponent is exactly max$(p,q)$. To see this, consider two strictly positive sequences: $a_n, b_n &gt;0$. Then trivially $$a*b(n) = \sum_{k\in \mathbb Z} a_{n-k} b_k \ge a_n b_0 ;$$ and similarly $a*b(n) \ge b_n a_0$. Then $a*b(n)$ cannot go to zero faster than the slower between $a$ and $b$. @Davide: if you want, you should post an answer. http://mathoverflow.net/questions/72232/equivalent-metrics-on-frechet-spaces-and-lipschitz-maps/72249#72249 Comment by Angelo Lucia Angelo Lucia 2011-08-08T08:39:48Z 2011-08-08T08:39:48Z ...but it still has a unique fixed point, right? http://mathoverflow.net/questions/72232/equivalent-metrics-on-frechet-spaces-and-lipschitz-maps/72249#72249 Comment by Angelo Lucia Angelo Lucia 2011-08-07T21:29:41Z 2011-08-07T21:29:41Z I meant <code>$\sum 2^{-n} (\left\lVert x-y \right\rVert \wedge k)$</code> on <code>$Y$</code>. http://mathoverflow.net/questions/72232/equivalent-metrics-on-frechet-spaces-and-lipschitz-maps/72249#72249 Comment by Angelo Lucia Angelo Lucia 2011-08-07T20:03:10Z 2011-08-07T20:03:10Z I understand the idea, but I would say that if I take <code>$\sum 2^{-n} ({\left\lVert x-y\right\rVert} \wedge 1)$</code> on <code>$X$</code>, I need <code>$\sum 2^{-n} ({\left\lVert x-y\right\rVert} \wedge 1/k)$</code> on <code>$Y$</code>. Anyway... what about maps from <code>$X$</code> into himself? Are maps <code>$f:X\toX$</code> satisfying <code>$\left\lVert f(x) -f(y)\right\rVert \le k \left \lVert x-y \right \rVert$</code>, with <code>$k&lt;1$</code>, distance-contraction (and thus have a unique fixed point)? maybe this is a different question...