User b-b - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T20:05:07Z http://mathoverflow.net/feeds/user/26382 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106895/extending-operator Extending operator B-B 2012-09-11T08:58:03Z 2012-10-08T10:01:40Z <p>Seeley's extension operator it is a linear continuous mapping $E: C^\infty([0, \infty)) \rightarrow C^\infty(\mathbb R)$ with the property $E(g)|_{[0, \infty))}=g$ for $g \in C^\infty([0,\infty))$. </p> <p>Let $(a_n), (b_n)$ are real sequences s.t.</p> <ul> <li><p>$b_n&lt;0$, $b_n \rightarrow -\infty$,</p></li> <li><p>$\sum_{n=1}^\infty |a_n| |b_n|^m &lt;\infty \ for m=0,1,2....$,</p></li> <li>$\sum_{n=1}^\infty a_n b_n^m =1 \ for \ m=0,1,2....$</li> </ul> <p>and let $h\in C_c(\mathbb R)$ be s. t. $h(x)=1$ for $x \in [0,1]$, $h(x)=0$ for $x \geq 2$. </p> <p>Operator $E(g)(x)=\sum_{n=1}^\infty a_n h(b_n x) g(b_n y) \ for \ x&lt;0$ and $E(g)(x)=g(x) \ for \ x\geq 0$ is an example of Seeley extension operator.</p> <p>Does there exist an operator $E$, with similar properties, from $C^\infty([0,c))$ into $C^\infty(-c,c)$ which is linear continuous s.t. $E(g)|_{[0, c))}=g$ for $g \in C^\infty ([0,c)$ ?</p> http://mathoverflow.net/questions/107980/convolution-of-sequences Convolution of sequences B-B 2012-09-24T14:16:33Z 2012-09-24T17:51:25Z <p>Let for given real sequences $(a_n)_{n \in \mathbb Z}, (b_n)_{n \in \mathbb Z}$, $c_n:=\sum_{k\in \mathbb Z} a_k b_{n-k}$ for $n \in \mathbb Z$ be the convolution of sequences $(a_n)$, $(b_n)$. </p> <p>For classical convolution if one of two functions is in $L^p$, the second in $L^q$, where $1\leq p,q &lt;\infty$ then their convolution $f*g$ is in $L^r$, where $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}-1$. Is it some similar type theorem true for convolution of sequences?</p> http://mathoverflow.net/questions/106895/extending-operator Comment by B-B B-B 2012-09-11T09:57:37Z 2012-09-11T09:57:37Z Thanks. In the first approach, by modyfing the proof, maybe there is a problem with finding seguences $a_n$, $b_n$ satisfying above conditions (with $b_n \rightarrow -c$ instead $-\infty$)?