Convolutions and Toeplitz Operators - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T17:11:39Z http://mathoverflow.net/feeds/question/22697 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/22697/convolutions-and-toeplitz-operators Convolutions and Toeplitz Operators Leandro 2010-04-27T08:02:52Z 2010-07-01T05:11:13Z <p>Let be $d>0$ an integer number and consider the Cartesian product $\mathbb Z^d$ as metric space, with the distance between $x,y\in\mathbb Z^d$ given by <code>$\|x-y\|_1=\sum_{j=0}^d|x_j-y_j|$</code>. </p> <p>Let be $g:[0,\infty)\to\mathbb [0,\infty)$ a function having the two following properties:</p> <p>1) $\sum_{z\in \mathbb Z^d}g(\|z\|_1)$ is convergent;</p> <p>2) there is a positive constant $K\in \mathbb R$ (which depends only on $g$) such that for any $x,y\in\mathbb Z^d$, we have<br> $$\sum_{z\in\mathbb Z^d}g(\|x-z\|_1)g(\|z-y\|_1)\leq K g(\|x-y\|_1),$$</p> <p><strong> Question:</strong> Can we determine lower bounds for the ratio decay of $g(\|x-y\|_1)$ when $\|x-y\|_1$ goes to infinity ?</p> <p><strong> Examples: </strong></p> <p>Ex1: For any $\varepsilon>0$ $$g(\|z\|_1)=\frac{1}{1+\|z\|_1^{d+\varepsilon}}$$ has the properties 1 and 2. </p> <p>For the other hand, $$g(\|z\|_1)=e^{-r\|z\|_1},$$ where $r>0$, breaks the property 2.</p> <p><strong>Edit: </strong></p> <p>I added the Toeplitz operator tag, because of the asymptotic behavior for $g$ (in terms of the lower bounds) could be obtained thinking $g(\|x-y\|_1)$ as matrix elements of a Toeplitz operator $A:L^p(\mathbb Z^d,2^{\mathbb Z^d},\sharp)\to L^p(\mathbb Z^d,2^{\mathbb Z^d},\sharp)$. In fact, in this point of view, we ask for lower bounds for the entries of a Toeplitz operator satisfying <code>$(A^2)_{xy}\leq K A_{xy}$</code>, where <code>$(A^2)_{xy}$</code> is the $xy$ element of the matrix $A^2$.</p> <p>Remark: The issues pointed out by Thomas Kragh in the comments were fixed by not considering $g$ as a function of space $\mathbb Z^d$ and requiring it to be positive.</p> <p>Any reference or help, even for partial answer is very welcome. </p>