Pointwise bounds for Dirichlet kernel over truncated lattice - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T01:46:39Z http://mathoverflow.net/feeds/question/76592 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/76592/pointwise-bounds-for-dirichlet-kernel-over-truncated-lattice Pointwise bounds for Dirichlet kernel over truncated lattice Marcin Kotowski 2011-09-28T02:28:38Z 2011-09-28T02:28:38Z <p>In 1 dimension, a "one-sided" Dirichlet kernel $D_N(x)=\sum_{k=0}^{k=N}e^{\frac{2\pi}{N}ikx}$ has its module sharply peaked around points corresponding, roughly, to the "dual lattice" $N\cdot\mathbb{Z}$ (which area really integer points in this case).</p> <p>Now consider a lattice $\Lambda \subseteq \mathbb{Z}^2$. Let $\Lambda_N = \Lambda \cap [0,N]^2$ and a Dirichlet kernel restricted to this truncated lattice: $D_{\Lambda_N}(x)=\sum_{v \in \Lambda_N}e^{i\langle x,v\rangle}$.</p> <p>We would expect that $|D_{\Lambda_N}|$ will be peaked around points from $N\cdot\Lambda^{\ast}$, where $\Lambda^{\ast}$ is the dual lattice. However, there is no direct way to perform the summation and obtain a compact formula as in 1D case. Is there any way to estimate $|D_{\Lambda_N}(w)|$ from below, for $w \approx N\cdot\Lambda^{\ast}$? </p>