DirchletKernel 2004-02-26 06:23:38 2007-07-07 03:27:59 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here The \emph{Dirichlet \PMlinkescapetext{kernel}} $D_n$ of order $n$ is defined as $$D_n(t)=\sum_{k=-n}^ne^{ikt}.$$ It can be represented as $$D_n(t)=\frac{\sin\left(n+\frac{1}{2}\right)t}{\sin\frac{t}{2}}.$$ \textbf{Proof:} It is \begin{align*} \sum_{k=-n}^ne^{ikt}&= e^{-int}\frac{1-e^{i(2n+1)t}}{1-e^{it}}\\ &=\frac{e^{i\left(n+\frac{1}{2}\right)t}-e^{-i\left(n+\frac{1}{2}\right)t}} {e^{i\frac{t}{2}}-e^{-i\frac{t}{2}}}\\ &=\frac{\sin\left(n+\frac{1}{2}\right)t}{\sin\frac{t}{2}}.\qquad\qquad\Box \end{align*} The Dirichlet kernel arises in the analysis of periodic functions because for any function $f$ of period $2\pi$, the convolution of $D_N$ and $f$ results in the Fourier-series approximation of order $n$: $$(D_N*f)(x)=\frac{1}{2\pi}\int_{-\pi}^\pi f(y)D_n(x-y)dy=\sum_{k=-n}^n\hat{f}(k)e^{ikx}.$$