|
|
|
Viewing Version
2
of
'Fejer kernel'
|
[ view 'Fejer kernel'
|
back to history
]
| Title of object: |
Fejer kernel |
| Canonical Name: |
FejerKernel |
| Type: |
Definition |
| Created on: |
2004-02-26 06:44:32 |
| Modified on: |
2004-04-14 02:59:49 |
| Classification: |
msc:26A30 |
Preamble:
% 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 |
Content:
The Fejer kernel $F_n$ of order $n$ is defined as
$$F_n(t)=\frac{1}{n}\sum_{k=0}^{n-1}D_k(t),$$
where $D_n$ is the Dirichlet kernel of order $n$. The Fejer kernel can be written as
\begin{equation}\label{eq:rep}
F_n(t)=\frac{1}{n}\left(\frac{\sin\frac{nt}{2}}{\sin\frac{t}{2}}\right)^2.
\end{equation}
\textbf{Proof:} Since
$$D_n(t)=\frac{\sin\left(\left(n+\frac{1}{2}\right)t\right)}{\sin\frac{t}{2}}$$
we have
$$\sin\frac{t}{2}D_n(t)=\sin\left(\left(n+\frac{1}{2}\right)t\right).$$
Therefore
\begin{align*}
n\sin^2\frac{t}{2}F_n(t)& =\sum_{k=0}^{n-1}\sin\left(\left(k+\frac{1}{2}\right)t\right)\sin\frac{t}{2}\\
&=\frac{1}{2}\sum_{k=0}^{n-1}(\cos kt-\cos((k+1)t)\\
&=\frac{1}{2}(1-\cos nt)\\
&=\sin^2\frac{nt}{2}.
\end{align*}
From this follows equation (\ref{eq:rep}). |
|
|
|
|
|
