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Delta Sequence

A Sequence of strongly peaked functions for which

\begin{displaymath}
\lim_{n\to \infty} \int_{-\infty}^\infty \delta_n(x)f(x)\,dx = f(n)
\end{displaymath} (1)

so that in the limit as $n \to \infty $, the sequences become Delta Functions. Examples include
$\displaystyle \delta_n(x)$ $\textstyle =$ $\displaystyle \left\{\begin{array}{ll} 0 & \mbox{$x < - {1\over 2n}$}\\  n & \m...
...\over 2n} < x < {1\over 2n}$}\\  0 & \mbox{$x > {1\over 2n}$}\end{array}\right.$ (2)
  $\textstyle =$ $\displaystyle {n\over \sqrt{\pi}} e^{-n^2x^2}$ (3)
  $\textstyle =$ $\displaystyle {n\over \pi} {\rm sinc}(ax) \equiv {\sin(nx)\over \pi x}$ (4)
  $\textstyle =$ $\displaystyle {1\over \pi x}{e^{inx}-e^{-inx}\over 2i}$ (5)
  $\textstyle =$ $\displaystyle {1\over 2\pi i x}\left[{e^{ixt}}\right]_{-n}^n$ (6)
  $\textstyle =$ $\displaystyle {1\over 2\pi}\int_{-n}^n e^{ixt}\,dt$ (7)
  $\textstyle =$ $\displaystyle {1\over 2\pi} {\sin\left[{\left({n+{1\over 2}}\right)x}\right]\over \sin \left({{1\over 2} x}\right)},$ (8)

where (8) is known as the Dirichlet Kernel.




© 1996-9 Eric W. Weisstein
1999-05-24