Delta Sequence

A Sequence of strongly peaked functions for which

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

so that in the limit as $n \to \infty$, the sequences become Delta Functions. Examples include

$$\delta_n(x) = \left\{\begin{array}{ll} 0 & \mbox{$x < - {1\over 2n}$}\\ n & \m... ...\over 2n} < x < {1\over 2n}$}\\ 0 & \mbox{$x > {1\over 2n}$}\end{array}\right.$$ (2)

$$= {n\over \sqrt{\pi}} e^{-n^2x^2}$$ (3)

$$= {n\over \pi} {\rm sinc}(ax) \equiv {\sin(nx)\over \pi x}$$ (4)

$$= {1\over \pi x}{e^{inx}-e^{-inx}\over 2i}$$ (5)

$$= {1\over 2\pi i x}\left[{e^{ixt}}\right]_{-n}^n$$ (6)

$$= {1\over 2\pi}\int_{-n}^n e^{ixt}\,dt$$ (7)

$$= {1\over 2\pi} {\sin\left[{\left({n+{1\over 2}}\right)x}\right]\over \sin \left({{1\over 2} x}\right)},$$ (8)

where (8) is known as the Dirichlet Kernel.