Definition:Delta Sequence

Definition

Let $\sequence {\map {\delta_n} x}_{n \mathop \in \N}$ be a sequence of continuous real-valued functions.

Let $\phi \in \map {C^\infty} \R$ be a smooth function with a compact support.

Suppose:

$\ds \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \map {\delta_n} x \map \phi x = \map \phi 0$

Then $\sequence {\map {\delta_n} x}_{n \mathop \in \N}$ is called the delta sequence.

Further research is required in order to fill out the details. In particular: Need to check whether conditions on the sequence, $\phi$, and normalization can be relaxed. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by finding out more. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Research}} from the code.

Notes

The name reflects the fact that functions $\map {\delta_n} x$ in the distributional sense approach Dirac delta distribution as $n$ approaches infinity.

However, in the sense of ordinary functions the sequence $\map {\delta_n} x$ has no limit because Riemann integrable Dirac function does not exist.

Also see

• Results about the delta sequence can be found here.

Sources

• 1970: George Arfken: Mathematical Methods for Physicists (2nd ed.): Chapter $8$. Second-order differential equations $8.6$ Nonhomogeneous equation -- Green's function