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.

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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

• 1964: I.M. Gel'fand and G.E. Shilov: Generalized Functions, Volume I: Properties and Operations: Chapter $1$. Section $2.5$. Delta-Convergent Sequences
• 1970: George Arfken: Mathematical Methods for Physicists (2nd ed.): Chapter $8$. Second-order differential equations $8.6$ Nonhomogeneous equation -- Green's function