# Definition:Delta Sequence

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## 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.

## Sources

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