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.
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