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The Gaussian family is an $L^{1}$ approximate identity

Example

For $ε > 0$ , define the normalized Gaussian on $R^{n}$ by

$G_{ε} (x) : = (2 π ε^{2})^{- n / 2} e^{- ∣ x ∣^{2} / (2 ε^{2})} .$

Then $(G_{ε})_{ε > 0}$ is an $L^{1}$ approximate identity.

Facts & Assumptions

Given: The Gaussian family $(G_{ε})$ .

[L1]

An $L^{1}$ approximate identity is defined in An $L^{1}$ approximate identity on $R^{n}$ .

Verification

technique · direct

1.1L1givenalgebra

The change of variables $u = x / ε$ gives [L1, given, algebra] $\int_{R^{n}} G_{ε} (x) d x = \int_{R^{n}} (2 π)^{- n / 2} e^{- ∣ u ∣^{2} / 2} d u = 1,$ so every kernel has mass one and $∥ G_{ε} ∥_{1} = 1$ .

2.1step 1.1algebra

For every $δ > 0$ , [step 1.1, algebra] $\int_{∣ x ∣ > δ} G_{ε} (x) d x = \int_{∣ u ∣ > δ / ε} (2 π)^{- n / 2} e^{- ∣ u ∣^{2} / 2} d u ⟶ 0$ as $ε \to 0^{+}$ , because the integration region escapes to infinity against an integrable Gaussian tail.

3.1L1step 1.1step 2.1∎

Steps 1.1 and 2.1 verify the defining clauses of [L1], so the Gaussian [L1, step 1.1, step 2.1] family is an $L^{1}$ approximate identity even though it is not compactly supported.

Depends on

An $L^1$ approximate identity on $\mathbb{R}^n$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Terence Tao, An Introduction to Measure Theory (standard reference, not scraped)

The Gaussian family is an L1L1 approximate identity