ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 2026-09-01

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Mollifying $1_{[0, 1]}$ at two scales

Example

Let $φ \in C_{c}^{\infty} (R)$ satisfy $\int φ = 1$ and $supp (φ) \subseteq [- 1, 1]$ . For $f : = 1_{[0, 1]}$ and $ε > 0$ ,

$(f * φ_{ε}) (x) = \int_{0}^{1} φ_{ε} (x - y) d y = \int_{(x - 1) / ε}^{x / ε} φ (u) d u .$

For $ε = 1 / 4$ and $ε = 1 / 8$ , the graph is $0$ outside $[- ε, 1 + ε]$ , equals $1$ on $[ε, 1 - ε]$ , and has only the two rounded boundary layers of width $ε$ .

Facts & Assumptions

Given: A compactly supported unit-mass bump $φ$ and $f = 1_{[0, 1]}$ .

[L1]

Mollifier families are approximate identities (A unit-mass smooth bump generates an $L^{1}$ approximate identity).

[L2]

Convolution with a mollifier is smooth (Convolution with a mollifier is smooth, and derivatives pass under the integral sign).

[L3]

Approximate identities converge uniformly on compacta for bounded continuous functions; on the interior plateau here the integral is exactly one by direct support control ( $L^{1}$ approximate identities converge uniformly on compacta for bounded continuous functions).

Verification

technique · direct

1.1

The change of variables $u = (x - y) / ε$ gives the displayed formula. [L1, given, algebra] Because $supp (φ_{ε}) \subseteq [- ε, ε]$ , the convolution vanishes unless the interval $[0, 1]$ meets $[x - ε, x + ε]$ , namely unless $x \in [- ε, 1 + ε]$ .

L1givenalgebra

2.1

If $x \in [ε, 1 - ε]$ , then the whole support of [L1, step 1.1, algebra] $φ_{ε} (x - \cdot)$ lies inside $[0, 1]$ , so $(f * φ_{ε}) (x) = \int φ_{ε} = 1.$ Near $x = 0$ and $x = 1$ , only part of the kernel fits inside $[0, 1]$ , producing the two smooth transition layers.

L1step 1.1algebra

3.1

By [L2], every $f * φ_{ε}$ is smooth; the cases [L2, L3, step 2.1] $ε = 1 / 4$ and $ε = 1 / 8$ differ only in the width of the two boundary layers, with the smaller $ε$ giving the sharper transition.

L2L3step 2.1∎

Depends on

Used by

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Dependency tree · two levels

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Sources

Richard L. Wheeden and Antoni Zygmund, Measure and Integral: An Introduction to Real Analysis (standard reference, not scraped)

Mollifying 1[0,1]1[0,1]​ at two scales