ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passverified 2026-09-23 (gpt-6-sol)

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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.
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$1_{[0, 1]} * 1_{[0, 1]}$ is the tent function

Example

Let $f : = 1_{[0, 1]}$ on $R$ . Then

$(f * f) (x) = \int_{R} 1_{[0, 1]} (x - y) 1_{[0, 1]} (y) d y$

is the tent function

$(f * f) (x) = {\begin{cases} 0, & x \leq 0, \\ x, & 0 \leq x \leq 1, \\ 2 - x, & 1 \leq x \leq 2, \\ 0, & x \geq 2. \end{cases}$

Facts & Assumptions

Given: The indicator $f = 1_{[0, 1]}$ .

Verification

technique · direct

1.1givenalgebra

For fixed $x$ , the integrand is $1$ exactly when $y \in [0, 1] \cap [x - 1, x]$ and zero otherwise. This bounded-support integrand is integrable for every $x$ , and $(f * f) (x)$ is the length of that overlap interval.

2.1step 1.1algebra

If $0 \leq x \leq 1$ , the overlap is $[0, x]$ , so $(f * f) (x) = x$ . If [step 1.1, algebra] $1 \leq x \leq 2$ , the overlap is $[x - 1, 1]$ , so $(f * f) (x) = 2 - x$ . For $x \leq 0$ or $x \geq 2$ , there is no overlap, so $(f * f) (x) = 0$ .

3.1step 2.1∎

The explicit formula is positive precisely on $(0, 2)$ , so its support is $[0, 2] = [0, 1] + [0, 1]$ , as claimed by the displayed tent function.

Used by

Nothing in the library uses this result yet.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Walter Rudin, Real and Complex Analysis, 3rd ed. (standard reference, not scraped)

1[0,1]∗1[0,1]1[0,1]​∗1[0,1]​ is the tent function