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The dyadic simple approximants for $x^{2}$ on $[0, 2]$ at $k = 1$ and $k = 2$

Example

For $f (x) = x^{2}$ on $[0, 2]$ , the explicit dyadic truncations are:

$s_{1} (x) = {\begin{cases} 0, & 0 \leq x < 1 / \sqrt{2}, \\ 1 / 2, & 1 / \sqrt{2} \leq x < 1, \\ 1, & 1 \leq x \leq 2, \end{cases}$

and

$s_{2} (x) = {\begin{cases} 0, & 0 \leq x < 1 / 2, \\ 1 / 4, & 1 / 2 \leq x < \sqrt{2} / 2, \\ 1 / 2, & \sqrt{2} / 2 \leq x < \sqrt{3} / 2, \\ 3 / 4, & \sqrt{3} / 2 \leq x < 1, \\ 1, & 1 \leq x < \sqrt{5} / 2, \\ 5 / 4, & \sqrt{5} / 2 \leq x < \sqrt{6} / 2, \\ 3 / 2, & \sqrt{6} / 2 \leq x < \sqrt{7} / 2, \\ 7 / 4, & \sqrt{7} / 2 \leq x < \sqrt{2}, \\ 2, & \sqrt{2} \leq x \leq 2. \end{cases}$

Facts & Assumptions

Given: The nonnegative measurable function $f (x) = x^{2}$ on $[0, 2]$ and the explicit dyadic truncations from the simple-approximation theorem.

[L1]

For a nonnegative measurable function, the approximants are

$s_{k} = \sum_{j < k 2^{k}} j 2^{- k} 1_{{j 2^{- k} \leq f < (j + 1) 2^{- k}}} + k 1_{{f \geq k}} .$

(Every nonnegative measurable function admits an explicit increasing sequence of simple approximations)

Verification

technique · direct

1.1

For $k = 1$ , the two dyadic cells are [L1, algebra] $0 \leq x^{2} < 1 / 2$ and $1 / 2 \leq x^{2} < 1$ , while the truncation cell is $x^{2} \geq 1$ . On $[0, 2]$ these become exactly the three intervals displayed in the first formula.

L1algebra

2.1

For $k = 2$ , the eight nontrivial dyadic cells are [step 1.1, L1, algebra] $j / 4 \leq x^{2} < (j + 1) / 4$ for $j = 0, \dots, 7$ , together with the truncation cell $x^{2} \geq 2$ . On $[0, 2]$ these become $[0, 1 / 2)$ , $[1 / 2, \sqrt{2} / 2)$ , $[\sqrt{2} / 2, \sqrt{3} / 2)$ , $[\sqrt{3} / 2, 1)$ , $[1, \sqrt{5} / 2)$ , $[\sqrt{5} / 2, \sqrt{6} / 2)$ , $[\sqrt{6} / 2, \sqrt{7} / 2)$ , $[\sqrt{7} / 2, \sqrt{2})$ , and $[\sqrt{2}, 2]$ , exactly as displayed.

step 1.1L1algebra∎

Depends on

Every nonnegative measurable function admits an explicit increasing sequence of simple approximations

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

Sheldon Axler, Measure, Integration and Real Analysis, Theorem 2.89 (standard reference, not scraped)

The dyadic simple approximants for x2x2 on [0,2][0,2] at k=1k=1 and k=2k=2