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The Weierstrass function with $a = 1 / 2$ and $b = 15$

Example

The explicit series

$W (x) : = \sum_{n = 0}^{\infty} 2^{- n} \cos (15^{n} π x)$

converges uniformly on $R$ , is continuous at every real point, and is differentiable at no real point.

Facts & Assumptions

Given: The parameters $a = 1 / 2$ and $b = 15$ .

[L1]

The unique smallest positive zero $γ$ of cosine lies in $(0, 2)$ (Cosine has a smallest positive zero, lying strictly between zero and two).

[L2]

The number $π$ is defined by $π = 2 γ$ (Pi as twice the smallest positive zero of cosine).

[L3]

The classical Weierstrass function is $W_{a, b} (x) = \sum_{n = 0}^{\infty} a^{n} \cos (b^{n} π x)$ (The classical Weierstrass function).

[L4]

If $0 < a < 1$ , $b > 1$ is an odd integer, and $a b > 1 + 3 π / 2$ , then $W_{a, b}$ is continuous everywhere and differentiable nowhere (Under $a b > 1 + 3 π / 2$ , the classical Weierstrass function is continuous everywhere and differentiable nowhere).

[L5]

For $0 < a < 1$ and odd integer $b > 1$ , the defining Weierstrass series converges uniformly on $R$ (The classical Weierstrass series converges uniformly to a continuous function).

Verification

technique · direct

1.1

The integer $15$ is odd, $0 < 1 / 2 < 1$ , and $15 > 1$ . By [L1] and [L2], $π = 2 γ < 4$ , so $1 + \frac{3 π}{2} < 7 < \frac{15}{2} = a b .$

L1L2algebra

1.2

Substituting $a = 1 / 2$ and $b = 15$ in [L3] gives exactly the displayed series, including its $n = 0$ term.

L3construct

2.1

Step 1.1 verifies every hypothesis of [L4]. The series in step 1.2 converges uniformly by [L5], and [L4] makes its sum continuous everywhere and differentiable nowhere.

step 1.1step 1.2L4L5∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

36 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

Jeff Calder, Weierstrass's Non-Differentiable Function, Theorem 1 (standard reference, not scraped)

The Weierstrass function with a=1/2a=1/2 and b=15b=15