Difference between revisions of "Knopp function"
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(Created page with "Let $a \in (0,1), ab > 4,$ and $b>1$ an even integer. Define the Knopp function $K \colon \mathbb{R} \rightarrow \mathbb{R}$ by $$K_{a,b}(x)=\displaystyle\sum_{k=0}^{\infty} a...") |
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| − | Let $a \in (0,1) | + | Let $a \in (0,1)$ $ab > 1$. Define the Knopp function $K \colon \mathbb{R} \rightarrow \mathbb{R}$ by |
| − | $$K_{a,b}(x)=\displaystyle\sum_{k=0}^{\infty} a^k \ | + | $$K_{a,b}(x)=\displaystyle\sum_{k=0}^{\infty} a^k \mathrm{dist}_{\mathbb{Z}} \left( b^k x \right),$$ |
| − | where $\ | + | where $\mathrm{dist}_{\mathbb{Z}}$ denotes the [[distance to integers]] function. |
=Properties= | =Properties= | ||
| − | + | [[Knopp function is continuous]]<br /> | |
| − | + | [[Knopp function is nowhere differentiable]]<br /> | |
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| − | </ | ||
| − | + | =See Also= | |
| − | + | [[Takagi function]]<br /> | |
| − | + | [[van der Waerden function]] | |
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=References= | =References= | ||
[https://pure.ltu.se/ws/files/30923977/LTU-EX-03320-SE.pdf] | [https://pure.ltu.se/ws/files/30923977/LTU-EX-03320-SE.pdf] | ||
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| + | [[Category:SpecialFunction]] | ||
Latest revision as of 03:31, 27 October 2016
Let $a \in (0,1)$ $ab > 1$. Define the Knopp function $K \colon \mathbb{R} \rightarrow \mathbb{R}$ by $$K_{a,b}(x)=\displaystyle\sum_{k=0}^{\infty} a^k \mathrm{dist}_{\mathbb{Z}} \left( b^k x \right),$$ where $\mathrm{dist}_{\mathbb{Z}}$ denotes the distance to integers function.
Properties
Knopp function is continuous
Knopp function is nowhere differentiable
See Also
Takagi function
van der Waerden function
