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I have a homework problem which consists of two parts, the first of which I have been staring at for several days with very little (constructive) progress.

I need to show that the function $f(t) = \sum_{n=1}^{\infty}\frac{\cos(3^{n}t)}{3^{n\alpha}}\in\Lambda^{\alpha}$ when $0 < \alpha \leq 1$. The second part is to show that if $\alpha < \beta < 1$, then $f\notin\Lambda^{\beta}$, but I'll worry about the second part later.

I tried considering $ \begin{eqnarray*} |f(t+h) - f(t)| &=& \left|\sum_{n=1}^{\infty}\frac{\cos(3^{n}(t + h))}{3^{n\alpha}} - \sum_{n=1}^{\infty}\frac{\cos(3^{n}t)}{3^{n\alpha}}\right|\\ &=& \left|\sum_{n=1}^{\infty}\frac{\cos(3^{n}t + 3^{n}h)}{3^{n\alpha}} - \sum_{n=1}^{\infty}\frac{\cos(3^{n}t)}{3^{n\alpha}}\right|\\ &=& \left|\sum_{n=1}^{\infty}\frac{\cos(3^{n}t)\cos(3^{n}h) - \sin(3^{n}t)\sin(3^{n}h)}{3^{n\alpha}} - \sum_{n=1}^{\infty}\frac{\cos(3^{n}t)}{3^{n\alpha}}\right|\\ &=& \left|\sum_{n=1}^{\infty}\frac{\cos(3^{n}t)\cos(3^{n}h) - \sin(3^{n}t)\sin(3^{n}h)}{3^{n\alpha}} - \sum_{n=1}^{\infty}\frac{\cos(3^{n}t)}{3^{n\alpha}}\right|\\ &=& \left|\sum_{n=1}^{\infty}\frac{\cos(3^{n}t)\cos(3^{n}h) - \sin(3^{n}t)\sin(3^{n}h) - \cos(3^{n}t)}{3^{n\alpha}}\right|\\ &\leq& \sum_{n=1}^{\infty}\left|\frac{\cos(3^{n}t)\cos(3^{n}h) - \sin(3^{n}t)\sin(3^{n}h) - \cos(3^{n}t)}{3^{n\alpha}}\right|\\ \end{eqnarray*} $

EDIT: Removed the last half - dozen lines which turned out to be completely non-constructive.

Now I'm not sure if I'm even remotely close to going down the right path, but if I could get this manipulated into something of the form $C^{\alpha}$ I'd be done. But I just can't seem to go any further. Any suggestions?

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    Right. In fact the $h$ was lost a long time ago. So taking the $sin/cos$ functions and bounding them by $1$ is going too far I guess.2011-10-05

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Use the Mean Value Theorem on the terms with $n: $ \begin{align} |f(t+h)-f(t)|&\le\sum_{n=1}^\infty\left|\frac{\cos(3^n(t+h))-\cos(3^nt)}{3^{n\alpha}}\right|\\ &=\sum_{n=1}^{N-1}\left|\frac{3^nh\sin(3^n(t+\eta_n))}{3^{n\alpha}}\right|+\sum_{n=N}^\infty\left|\frac{\cos(3^n(t+h))-\cos(3^nt)}{3^{n\alpha}}\right|\\ &\le|h|\frac{3^{N(1-\alpha)}-3^{1-\alpha}}{3^{1-\alpha}-1}+2\frac{\frac{1}{3^{N\alpha}}}{1-\frac{1}{3^{\alpha}}}\tag{1} \end{align} $ Choose $N$ so that $|h|\sim3^{-N}$. Then $|h|3^{N(1-\alpha)}\sim|h|^\alpha$ and $\frac{1}{3^{N\alpha}}\sim|h|^\alpha$. Thus, the right side of $(1)\sim|h|^\alpha$.

To be more precise, let $N=\lfloor\log_3(\frac{1}{h})\rfloor$. Then $|h|3^{N(1-\alpha)}\le|h|^\alpha$ and $\frac{1}{3^{N\alpha}}\le3|h|^\alpha$. Thus, $ |f(t+h)-f(t)|\le\left(\frac{1}{3^{1-\alpha}-1}+\frac{6}{1-\frac{1}{3^{\alpha}}}\right)|h|^\alpha\tag{2} $ Note that the $\Lambda_\alpha$-norm in $(2)$ blows up near $\alpha=0$ and $\alpha=1$.

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    Well when you put it like that, how can I disagree? :P I guess I'm just so used to working with things between $|\cdot|$ signs that I imagine them there when they aren't. Thanks again.2011-10-07