Example for deterministic function with unbounded total variation and bounded quadratic variation - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T12:14:50Z http://mathoverflow.net/feeds/question/23104 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23104/example-for-deterministic-function-with-unbounded-total-variation-and-bounded-qua Example for deterministic function with unbounded total variation and bounded quadratic variation vonjd 2010-04-30T13:20:07Z 2010-04-30T14:20:36Z <p>It is well known that e.g. $sin(1/x)$ is of <a href="http://en.wikipedia.org/wiki/Bounded_variation#Examples" rel="nofollow">unbounded total variation</a> (in the interval [0,1] assuming $f(0)=0$). (Preliminary numerical tests suggest that) it is also of unbounded <a href="http://en.wikipedia.org/wiki/Quadratic_variation" rel="nofollow">quadratic variation</a>. $x\ sin(1/x)$ is of unbounded total variation too but (preliminary numerical tests suggest that) it is of zero quadratic variation. </p> <p><strong>My question:</strong> How to construct a deterministic function with unbounded total variation and bounded (non zero) quadratic variation along these lines? I don't want to have a function which is defined by a sum of terms (like the <a href="http://en.wikipedia.org/wiki/Weierstrass_function" rel="nofollow">Weierstrass function</a>) but one which is defined straight forward like the two above mentioned examples. References (if available) would also be appreciated!</p> <p><strong>Addendum:</strong> If some of these conjectures are not true please tell me. And please tell me also if it is not possible to construct such a deterministic function (and why not).</p> http://mathoverflow.net/questions/23104/example-for-deterministic-function-with-unbounded-total-variation-and-bounded-qua/23109#23109 Answer by Gerald Edgar for Example for deterministic function with unbounded total variation and bounded quadratic variation Gerald Edgar 2010-04-30T13:54:54Z 2010-04-30T14:20:36Z <p>Function $x^a\sin(1/x)$ on $(0,1]$ has bounded variation iff $a>1$ and finite quadratic variation iff $a>1/2$. So for your example, take $1/2 &lt; a \le 1$. It looks like more than numerical tests may be needed to decide questions like this... </p> <p>I took "quadratic variation" to mean $$\sup \sum_{j=1}^n |f(x_j)-f(x_{j-1})|^2$$ with sup over all finite sequences $x_0\lt x_1\lt\cdots\lt x_n$ in $(0,1]$. Perhaps you meant something else? </p> <p><strong>addition</strong><br> On the other hand, maybe you mean $$\lim_{\delta \to 0}\sup \sum_{j=1}^n |f(x_j)-f(x_{j-1})|^2$$ with sup over all finite sequences $x_0\lt x_1\lt\cdots\lt x_n$ in $(0,1]$ such that $x_j-x_{j-1}&lt;\delta$. In that case, on any inverval where $f$ is monotone it has quadratic variation $0$, so you won't find any such simple example.</p>