Nice orthonormal basis for L^2(Cantor set) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T14:21:29Z http://mathoverflow.net/feeds/question/34402 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/34402/nice-orthonormal-basis-for-l2cantor-set Nice orthonormal basis for L^2(Cantor set) Alvin Kerber 2010-08-03T15:22:43Z 2010-08-03T19:10:10Z <p>Let X be the Cantor set, which we view as the space $2^\mathbb{N}$ (the set of all infinite binary sequences), equipped with the product topology. We can construct a Borel probability measure $\mu$ on this space by defining $\mu(C_{a_i})=1/2$, where the $C_{a_i}=\{x\in X | x_i=a_i\}$ are the open subbase cylinders of the product topology, and extending to a $\sigma$-algebra in the standard fashion.</p> <p>Now, consider the Hilbert space $L^2(X,\mu)$. We can obtain orthonormal bases for it using the measure-space isomorphism between $(X,\mu)$ and $[0,1]$ (with Lesbesgue measure) via the binary decimal representations of real numbers. However, the ordinary bases (e.g., the trigonometric basis) on $L^2([0,1])$ are quite ugly when viewed on the Cantor set.</p> <blockquote> <p>Is there an orthonormal basis for $L^2(X,\mu)$ with nice properties (continuity? simply expressible functions?) relative to the structure of the Cantor set?</p> </blockquote> http://mathoverflow.net/questions/34402/nice-orthonormal-basis-for-l2cantor-set/34406#34406 Answer by Willie Wong for Nice orthonormal basis for L^2(Cantor set) Willie Wong 2010-08-03T15:39:32Z 2010-08-03T15:39:32Z <p>Perhaps the Walsh functions? <a href="http://en.wikipedia.org/wiki/Walsh_function" rel="nofollow">http://en.wikipedia.org/wiki/Walsh_function</a></p> <p>They are defined by dyadic intervals on $L^2([0,1])$, so is relatively well behaved under binary decimal representations. And thus should give a fairly nice description of you $L^2(X,\mu)$.</p> http://mathoverflow.net/questions/34402/nice-orthonormal-basis-for-l2cantor-set/34409#34409 Answer by BS for Nice orthonormal basis for L^2(Cantor set) BS 2010-08-03T16:20:25Z 2010-08-03T19:10:10Z <p>Since the Cantor set with your measure is also the compact group $(\mathbb{Z}/2)^\mathbb{N}$ with Haar measure, a natural orthonormal basis is the (continuous) characters $\alpha:X\to S^1$, namely the finite products of coordinates $c_n(x)$, $n\in\mathbb{N}$ if you view $\mathbb{Z}/2$ as <code>{-1,1}</code>. These form the discrete group $(\mathbb{Z}/2)^{(\mathbb{N})}$. </p> <p>If you view $X$ as the Cantor middle third, $c_n(x)$ corresponds to $a_n(x)-1$, the $n$-th base $3$ digit of $x$ minus 1 (all digits are 0 or 2 by definition). These correspond to Walsh functions mentioned by Willie Wong when you use the measure isomorphism $X\to I$, which maps $x$ to ${1\over2} \sum_n a_n(x) 2^{-n}$.</p> <p>Another possible model is $\mathbb{Z}_2$, the compact group of <a href="http://en.wikipedia.org/wiki/2-adic_integers" rel="nofollow">2-adic integers</a>, and the characters are then identified to power-of-two roots of unity, forming a group isomorphic to $\mathbb{Z}[1/2]/\mathbb{Z}$. <s> This seems to lead to the same basis, although indexed differently.</s> EDIT: as remarked by Greg Kuperberg in a comment, this can't be true.</p> http://mathoverflow.net/questions/34402/nice-orthonormal-basis-for-l2cantor-set/34411#34411 Answer by Vagabond for Nice orthonormal basis for L^2(Cantor set) Vagabond 2010-08-03T16:43:18Z 2010-08-03T17:56:08Z <p>Though not always, for certain Cantor measures $\mu$ there exists orthonormal basis for $L^2(\mu)$ consisting of complex exponentials <code>$\{e^{2 \pi i \lambda_n t}: \lambda \in \Lambda \}$</code> where $\Lambda \subset \mathbb R$. These are called spectral cantor measures.</p> <p>see the following papers for details</p> <p>P. E. T. Jorgensen and S. Pedersen , Dense analytic subspaces in fractal L2-spaces. J. Anal. Math. 75 (1998), pp. 185–228. <a href="http://www.springerlink.com/content/211651p66833m7j7/" rel="nofollow">http://www.springerlink.com/content/211651p66833m7j7/</a></p> <p>R. Strichartz , Remarks on Dense analytic subspaces in fractal L2-spaces. J. Anal. Math. 75 (1998), pp. 229–231 <a href="http://www.springerlink.com/content/a53685h15g17x509/" rel="nofollow">http://www.springerlink.com/content/a53685h15g17x509/</a></p> <p>Izabella Laba and Yang Wang On Spectral Cantor Measures Journal of Functional Analysis Volume 193, Issue 2, 20 August 2002, Pages 409-420 <a href="http://www.math.ubc.ca/~ilaba/preprints/meas1.dvi" rel="nofollow">http://www.math.ubc.ca/~ilaba/preprints/meas1.dvi</a></p>