Holomorphic function with a.e. vanishing radial boundary limits - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T14:53:40Z http://mathoverflow.net/feeds/question/52299 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/52299/holomorphic-function-with-a-e-vanishing-radial-boundary-limits Holomorphic function with a.e. vanishing radial boundary limits CJ 2011-01-17T07:49:31Z 2011-07-31T19:13:58Z <p>Hello everybody.</p> <p>I'm looking for an "easy" example of a (non-zero) holomorphic function $f$ with almost everywhere vanishing radial boundary limits: $\lim\limits_{r \rightarrow 1-} f(re^{i\phi})=0$.</p> <p>Does anyone know such an example.</p> <p>Best CJ</p> http://mathoverflow.net/questions/52299/holomorphic-function-with-a-e-vanishing-radial-boundary-limits/52301#52301 Answer by Andrey Rekalo for Holomorphic function with a.e. vanishing radial boundary limits Andrey Rekalo 2011-01-17T09:57:59Z 2011-01-17T09:57:59Z <p>I am not sure if this would qualify as 'easy' but the first example of such a function was constructed by Lusin. It can be found in N. Lusin, J. Priwaloff, <a href="http://archive.numdam.org/ARCHIVE/ASENS/ASENS_1925_3_42_/ASENS_1925_3_42__143_0/ASENS_1925_3_42__143_0.pdf" rel="nofollow">Sur l'unicité et la multiplicité des fonctions analytiques</a>, <em>Ann. Sci. École Norm. Sup.</em> (3), 1925, p. 143-191 (see p. 185).</p> http://mathoverflow.net/questions/52299/holomorphic-function-with-a-e-vanishing-radial-boundary-limits/52304#52304 Answer by GH for Holomorphic function with a.e. vanishing radial boundary limits GH 2011-01-17T10:38:36Z 2011-07-31T19:13:58Z <p>To complement Andrey Rekalo's response, Lusin's construction was generalized by Bagemihl and Seidel (Math. Zeitschrift 61 (1954), online <a href="http://www.digizeitschriften.de/main/dms/img/?PPN=GDZPPN002384698" rel="nofollow">here</a>). See their Corollary 4 whose proof takes about 2 pages, much less than the original one by Lusin-Priwaloff. Of course the proof relies on Mergelyan's famous approximation theorem for which see Section 20.5 in Rudin: Real and Complex Analysis.</p> <p><strong>EDIT:</strong> Lvriemsurf asked in a comment if we can replace "almost every angle" by "every angle" in the construction. The answer is "no", as follows from the <a href="http://eom.springer.de/l/l061060.htm" rel="nofollow">Lusin-Priwaloff theorems</a>.</p> http://mathoverflow.net/questions/52299/holomorphic-function-with-a-e-vanishing-radial-boundary-limits/52357#52357 Answer by David Hansen for Holomorphic function with a.e. vanishing radial boundary limits David Hansen 2011-01-17T22:38:51Z 2011-01-18T00:28:50Z <p>According to a footnote in the famous Hardy-Ramanujan paper "Asymptotic formulae in combinatory analysis", the function $f(q)=\prod_{n=1}^{\infty}\frac{1}{1-q^n}$ vanishes like $f(re^{i\theta})=o((1-r)^{1/4-\varepsilon})$ for almost all $\theta$. No proof is given, though I can't imagine Hardy would have made a statement like this without a proof in his pocket.</p> <p><strong>Edit:</strong> This isn't actually hard to guess at. By Euler's pengatonal number theorem, we have $f(q)^{-1}=\sum_{n\in \mathbf{Z}}(-1)^{n}q^{n(3n-1)/2}$, so Plancherel gives</p> <p>$\int_{0}^{2\pi}|f(re^{i\theta})|^{-2}d\theta=2\pi\sum_{n\in \mathbf{Z}}r^{n(3n-1)} \sim 2 \pi^{3/2}3^{-1/2}(1-r)^{-1/2}.$</p>