A question about the limit of a sequence of pointwise convergent analytic funtions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T07:52:38Z http://mathoverflow.net/feeds/question/117633 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/117633/a-question-about-the-limit-of-a-sequence-of-pointwise-convergent-analytic-funtion A question about the limit of a sequence of pointwise convergent analytic funtions woodbass 2012-12-30T14:36:02Z 2012-12-31T14:23:49Z <p>Question: Let ${f_n}$ be a sequence of analytic functions on the unit disk $\Delta$ and suppose that $f_n$ converges to a continuous function $f$ on $\Delta$ pointwisely. (1) Can we say that $f$ is analytic on $\Delta$? (2) If $f$ is analytic, is the convergence $\underline{locally}$ uniform on $\Delta$? (Note: I add the words "locally" due to obvious reason.)</p> <p>If we do not assume that the limit function $f$ is continuous (of course $f$ is measurable) in advance, (3) can we say that $f$ is continuous? [I number this new question by (3)]</p> <p>Note that evrey measurbale functon on $\Delta$ can be the limit of a sequence of analytic functions in the Lebesgue sense (i.e. almost everywhere).</p> http://mathoverflow.net/questions/117633/a-question-about-the-limit-of-a-sequence-of-pointwise-convergent-analytic-funtion/117637#117637 Answer by GH for A question about the limit of a sequence of pointwise convergent analytic funtions GH 2012-12-30T15:04:22Z 2012-12-30T15:04:22Z <p>I suspect the answer is no in general, but if you assume local boundedness then (1) is true and, for (2), convergence is locally uniform. This is a special case of the Vitali-Porter theorem, see <a href="http://www.math.qc.edu/~zakeri/mat704/ch3_3_19_2011.pdf" rel="nofollow">here</a>.</p> http://mathoverflow.net/questions/117633/a-question-about-the-limit-of-a-sequence-of-pointwise-convergent-analytic-funtion/117643#117643 Answer by Malik Younsi for A question about the limit of a sequence of pointwise convergent analytic funtions Malik Younsi 2012-12-30T15:38:07Z 2012-12-30T20:48:55Z <p>Of course (1) does not imply (2) : the functions $f_n:=z^n$ converge pointwisely to $0$ on the unit disk, but the convergence is not uniform. </p> <p>In fact, assuming (1), the convergence need not be locally uniform : using Runge's Theorem, it is possible to find a sequence of polynomials $p_n$ such that $p_n \rightarrow 0$ pointwisely on the unit disk, but the convergence is not uniform in any neighborhood of $0$ : see e.g. <a href="http://math.stackexchange.com/questions/113240/pointwise-convergence-of-sequences-of-holomorphic-functions-to-holomorphic-funct" rel="nofollow">this question</a></p> <p><strong>EDIT</strong></p> <p>A similar argument (using Runge's Theorem) answers your question (3) negatively : It is possible to construct a sequence of polynomials $p_n$'s with $p_n(z) \rightarrow 0$ for every $z \neq 0$ but $p_n(0) \rightarrow 1$. </p> http://mathoverflow.net/questions/117633/a-question-about-the-limit-of-a-sequence-of-pointwise-convergent-analytic-funtion/117644#117644 Answer by Alexandre Eremenko for A question about the limit of a sequence of pointwise convergent analytic funtions Alexandre Eremenko 2012-12-30T15:40:54Z 2012-12-30T17:12:53Z <p>These questions were investigated by Osgood, Montel and Lavrentiev, Sur les fonctions d'une variable complexe representable par des series de polynomes, Paris 1936. (There is a Russian translation in his selected Works available free on Internet). If you prefer German, see Hartogs and Rosenthal, Uber Folgen analytischer Funktionen, Math Ann., 1928, 100, 212-263, and 1932, 104, 606-610.</p> <p>In general, if a sequence of polynomials converges pointwise in a region $D$, then the limit function is analytic except for a closed nowhere dense set $E$ (Osgood). Montel proved that $E$ is a perfect set whose union with the complement of the disc is connected. Lavrentiev completely characterized the sets $E$ that can occur, and proved that every function of the first Baire class which is analytic outside $E$ is a pointwise limit of polynomials.</p> <p>Thus every continuous function, analytic outside $E$ can be obtained as a limit of polynomials. Convergence outside $E$ is locally uniform.</p> <p>This answers all your questions.</p> <p>By the way, similar problems for harmonic functions (characterization of their pointwise limits) is still not solved completely.</p> <p>EDIT. For example, any simple curve in the unit disc, going from $0$ to $1$, satisfies the Lavrentiev condition. Taking this curve with positive area, we can construct a continuous function $f$ in the unit disc, which is analytic outside the curve but not analytic in the unit disc. This function will be a pointwise limit of polynomials.</p>