A question about the limit of a sequence of pointwise convergent analytic funtions - MathOverflow

A question about the limit of a sequence of pointwise convergent analytic funtions

2012-12-30T14:36:02Z

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.)

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)]

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).

Answer by GH for A question about the limit of a sequence of pointwise convergent analytic funtions

2012-12-30T15:04:22Z

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 here.

Answer by Malik Younsi for A question about the limit of a sequence of pointwise convergent analytic funtions

2012-12-30T15:38:07Z

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.

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. this question

EDIT

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 $p_n(0) \rightarrow 1$.

Answer by Alexandre Eremenko for A question about the limit of a sequence of pointwise convergent analytic funtions

2012-12-30T15:40:54Z

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.

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.

Thus every continuous function, analytic outside $E$ can be obtained as a limit of polynomials. Convergence outside $E$ is locally uniform.

This answers all your questions.

By the way, similar problems for harmonic functions (characterization of their pointwise limits) is still not solved completely.

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.