Sheng Li (gmachine1729) wrote,
Sheng Li

More mathematical struggles

Originally published at 狗和留美者不得入内. You can comment here or there.

Math is hard. It wrecks my self-esteem, and at times, it makes me feel an utter loser, who simply isn’t smart enough, who is a league if not multiple away from the big name mathematicians, who come up with much if not most of the most original results in mathematics. There are times when the formalism within the mathematics looks, perhaps superficially out of lack of perception no the part of its viewer, so excruciatingly complex and dry, and that one is inclined to simply go: this is too hard, give up. I’ve felt that, and I think just about everyone, no matter how smart, has, to some extent. Over time, I’ve come to realize that the dirty details tend to be a natural product of a few main ideas behind the proof, and once such ideas as grasped, every detail can easily be seen to have its rightful place within the entire construction. There was a time when I felt demoralized or slightly baffled upon seeing this answer of Ron Maimon that can totally come across as intellectually too presumptuous, from a guy too smart who never had to struggle like all us ordinary folks, from a guy who takes for granted as routine what is a slog for most, without being metacognitively aware enough to appreciate that he is of a totally different beast. In this, stood out the following quote:

You need to learn to “unpack” proofs into the construction that is involved, to know what the proof is saying really. It is no good to memorize the proof, you need to understand the construction, and this will motivate the proof.

What he means by this, as far as I can tell, is that one should try to reverse engineer the source of the proof, the path or motivation that brought to it its discoverer. This, per convention of terseness in mathematical literature, is usually obfuscated, and it is the reader who is expected to uncover it himself. In any case, one finds that in mathematics, or any deep intellectual discipline, it is largely up to the learner himself  to form the right mental picture, which cannot be done with any form of explanation on too dull a pupil who lacks the inner drive.

I was disappointed yesterday, struggling to solve a problem that I had failed to solve back in 2014, the solution of which I had back then read, and even written up for myself, but which had evanesced entirely from my unretentive memory. It is my hope that this doesn’t happen again.

Apparently, there is a universal entire function f such that only any compact set for any entire g, for any \epsilon > 0, there is a c > 0 such that |f(z+c) - g(z)| < \epsilon.

This looks initially entirely elusive, but once one realizes that entire functions can be approximated to arbitrary precision via a countable set of polynomials, namely (\mathbb{Q}+i\mathbb{Q})[z], it is hinted that it suffices to approximate each of these polynomials arbitrarily closely on an arbitrarily large disk. This lends to taking a sequence of that dense set of polynomials (call it \{P_j\}_{j=0}^{\infty}) with each distinct element in it occurring an infinite number of times, and constructing a uniformly convergent everywhere series such that the jth partial sum approximates the jth polynomial of the sequence to a degree of closeness that goes to zero as j \to \infty when translated some c_j > 0 to the right. Denote this as f(z) = \displaystyle\sum_{i=0}^\infty Q_i(z). The individual Q_is we can obtain via Runge’s theorem.

Range’s theorem states that for any compact subset K of \mathbb{C} and function f holomorphic on some open set containing K, for any set A containing at least one element in each connected component of \mathbb{C} \setminus K, there is a sequence of rational functions with all poles in A that converges uniformly to f on K. This is shown partly by taking Riemann sums of the integral associated with Cauchy’s integral formula on some closed piecewise-linear contour \Gamma in the open set that contains K in its interior.

We associate each P_j in our sequence with disk D_j with center c_j > 0 and radius j with no intersections between the disks. One observes that out of infinite occurrence of each of the elements of P_j, we can approximate on an infinitely large disk (the disks among \{D_j\}_{j=0}^{\infty} corresponding to each polynomial in our dense set is infinite and thus unbounded in radius) and with arbitrary degree of precision (by having the \epsilon of approximation go to 0 as j \to \infty).

We have disks about the origin E_j containing D_i, 0 \leq i \leq j and not containing any D_i, i > j, with E_j \subset E_{j+1} and |Q_j| < \frac{1}{2^j} on E_{j-1} for j \geq 1. This way, the tail \displaystyle\sum_{j=n}^{\infty} Q_j uniformly convergent on E_j to what goes to 0 as n \to \infty. The containment relation with respect to the D_js also makes it such that the radii of E_j go to \infty, necessary for full coverage of \mathbb{C}.

We first let Q_0 = P_0 and then we obtain Q_j via Runge’s theorem for any \epsilon > 0 that satisfies |Q_j(z) - (P_j(z-c_j) - \displaystyle\sum_{i=0}^{j-1} Q_i(z))| < \epsilon on D_j in addition to the aforementioned |Q_j| < \frac{1}{2^j} on E_{j-1}. Here we have the holomorphic function approximated as P_j(z-c_j) - \displaystyle\sum_{i=0}^{j-1} Q_i(z) on D_j and 0 on E_{j-1}, separate on these two disjoint sets. Note that there are still poles of the rational approximation from Runge’s theorem, which is problematic, as our universal function must be entire. This is easy to resolve by modifying Q to be polynomials which uniformly approximate on D_j \cup E_{j-1} (by taking partial sum of the series expansion that is analytic on that region).

Take any entire g and some r > 0. We have for each element of our dense set some smallest j at which it occurs for which \displaystyle\sum_{i=j}^{\infty} \frac{1}{2^i} < \epsilon, and by density, one of them is such that it differs from g on |z| \leq r by less than \epsilon uniformly. In sum, we have for some j on |z| \leq r

|g(z) - P_j(z)| < \epsilon,

|\displaystyle\sum_{i=0}^j Q(z+c_j) - P_j(z)| < \epsilon,


|f(z+c_j) - \displaystyle\sum_{i=0}^j Q(z+c_j)| < \displaystyle\sum_{i=j}^{\infty} \frac{1}{2^i} < \epsilon.

Combining the three with triangle equality yields that |f(z+c_j) - g(z)| < 3\epsilon on our desired disk.

Tags: analysis, complex analysis, 数学/математика

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