On Mon, Jun 4, 2012 at 6:44 PM, Bill Gosper <billgosper@gmail.com> wrote:
The Fourier series mentioned below is a specialization of the one in gosper.org/fst.pdf, which alternatively specializes to the Koch's Snowflake, Peano's spacefiller, etc., according to the parameter s. For convergence, the geometry of the recursion underlying the construction of the Fourier coefficients requires |s-i| < 2, but at s= √3, the infinite product telescopes, revealing that series convergence also extends to this boundary point. A couple of days ago I thought I noticed convergence beyond |s-i| = 2, but common sense and recent experiments at Julian's house convinced me I was hallucinating. In the process, we produced some pleasantly Picassoid squiggle <http://gosper.org/snowfourier.pdf>s. (Or maybe just Dr. Seuss.) The function futst(t,s,m,cutoff) traces the outline of the curve repeated around a regular m-gon as 0≤t≤2m, low-pass (t=time) filtered to the cutoff frequency (default ±69 harmonics).
Thus lacking evidence of anomalous convergence, I was puzzled to receive just now (again, quoting without permission):
You may have figured this out already, but just in case: the reason that it can converge while the fractal doesn't is that you found the series from an integral, but you don't actually compute the integral. Instead you compute a sequence of Riemann sums (intervals of size 2^-n), but this only necessarily computes the integral when the function in question is continuous. The fractal curve is not defined, and thus is not continuous, so the Fourier series is just some (necessarily discontinuous on a dense set, as otherwise you could find an interval where it would be continuous and match the "fractal" at dyadic rational values) function. It may or may not have to pass through the same points as the "fractal" at dyadic rational values. I don't know why it matches the predictions for 1/3, but the problem seems approachable.
Julian
He's apparently right! Out[219]= {futst[1/3, s, m, 9999], (I + s)/(3 + I s) + Cot[π/m]}
In[227]:= %219 /. {9999 -> 99999, m -> 2., s -> 7/4.}
Out[227]= {0.586944 - 0.00655471 I, 0.580311 - 0.00518135 I}
where the Cot expression is the theoretical t=1/3 value for cutoff→∞. Trying 2 million terms, In[228]:= %219 /. {9999 -> 999999, m -> 2., s -> 7/4.}
NEVER RETURNS, because somewhere between 99999 and 999999, Sum arbitrarily, silently, and PERVERSELY changes 1/2. from .5 to 1/2 !
This is infuriating. This is evil. --rwg
So I forced the term calculation (fut) to use machine precision, conceding the ability for the series (futst) to control the precision. Trying 400000 terms, In[287]:= (Print[#1]; #2) & @@AbsoluteTiming[%219 /. {9999 -> 199999, m -> 2., s -> 7/4.}] During evaluation of In[287]:= 122.436966 Out[287]= {0.575792 - 0.0102995 I, 0.580311 - 0.00518135 I} And then 600000: In[288]:= (Print[#1]; #2) & @@ AbsoluteTiming[%219 /. {9999 -> 299999, m -> 2., s -> 7/4.}] During evaluation of In[288]:= 126.319336 Out[288]= {0.578973 - 0.0108453 I, 0.580311 - 0.00518135 I} And then 1.4 million: In[283]:= (Print[#1]; #2) & @@ AbsoluteTiming[%219 /. {9999 -> 699999, m -> 2., s -> 7/4.}] During evaluation of In[286]:= 2722.704824 Out[283]= {0.101321 ((8.87115 + 1.71512 I) - Limit[(-1)^k E^(1/3 I (0.5 + k) \[Pi]) fut[0.5 + k, 1.75], k -> 0, Assumptions -> True]), 0.580311 - 0.00518135 I} Somebody should be shot. Despite these efforts, Mathematica failed to conceal a surprise in the actual math: The series seems to get close and then simply dither, never actually converging! Upping the precision for 600000: In[290]:= (Print[#1]; #2) & @@ AbsoluteTiming[%219 /. {9999 -> 299999, m -> 2.`25, s -> 7/4.`25}] During evaluation of In[292]:= 1672.388773 Out[290]= {0.5789734583798222659864 - 0.0108452753013866670336 I, 0.580310880829015544041451 - 0.0051813471502590673575130 I} in perfect agreement with machine precision. Let's make some plots<http://gosper.org/snowfourier+.pdf> . %264 is the apparently convergent s=√3 case on an (m=2)gon. . %267 is part of the questionable s=7/4 case on an (m=3)gon. (Anatomists, not to mention trigonometrists, actually speak of trigons.) %268 is 288 terms of s=7/4. A cross between a bowling lane and a cricket pitch! Upping to 999 terms, %277 grows another row of "circles" and switch to pool. The One Ball was amputated by PlotRange->Automatic. %278 is the same, with finer plotting. Why fifteen loops? And what are those little tripods? Still with s=7/4, %279 takes 20000 series terms, grows another row of loops, and becomes UNDERsampled, while the tripods become indented triangles. %280 takes 200000 terms (after many hours), concentrating on 2<t<2.1, a short, off center subinterval. And it's still undersampled. Each loop is orbited hundreds of times. Perfunctory experiments with s=1.742 show very similar behavior. This is very peculiar non-convergence. This was a case where pictures actually clarified misleading numbers (unlike Mike Stay's recent 8^2=5*13 mention, or my old dysplotsia<http://www.tweedledum.com/rwg/dysplotsia.htm>post.) --rwg "It turns out that the purpose of computing was numbers after all. sorry for the confusion." --"W.R. Hamming" (Who, really?) (Which reminded Neil of "Computers are useless--they can only give you answers." --Picasso.) Too bad Mathworld is also hard to update.