Very nice piece. One thing jumped out at me, and it is sitting in your own narrative.
Every time analysis breaks here, the repair is the same move: bound the space. Pointwise convergence is badly behaved, so impose a norm. All functions is too big, so require the integral of |f|^p to be finite. The space is incomplete, so complete it, which is exactly forbidding Cauchy sequences from escaping. Stone-Weierstrass needs closed and bounded. And the space that works is literally the bounded functions.
So perhaps the shift was not "mathematics adopted rigor" so much as "mathematics learned to bound its spaces, and rigor came along for free." Epsilon-delta is not a companion to a bound. It is one.
Gibbs makes the point sharper than anything else in the post. The square wave is bounded between 0 and 1, but the partial sums are under no obligation to respect that, so they overshoot by about nine percent and never stop, narrower but never smaller. Fejér summation kills it entirely, and the reason is the whole story: the Fejér kernel is non-negative, so each Cesàro mean is a weighted average of the function's own values, and an average cannot escape the range of what it averages. The Dirichlet kernel is not non-negative, so the partial sums are free to leave the bound, and they always do.
Gibbs may not be a fact about Fourier series at all. It looks like what always happens when an unbounded method approximates a bounded object.
Out of curiosity, how more precisely did the Gibbs phenomenon show up with your oscilloscope? Your pictures seem (to me) to show an instability that happens when we do the reverse Fourier transform from {(1,1),(3,1/3),(5,1/5),…} to the square wave, not when we do our Fourier transform the other way (from the square wave to {(1,1),(3,1/3),(5,1/5),…}).
I'm afraid I cannot remember precisely; bear in mind, this was over 15 years ago. But I can tell you that that brand of oscilloscope could not produce an exact square wave; it approximated it using a composition of sine waves. Which is fine, as long as you don't look at it too closely. (This is how my lab instructor explained it, anyway.)
Very nice piece. One thing jumped out at me, and it is sitting in your own narrative.
Every time analysis breaks here, the repair is the same move: bound the space. Pointwise convergence is badly behaved, so impose a norm. All functions is too big, so require the integral of |f|^p to be finite. The space is incomplete, so complete it, which is exactly forbidding Cauchy sequences from escaping. Stone-Weierstrass needs closed and bounded. And the space that works is literally the bounded functions.
So perhaps the shift was not "mathematics adopted rigor" so much as "mathematics learned to bound its spaces, and rigor came along for free." Epsilon-delta is not a companion to a bound. It is one.
Gibbs makes the point sharper than anything else in the post. The square wave is bounded between 0 and 1, but the partial sums are under no obligation to respect that, so they overshoot by about nine percent and never stop, narrower but never smaller. Fejér summation kills it entirely, and the reason is the whole story: the Fejér kernel is non-negative, so each Cesàro mean is a weighted average of the function's own values, and an average cannot escape the range of what it averages. The Dirichlet kernel is not non-negative, so the partial sums are free to leave the bound, and they always do.
Gibbs may not be a fact about Fourier series at all. It looks like what always happens when an unbounded method approximates a bounded object.
Would read the Hilbert space follow-up.
Danny
Thank you!
You are most welcome.
Out of curiosity, how more precisely did the Gibbs phenomenon show up with your oscilloscope? Your pictures seem (to me) to show an instability that happens when we do the reverse Fourier transform from {(1,1),(3,1/3),(5,1/5),…} to the square wave, not when we do our Fourier transform the other way (from the square wave to {(1,1),(3,1/3),(5,1/5),…}).
I'm afraid I cannot remember precisely; bear in mind, this was over 15 years ago. But I can tell you that that brand of oscilloscope could not produce an exact square wave; it approximated it using a composition of sine waves. Which is fine, as long as you don't look at it too closely. (This is how my lab instructor explained it, anyway.)