[throwawaymath]

Unless I'm severely misunderstanding you, a discrete set (or function) cannot be a dual of a continuous set (or function). If nothing else, the former is countable and the latter is uncountable; there can be no isomorphism between the two.

[adamnemecek]

Unless I'm severely misunderstanding you, a discrete set (or function) cannot be a dual of a continuous set (or function). If nothing else, the former is countable and the latter is uncountable; there can be no isomorphism between the two.

There is. Discrete samples are samples of the continuous space. In order to capture the continous space, you only really need to capture a particular set of samples that you then interpolate between.

The way I interpret isomorphism in this context is if you can capture one space in the other and then convert to the other without a loss of information.

Imagine a polynomial (in the smooth space). You can capture a particular set of points that uniquely determines the polynomial. In some circumstances you can use these samples to reconstruct the original polynomial by interpolating between any of the two points.

[throwawaymath]

Okay, I think I understand what you're getting at. But if you've taken a set of points from a continuous set (like an interval on the reals) and you can put those in bijection with a discrete set, then by definition your subset of the continuous set isn't continuous. It must be discrete.

More succinctly, you actually can't draw an isomorphism between discrete and continuous spaces without losing information from the continuous space.

[adamnemecek]

Here's the thing, your description of the continuous set is already discretized. If we say an interval 4-6 we have captured the continuous space using only two numbers.

I know that this is a silly argument in some sense but this is something that you do naturally that you don't even think about it.

Do you see what I'm getting at? You capture extrema in the discrete space and the interpolate in the smooth space to recreate the smooth curve.

[wenc]

> Imagine a polynomial (in the smooth space). You can capture a particular set of points that uniquely determines the polynomial. In some circumstances you can use these samples to reconstruct the original polynomial by interpolating between any of the two points.

This is not duality though and you do lose information.

For instance let’s say one has a cubic polynomial and one samples 5 points from it and stores those points. If one didn’t know the original order was cubic, and if one tried to interpolate over the 5 points to fit a quintic, that would be an incorrect reconstruction.

[adamnemecek]

You don't lose information if you pick your points correctly (you store only the extrema). In the cubic case, you need the two extrema (one minimum and one maximum and you need to know whether each extremum is a min or max) and then interpolate between them.

[wenc]

Unfortunately this is incorrect. Extrema do not always exist (consider y=x^3) and they do not uniquely define a polynomial (y=x^2 and y=x^4 both have minima at x=0).

[adamnemecek]

Unfortunately this is incorrect. Those minima are not the same. Remember that dual points have a real part and a dual part that indicates the rate of change at that point. The real part is the same but the dual is different.

[wenc]

My point is it’s not possible to uniquely reconstruct an arbitrary polynomial by just knowing the extrema because there may be information loss in the general case. I will stop here.