09/11/2014, 02:13 PM
(This post was last modified: 09/11/2014, 04:02 PM by sheldonison.)

Yeah, long way to go, and I'm not sure I'm up for the task of making it all rigorous, and figuring out what's new and what the relevant prior mathematical work is. So far, I've only done the calculations for three function, , for the last case, I have a closed form solution, that I can rigorously prove, post#85.

They all behave rather differently, so claiming there is some sort of unified interpolation theory seems a bit of a stretch... But in all cases, we are trying to work with interpolating a function that eventually has all positive derivatives, and as you rotate around the unit circle (perhaps multiple times), the function eventually gets arbitrarily small in magnitude. And I think maybe that's the unifying thread. Also, the error term is proportional to 1/x, because we generate an entire Taylor series, but the function we are generating an asymptotic for has a Laurent series, since it is not entire.

For the ln(x) or case, we have conjectured equivalent integral formation,

f(z) =

Since f(z) is entire, if we plug exp(z)f(z) into our interpolation function, we get out exactly exp(z)f(z). Also, if we plug into our interpolation engine, we also get an entire function. The difference of two entire functions has to be an entire function; in this case, conjectured equal to zero. So, if I can prove exp(z)f(z) eventually has all positive derivatives, then I think I could rigorously prove they are equivalent, since I'm just taking the "fake" Cauchy integral...

Speaking of all positive derivatives, another question is the Gaussian approximation, post#16, which can approximate all of these functions reasonably well, accurate for to one part per thousand or better for large n. In fact, for my third example here, the Gaussian approximation is coincidentally, exact! However, like the exp^{0.5} case, for this function we wrap the function we are interpolating around the unit circle multiple times, or an infinite number of times in this case, where we choose n to get optimal results. So claiming there is some sort of unified interpolation theory again seems like a huge bit of a stretch...

summing up multiple loops around the unit circle sometimes helps convergence. Integrate instead of

They all behave rather differently, so claiming there is some sort of unified interpolation theory seems a bit of a stretch... But in all cases, we are trying to work with interpolating a function that eventually has all positive derivatives, and as you rotate around the unit circle (perhaps multiple times), the function eventually gets arbitrarily small in magnitude. And I think maybe that's the unifying thread. Also, the error term is proportional to 1/x, because we generate an entire Taylor series, but the function we are generating an asymptotic for has a Laurent series, since it is not entire.

For the ln(x) or case, we have conjectured equivalent integral formation,

f(z) =

Since f(z) is entire, if we plug exp(z)f(z) into our interpolation function, we get out exactly exp(z)f(z). Also, if we plug into our interpolation engine, we also get an entire function. The difference of two entire functions has to be an entire function; in this case, conjectured equal to zero. So, if I can prove exp(z)f(z) eventually has all positive derivatives, then I think I could rigorously prove they are equivalent, since I'm just taking the "fake" Cauchy integral...

Speaking of all positive derivatives, another question is the Gaussian approximation, post#16, which can approximate all of these functions reasonably well, accurate for to one part per thousand or better for large n. In fact, for my third example here, the Gaussian approximation is coincidentally, exact! However, like the exp^{0.5} case, for this function we wrap the function we are interpolating around the unit circle multiple times, or an infinite number of times in this case, where we choose n to get optimal results. So claiming there is some sort of unified interpolation theory again seems like a huge bit of a stretch...

summing up multiple loops around the unit circle sometimes helps convergence. Integrate instead of

- Sheldon