Equations for Kneser sexp algorithm

[sheldonison]

Its definitely a uniqueness criterion... any sexp(z) solution would either be the unique solution, with \( \theta(z) \) exponentially decaying to a constant as \( z\to+i\infty \), or else if it is any other solution, than \( \theta(z) \) grows exponentially as \( z\to+i\infty \).

That screams for a proof, does it?

For purposes of this proof, we assume we have two different real valued sexp solutions. It is assumed that the first sexp_a solution follows the uniqueness criteria, and then a proof is given that the second solution does not follow the uniqueness criteria.

\( \text{sexp}_a(z)=\text{superf}(z+\theta_a(z)) \)
\( \text{sexp}_b(z)=\text{superf}(z+\theta_b(z)) \)

The uniqueness criteria is that the 1-periodic function \( \theta_a(z) \) exponentially decays to a constant as \( z\to+i\infty \), meaning that there all of the individual terms for n>=1 in \( \theta_a(z) \) exponentially decay as \( z\to+i\infty \). It will be shown that \( \theta_b(z) \) does not decay to a constant as \( z\to+i\infty \). So there cannot be two different solutions, where \( \theta_a(z) \) and \( \theta_b(z) \) which both decay to a constant as \( z\to+i\infty \), and therefore, sexp_a(z) is a unique solution.

\( \theta_a(z) = a_0 + \sum_{n=1}^{\infty} a_n \exp(2n\pi i z) \)

It can be trivially proven that terms of the form \( a_n \exp(2n\pi i z) \) decay to zero as \( z\to+i\infty \).

Since sexp_a and sexp_b are both real valued, then there exists another real valued 1-periodic function, linking sexp_b to sexp_a, which will be called \( \theta_c(z) \).
\( \text{sexp_b}(z) = \text{sexp_a(z+\theta_c(z)) \)

Because \( \theta_c(z) \) is real valued at the real axis, it must be represented as a sum of this form, where at the real axis, the 2nd set of exponential summation terms are the complex conjugate of the first set of exponential summation terms, so that their sum is a real number. Any real valued 1-periodic function can be represented in this form.
\( \theta_c(z) = c_0 + \sum_{n=1}^{\infty} c_n \exp(2n\pi i z) +
\sum_{n=1}^{\infty} \overline{c_n} \exp(-2n\pi i z) \)

For \( \theta_c(z) \), there are two sets of exponential terms. Terms of the form \( c_n \exp(2n\pi i z) \) decay to zero as \( z\to+i\infty \), and likewise it can be trivally shown that terms of the form \( \overline{c_n} \exp(-2n\pi i z) \) grow exponentially as \( z\to+i\infty \). So, as \( z\to+i\infty \), the behavior of \( \theta_c(z) \) is determined only by the terms \( \overline{c_n} \exp(-2n\pi i z) \).

The next step is to generate an equation for \( \theta_b(z) \), by using the equation for sexp_b(z) in terms of sexp_a(z) and \( \theta_c(z) \). Then substitute the equation for sexp_a(z) in terms of the superfunction, and take the inverse superfunction, which gives an equation \( \theta_b(z) \).

\( \text{sexp}_b(z) = \text{sexp}_a(z+\theta_c(z)) \)
\( \text{sexp}_a(z) = \text{superf}(z+\theta_a(z)) \)
Then substituting \( z+\theta_c(z) \) from the first equation into the second equation in place of (z) to get:
\( \text{sexp}_b(z) = \text{sexp}_a(z+\theta_c(z)) = \text{superf}(z+\theta_c(z)+\theta_a(z+\theta_c(z))) \)
Then, notice that this equation can be compared to the other equation for sexp_b(z), which allows us to get an equation for \( \theta_b(z) \).
\( \text{sexp}_b(z) = \text{superf}(z+\theta_c(z)+\theta_a(z+\theta_c(z))) = \text{superf}(z+\theta_b(z)) \)
Taking the inverse superfunction of both sides, results in this equation
\( z+\theta_c(z)+\theta_a(z+\theta_c(z)) = z+\theta_b(z) \)
\( \theta_b(z) = \theta_c(z)+\theta_a(z+\theta_c(z)) \)

So, now there is an equation for the 1-periodic \( \theta_b(z) \) in terms of the 1-periodic \( \theta_c(z) \) and \( \theta_a(z) \). The properties of \( \theta_c(z) \) and \( \theta_a(z) \) can be used to prove that \( \theta_b(z) \) does not decay to a constant, but rather becomes a function whose amplitude grows arbitrarily large as \( z\to+i\infty \), which means that sexp_b(z) does not meet the uniqueness criteria that defines sexp_a(z).

First of all, assume that at least some of the c_n terms (n>=1) in \( \theta_c(z) \) are non-zero. Otherwise theta_c(z) would be the identity, and sexp_b(z) would be equivalent to sexp_a(z).

Now, we go back to the equation for \( \theta_b(z) \)
\( \theta_b(z) = \theta_c(z)+\theta_a(z+\theta_c(z)) \)
We know that \( \theta_c(z) \) grows exponentially as \( z\to i\infty \) since the \( \overline{c_n} \exp(-2n\pi i z) \) terms in the 1-periodic function \( \theta_c(z) \) all grow exponentially. But what about \( \theta_a(z+\theta_c(z)) \)? We need to show that \( \theta_a(z+\theta_c(z)) \) does not somehow cancel out the exponential growth of \( \theta_c(z) \). We take advantage of the fact that as \( z\to i\infty \;\; \theta_a(z) \approx a_0 \), where the amplitude of the 1-periodic \( \theta_z(a) \) terms all decay exponentially as \( z\to i\infty \). Where \( \Im(z+\theta_c(z)) \) is large enough positive, then \( \theta_a(z+\theta_c(z)) \approx a_0 \) and \( \theta_b(z) \approx a_0 + \theta_c(z) \). Where \( \Im(z+\theta_c(z)) \) approaches the real axis, which happens when \( \Im(\theta_c(z)) \) is negative enough, then the equations are less clear, since \( \theta_a(z) \) has a singularity at the real axis for integer values of z, so \( \theta_a(z) \) is not predictable when \( z+i\Im(\theta_c(z)) \) approaches the real axis. But the overall function, \( \theta_b(z) \) for some arbitrarily large value of \( \Im(z)\to+i\infty \), can be shown to have an arbitrarily large amplitude, tracking \( \theta_b(z) \approx a_0 + \theta_c(z) \), as long as \( \Im(z+\theta_c(z)) \) is sufficiently large. So \( \theta_b(z) \) cannot be converging arbitrarily closely to a constant as \( \Im(z)\to+\infty \). That is to say, \( \theta_b(z) \) will cover the range of values for \( \approx a_0 + \theta_c(z) \), as opposed to converging to a constant, as we would expect if \( \theta_b(z) \) were expressible in the same form as \( \theta_a(z) \). Therefore, \( \theta_b(z) \) does not match the assumption, that sexp_b(z) as an alternative solution which also meets the uniqueness criteria.

It seems to me that the last paragraph is hard to follow, and that it probably needs to be rewritten and formalized. But, at the moment, I'm not quite sure how to do that. But at the very least, I think one can compare the complex somewhat unpredictable arbitrarily large amplitude non-converging behavior of \( \theta_b(z) \) as \( z\to+i\infty \) with the exponential decay of \( \theta_a(z) \) to a constant as \( z\to+i\infty \), and see the contradiction in assuming that sexp_b(z) is an alternative solution which meets the uniqueness critera. Therefore, there aren't multiple solutions which meet the uniqueness criteria.
- Sheldon