Harmonic entropy: Difference between revisions
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==Derivation== | ==Derivation== | ||
For now, our derivation is limited to the case of <math>\sqrt{nd}</math> Tenney-weighted rationals, although it may be possible to derive a similar result for <math>\max(n,d)</math> weighting as well. | For now, our derivation is limited to the case of <math>\sqrt{nd}</math> Tenney-weighted rationals, although it may be possible to derive a similar result for <math>\max(n,d)</math> weighting as well. | ||
Additionally, because it simplifies the derivation, we will use ''unreduced rationals'' in our basis set, meaning that we will even allow unreduced fractions such as <math>4/2</math> so long as <math>nd<N</math> for our bound N. The use of HE with unreduced rationals have previously been studied by Paul and shown to be not that much different than HE with reduced ones. However, we will easily show later unreduced and reduced rationals converge to the same thing in the limit as <math>N \to \infty</math>, up to a constant multiplicative scaling. | |||
===Analytic Continuation of the Convolution Kernel=== | ===Analytic Continuation of the Convolution Kernel=== | ||