Thursday, October 20, 2016

[sunifufr] Function from zero crossings

Given a list of zero-crossings on the X axis, construct a wavy function which is zero at only those points.  (Need to say "only", or else we can simply construct the zero function.  Or perhaps those are the only zeros inside the interval defined by the list; there may be further zeros outside the interval.)  This is an underconstrained problem.  Construct a "nice" function, for some definition of nice.

Splines are the obvious first idea.  An analytic function (infinitely differentiable) would be nicer.  Low order polynomial that has "large" absolute value between zeros but low maximum absolute value on the interval.

What if the list of zero-crossings is infinite, e.g., Riemann-Siegel Z function, or sine, or cosine?  Construct a sum of sines and cosines which, when truncated, get the zeros approximately correct, for some definition of approximate.

Original inspiration, the zero crossings of sin(x) sin(m*x) where m is an irrational number, have no periodic pattern.  The zeros are where a line of irrational slope intersect a regular grid of orthogonal lines.