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Cumulative Sampling

Hello, I have a very strange problem that I don't know if anyone can provide an answer for. If they can thats amazing, but information about what this type of problem is called, and what fields of mathematics are relevant to trying to solve it is equally as useful to me.

So I have a periodic function y, and I want to discretely integrate it such that the distance between successive points of integration increase and / or decrease proportionally to the current total of the integral.

Is it possible to find a relation between the current total of the integral and the distance between sample points that will maximize the integral over an integer number of periods of y? If so how do I systematically find that relation?

kyle
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  • "discretely integrate it" so, a summation? – Zubin Mukerjee Feb 14 '18 at 19:46
  • Your question is a little contradictory: "increase and / or decrease proportionally to the current total" vs. "find a relation between...". Are you just after the proportionality constant ? –  Feb 14 '18 at 19:50
  • You are wanting to maximize the integral/summation, correct? – Clayton Feb 14 '18 at 19:51
  • Are you dealing with the function where the integral over the period is zero, or nonzero? Because, if it is nonzero, then over the integral number $n$ of periods, the integral (regardless of the choice of points) will behave like $Cn+O(1)$ where $C=\int_I ydx$, and $I$ is the interval with the same length as the period - so the "main" term will not depend on the choice of points at all. –  Feb 14 '18 at 20:02
  • I don't know how to explain this more clearly, or use latex, but I will try. Say we have a sinusoidal function y(x), and we want to sample it at successive points xi, and add y(xi) to a cumulative total (integrate /or sum). The xi are not evenly spaced, xi is a function of the current cumulative total. Thus since y(x) takes on negative values as well as positive over its period, the distance between the xi will change. I want to find the function that gives the xi at which to sample y that will maximize the sum of y(xi), does that make sense? – kyle Feb 14 '18 at 22:45
  • Something like https://en.wikipedia.org/wiki/Adaptive_quadrature?wprov=sfti1 – user251257 Feb 14 '18 at 23:24
  • Adaptive algorithms for computing integrals is somewhat related but not what I am looking for. The point of an adaptive algorithm is to try to increase the accuracy of a discrete integral. I am doing something like this: sum_(i) = sum_(i-1) + y(x_i) . With x_i = F(sum_(i-1)). I want to find F such that the sum is maximized. Its easy to make a script and play with numbers to show that even if y is periodic and its continuous integral is zero over N periods you can get a nonzero result using this scheme of changing the distance between sample points, but I want the maximal nonzero result. – kyle Feb 14 '18 at 23:37

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