Getting a function that passes through 'n' points

Question

Is there a way to get a periodic function that passes through n arbitrary points?

Answer 1

This could be done with linear algebra. Let $p(x)$ be a periodic function with $n$ unknown coefficients, such as $p(x) = \sum_{i = 1}^n a_i \sin(n x)$. Then if the points you want to satisfy are $(x_i, y_i)$, the equations $p(x_i) = y_i$ create a system of $n$ linear equations with $n$ unknowns. This can be solved provided the corresponding matrix has nonzero determinant.