# interpolating a periodic time series

I have a bunch of readings that run every 4 hours, however each sensor has a different offset. One sensor might read at $t = 0,4,8,12,16,20$ and another senor reads at $t = 1,5,9,13,17,21$. This makes it difficult to compare the reading.

It would be nice to put them all on equal footing - "normalization". To do this, I am considering the following problem:

Let $f(t)$ be a periodic function with $f(t+24) = f(t)$. We are given $f(4k)$ for $k = 0,1,\dots,5$. How can we estimate $f(1)$?

Really I want to estimate $f(t)$ for all $0 \leq t < 24$ but you get the idea.

One approach I considered is using Fourier Transform. Let $\omega = e^{\pi i /3}$ be a sixth root of unity and write:

$$f(n) = \hat{f}(0)\cdot 1 + \hat{f}(1)\cdot \omega^{n/6} + \hat{f}(2)\cdot \omega^{2n/6} + \hat{f}(3)\cdot \omega^{3n/6} + \hat{f}(4)\cdot \omega^{4n/6} + \hat{f}(5)\cdot \omega^{5n/6}$$

Then I am asking for $f(n+\tfrac{1}{4})$ which I can estimate using this Fourier transform. Not sure if my scaling is correct.

Is this correct or are there alternative ways like linear interpolation that work better?