I have two vectors: $\vec{a}$ and $\vec{t}$: $a\_k$ is the sampling value taken at $t\_k$. I need to do DFT, but the sampling period is irregular. I've learned about Frames but unsure how to use the Duffin–Schaeffer's suggestion for a Frame-function: $\sqrt{\frac{t\_{n+1}-t\_{n-1}}{2T}}sinc(\frac{\pi}{T}[t-t\_n])$.


Does the following operation: $C\_n=\sum\_k{f(k) sinc(\frac{\pi}{T\_k}[t\_k-t\_n])}$, really do the work to get the coefficients ($C\_n$) ? (... pretty sure I misinterpreted the equation somehow ...)


Do I need the preceding $\sqrt{\frac{t\_{n+1}-t\_{n-1}}{2T}}$ ?



How to do DFT for irregular sampling period ?
There have been a number of publications about "Unequally Spaced FFTs" in the numerical analysis literature. These typically involve an automated sort of interpolation (usually Gaussian) to an equally spaced grid followed by an FFT. The methods come with error bounds specified. You could search under the names Rokhlin, Dutt, and Beylkin, with title words "Unequally spaced FFT." 


I located the following:


A. Dutt and V. Rokhlin, Fast Fourier transforms for nonequispaced data, SIAM J. Sci. Comput. 14, 1993, 1368–1393.


There is also work by Greg Beylkin, see for example:


<http://www.ipam.ucla.edu/publications/mgatut/mgatut_4992.pdf> 


and references therein.


Mind you, I have not used these directly, but have used curvelet transform codes that are based on similar algorithms.


Hope this helps a bit,


Tom


