9.4.4 The cubic spline  9.4 Interpolation and extrapolation  9.4.2 Lagrange interpolation

9.4.3 Whittacker's interpolation formula

For time series analysis, the discrete samples in a series were represented in section 8.8 by the function  

g(t) = f(t)I(t)  (9.26)

 

where

$$I(t)=\sum_m\delta(t-m\Delta T) \ \ .$$ (9.27)

 

The Fourier transform of the delta function was a series of delta functions in frequency space, and this led to mixing together of various frequency components or to aliasing. However, if the Fourier transform of f(t), $\tilde f(\nu)$, is zero for frequencies above the Nyquist frequency, no contamination of the signal by aliasing occurs, and the true Fourier transform can be recovered from

$$\tilde f(\nu) = \tilde g(\nu)\tilde h(\nu)$$ (9.28)

 

where

$${rl}\tilde h(\nu) = \Delta, \ \ \vert\nu\vert\le {{1}\over{2\Delta T}}$$

$$=0\ \ {\rm otherwise}.$$ (9.29)

 

The Fourier transform of $\tilde h(\nu)$ is

$$h(t) = {{\sin(\pi t/\Delta)}\over{\pi t/\Delta}}$$ (9.30)

 

so, in the case where there is no variance at frequencies higher than the Nyquist frequency, the underlying function can be recovered from the discrete series by using the formula

$$f(t) = \int_{-\infty}^\infty h(t')g(t-t')dt' \ \ .$$ (9.31)

 

This is Whittacker's interpolation formula, often used for interpolation between values of a time series for this reason. When (9.26) is substituted in (9.31) the result is

$$f(t) = \sum_{n=-\infty}^\infty {{\sin[\pi(t-n\Delta T)/\DeltaT]}\over{\pi(t-n\Delta T)/\Delta T}} f(n\Delta T) \ \ .$$ (9.32)

 

Because the terms decrease in magnitude as 1/n, and oscillate in sign, the summation converges fast enough to be practical. 

   
 9.4.4 The cubic spline  9.4 Interpolation and extrapolation  9.4.2 Lagrange interpolation