Special Theory Finishing Touches
$
\def \sinc {\operatorname{sinc}}
\def \slechts {\quad \Longleftrightarrow \quad}
$
So far so good about the General perspective, but the Special Theory is
still in the need of some finishing touches. One of the first is that the above
interpolation with $\sinc$ functions is by far not the only possibility to have
a (theoretically) error free recovery of a function from its samples. In fact,
any hat function with a band limited Fourier transform will do the job.
Such another bandlimited hat function has been found in the subsection
Inverse Fourier transform method. It is:
$$
p(x) = \frac{1}{\sigma}
\left[\frac{\sin(\pi/\sigma\, .\,x)}{\pi/\sigma\, .\,x}\right]^2
= \frac{1}{\sigma} \sinc^2\left(\frac{\pi}{\sigma} x\right)
$$ $$
\slechts A(y) = \left\{ \begin{array}{lll}
0 & \mbox{for} & y \le -2\pi/\sigma \\
1+y\sigma/(2\pi) & \mbox{for} & -2\pi/\sigma \le y \le 0 \\
1-y\sigma/(2\pi) & \mbox{for} & 0 \le y \le +2\pi/\sigma \\
0 & \mbox{for} & +2\pi/\sigma \le y
\end{array} \right.
$$
Resulting in this comb of squared sinc functions:
$$
\frac{\Delta}{\sigma} \sum_{L=-\infty}^{+\infty}
\sinc^2\left(\frac{\pi}{\sigma}\left[x-L\Delta\right]\right) =
1 + \sum_{k=1}^\infty A(k\omega)\,\cos(k\omega x)
$$
Because $\,A(y)\,$ is a triangle which is zero for $\,|y| \le 2\pi/\sigma\,$, it is
clear that $\,A(k\omega) = 0\,$ for all $\,k > 0\,$ iff:
$$
A(\omega) = 0 \slechts \frac{2\pi}{\Delta} \le \frac{2\pi}{\sigma}
\slechts \sigma \ge \Delta
$$
And the true miracle has happened again. There is no error present, at
all, in the following formula. Which thus holds exactly, for any
$\,\sigma \ge \Delta\,$:
$$
\sum_{k=-\infty}^{+\infty} \frac{\Delta}{\sigma}
\sinc^2\left(\frac{\pi}{\sigma}\left[ x - k \Delta \right]\right) = 1
$$
And we are not finished yet. As I have said, any hat function with a band
limited Fourier transform will do the job. There is a thread in the Usenet /
Google newsgroup sci.math written by this author and called Sum of inverse cubes.
Yes, that's what you can do with combs of hat functions ! The
reference - and my fooling around with the illusion of a great new discovery -
can be found in the newsgroup at
sum of inverse almost cubes.
Note. The result would only have been brand new if a closed expression would
have been found for e.g. $\,\sum_{k=1}^\infty 1/k^3\,$. This is "a bit" different
from the discovery at hand, which on the contrary is a well known result.