Editing Signal reconstruction (section)

== Popular reconstruction formulae ==
Perhaps the most widely used reconstruction formula is as follows. Let <math>\{ e_k \}</math> be a basis of <math>L^2</math> in the Hilbert space sense; for instance, one could use the eikonal

:<math>e_k(t):=e^{2\pi i k t}\,</math>,

although other choices are certainly possible. Note that here the index ''k'' can be any integer, even negative.

Then we can define a linear map ''R'' by

:<math>R(d_k)=e_k\,</math>

for each <math>k=\lfloor -n/2 \rfloor,...,\lfloor (n-1)/2 \rfloor</math>, where <math>(d_k)</math> is the basis of <math>\mathbb C^n</math> given by

:<math>d_k(j)=e^{2 \pi i j k \over n}</math>

(This is the usual discrete Fourier basis.)

The choice of range <math>k=\lfloor -n/2 \rfloor,...,\lfloor (n-1)/2 \rfloor</math> is somewhat arbitrary, although it satisfies the dimensionality requirement and reflects the usual notion that the most important information is contained in the low frequencies. In some cases, this is incorrect, so a different reconstruction formula needs to be chosen.

A similar approach can be obtained by using [[wavelet]]s instead of Hilbert bases. For many applications, the best approach is still not clear today.{{or?|date=December 2020}}