# Modification of Shannon wavelets for compactly supported maximally “square” wavelets?

1. The Shannon wavelets are maximally square in the sense that they are ideal band-pass filters in the Fourier domain.

2. Their amplitudes are literally box functions on frequency line and time shifts as we know correspond to phase ramps.

If we want to modify them in a way that aims at two goals:

1. Compact support. Every function (in time domain) is allowed only $$N$$ nonzero samples.

2. Maximal closeness (in $$L_2$$ sense in Fourier domain) to the ideal boxes described above.

I already have several numerical approaches as how to calculate such functions so I know it is possible to practically calculate them.

But can we prove or derive mathematically how they would be expressed or what properties they would have, using for example functional analysis and algebra?