Discrete transform; norms

Let

a = (a_1, a_2, a_3, ...)

be a sequence of real or complex numbers (understood to decay fairly
rapidly or have compact support). Define the transform xi of a as
follows:

xi_d = \sum_l a_{d l}

It is easy to see that (if everything decays nicely) this transform is
invertible:
a_d = \sum_l \mu(l) xi_{d l}

Questions:
(1) Does this transform has a name? Has it been studied?
(2) In particular, do we know how the l_2 norm behaves under the
transform? Weigh the l_2 norm as desired/needed. (Example: if
l_2(a):=\sum_d d a_d^2, and we are given that
a_1=1 and a_d has support on [1,D], how close can |a|_2 + |xi|_2 be to
1? Where have similar questions been treated?)

Harald
helfgott@xxxxxxxxxxxxxxxx

.