  We describe an efficient algorithm for computing the matrix vector products
that appear in the numerical resolution of boundary integral equations in 2
space dimension. This work is an extension of the so-called Sparse Cardinal
Sine Decomposition algorithm by Alouges et al., which is restricted to
three-dimensional setups. Although the approach is similar, significant
differences appear throughout the analysis of the method. Bessel decomposition,
in particular, yield longer series for the same accuracy. We propose a careful
study of the method that leads to a precise estimation of the complexity in
terms of the number of points and chosen accuracy. We also provide numerical
tests to demonstrate the efficiency of this approach. We give the compression
performance for a $N \times N$ linear system for several values $N$ up to
$10^7$ and report the computation time for the off-line and on-line parts of
our algorithm. We also include a toy application to sound canceling to further
illustrate the efficiency of our method.
