Real-space computation of $E$/$B$-mode maps I: Formalism, Compact
  Kernels, and Polarized Filaments
  We derive full-sky, real-space operators that convert between polarization
Stokes Q/U parameters and the coordinate-independent scalar E/B modes that are
widely used in Cosmic Microwave Background (CMB) and cosmic shear analysis. We
also derive real space operators that decompose the measured Stokes parameters
into those corresponding to E-modes and B-modes respectively, without ever
evaluating the scalar fields themselves. For all these real space operators we
show that the kernels split naturally into angular and radial parts and we show
explicitly how the radial extent of these kernels depends on the targeted
band-limit. The kernels can be interpreted either as a complex convolving beam
or as a Green's function when they are expressed in terms of the forward or
inverse rotation Euler angles. We show that an arbitrary radial function can
produce E/B-like maps, provided it vanishes at the origin and the antipodal
point. These maps are simply filtered versions of the standard E/B maps. We
argue that it is possible compute E/B maps in real space with a
compactly-supported kernel, an approach that can guarantee the avoidance of
known foreground regions and could be employed in a massively-parallel scheme
at high-resolution. We show that the spin raising and lowering operators
$\eth^2$/$\bar{\eth}^2$ are special cases of these generalized radial
functions, and present their band limited versions.
  The spatial structure of the real space operators provides great intuition
for the E/B structure of polarized, filamentary galactic foregrounds. We
predict a non-zero B-mode signature that is expected from polarized filaments
in the sky. This paper is the first part in a series of papers that explore
real-space computation of polarization modes and their applications.
