The structure of these functions is readily apparent from geometrical considerations. A single plane wave contributes to a range of angular scales from at to larger angles as , where (see Fig. 1). The power in of a single plane wave shown in Fig. 3(a) (top panel) drops to zero , has a concentration of power around and an extended low amplitude tail to .

Now if the plane wave is multiplied by an intrinsic angular dependence, the projected power changes. The key to understanding this effect is to note that the intrinsic angular behavior is related to power in as

Similarly, the structures of , and are apparent from
the intrinsic angular dependences
of the *G _{1}*

(14) |

There are two interesting consequences of this behavior.
The sharpness of the radial function around quantifies
how faithfully features in the *k*-space spectrum are preserved
in -space.
If all else is equal, this faithfulness increases with |*m*| for
*G*_{|m|}^{m}
due to aliasing suppression from
On the other hand, features in *G*_{|m|+1}^{m} are
washed out in comparison due projection suppression from the
factor.

Secondly, even if there are no contributions from long wavelength sources with , there will still be large angle anisotropies at which scale as

(15) |

The same arguments apply to the spin-2 functions with the
added complication of the appearance of two radial functions
and .
The addition of spin-2 angular momenta introduces
a
-contribution from except for *m*=0. For , the -contribution strongly
dominates over the -contributions;
whereas for , -contributions are slightly
larger than -contributions (see Fig. 4).
The ratios reach the asymptotic values of

(16) |

Now let us consider the low tail
of the spin-2 radial functions.
Unlike the
spin-0 projection, the spin-2 projection allows increasingly
more power at and/or ,i.e. , as
|*m*| increases (see Table 1 and note the factors of ).
In this limit, the power
in a plane wave fluctuation goes as

(17) |

Finally it is interesting to consider the cross power between
spin- and spin-2 sources because it will be used
to represent the temperature-polarization cross correlation.
Again interesting
geometric effects can be identified
(see Fig. 5).
For *m*=0, the power in correlates
(Fig. 5, top panel solid line, positive definite);
for *m*= 1,
oscillates (short dashed line)
and for *m*=2,
anticorrelates (long dashed line,
negative definite).
The cross power involves only due to the opposite parity of the modes.

These properties will become important in §
III and IVB
and translates into cross power contributions with
*opposite* sign between the scalar monopole temperature cross
polarization sources
and tensor quadrupole temperature cross polarization sources
[9].
Vector dipole temperature and polarization
sources do not contribute strongly to the cross power since
correlations and anticorrelations in will cancel when modes are
superimposed. The same is true of the scalar dipole
temperature cross polarization as is apparent from
Figs. 3 and 4.
The vector cross power is dominated by
quadrupole temperature and polarization sources
(Fig. 5 lower panel).