  We construct the spin flaglet transform, a wavelet transform to analyze spin
signals in three dimensions. Spin flaglets can probe signal content localized
simultaneously in space and frequency and, moreover, are separable so that
their angular and radial properties can be controlled independently. They are
particularly suited to analyzing of cosmological observations such as the weak
gravitational lensing of galaxies. Such observations have a unique 3D
geometrical setting since they are natively made on the sky, have spin angular
symmetries, and are extended in the radial direction by additional distance or
redshift information. Flaglets are constructed in the harmonic space defined by
the Fourier-Laguerre transform, previously defined for scalar functions and
extended here to signals with spin symmetries. Thanks to various sampling
theorems, both the Fourier-Laguerre and flaglet transforms are theoretically
exact when applied to bandlimited signals. In other words, in numerical
computations the only loss of information is due to the finite representation
of floating point numbers. We develop a 3D framework relating the weak lensing
power spectrum to covariances of flaglet coefficients. We suggest that the
resulting novel flaglet weak lensing estimator offers a powerful alternative to
common 2D and 3D approaches to accurately capture cosmological information.
While standard weak lensing analyses focus on either real or harmonic space
representations (i.e., correlation functions or Fourier-Bessel power spectra,
respectively), a wavelet approach inherits the advantages of both techniques,
where both complicated sky coverage and uncertainties associated with the
physical modeling of small scales can be handled effectively. Our codes to
compute the Fourier-Laguerre and flaglet transforms are made publicly
available.
