Fourier-Laguerre transform, convolution and wavelets on the ball

We review the Fourier-Laguerre transform, an alternative harmonic analysis on the three-dimensional ball to the usual Fourier-Bessel transform. The Fourier-Laguerre transform exhibits an exact quadrature rule and thus leads to a sampling theorem on the ball. We study the definition of convolution on the ball in this context, showing explicitly how translation on the radial line may be viewed as convolution with a shifted Dirac delta function. We review the exact Fourier-Laguerre wavelet transform on the ball, coined flaglets, and show that flaglets constitute a tight frame.