Fourier-Laguerre transform, convolution and wavelets on the ball

Understand

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.