Papers › Fast expansion into harmonics on the ball
Fast expansion into harmonics on the ball
Joe Kileel, Nicholas F. Marshall, Oscar Mickelin, Amit Singer
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We devise fast and provably accurate algorithms to transform between an N×N ×N Cartesian voxel representation of a three-dimensional function and its expansion into the {ball harmonics}, that is, the eigenbasis of the Dirichlet Laplacian on the unit ball in ℝ³. Given ε> 0, our algorithms achieve relative ℓ¹ - ℓ^∞ accuracy ε in time O(N³ (logN)² + N³ |logε|²), while the na\"{i}ve direct application of the expansion operators has time complexity O(N⁶). We illustrate our methods on numerical examples.
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