Low-Memory Evaluation of Associated Legendre Functions for Scattered-Point Spherical Harmonic Synthesis

In scattered-point spherical harmonic synthesis, repeated pointwise evaluation of associated Legendre functions is a major computational kernel. Recurrence methods use little memory but contain sequential degree dependence, whereas phase-function evaluators provide high accuracy using larger resident tables. We present a guarded hybrid evaluator that moves the high-accuracy reference evaluator to offline precomputation and replaces it at runtime with a compact three-dimensional Chebyshev table of smooth phase-related quantities. The surrogate path is used in numerically safe regions, and a C++ scaled recurrence with exponent tracking provides the fallback near turning points and in exponentially decaying regions. For bandlimits up to 4096, the runtime table is 3.66 MiB, about 60 times smaller than the public ALegendreEval table set used here. Across sparse scattered-point benchmarks, the method agrees with the reference evaluator at 10−11-level output differences and achieves representative speedups of 2–5 over an optimized C++ recurrence baseline; even unfavorable cases fall back to recurrence-level runtime. These results show that reference-level Legendre evaluation can be reorganized into a low-resident-memory kernel suitable for repeated scattered-point synthesis.