Beta-Kernel Truncation Bases for the Functional Renormalization Group: Complete Error Analysis and Adaptive Nyström Propagator Approximation

We construct a family of Beta-kernel basis functions u_m^(p)(t) = Cm,p t^m(1−t)^p for truncating the effective action and performing Nyström approximation of the regularised propagator in the Functional Renormalization Group (FRG). Working in one-dimensional quantum mechanics with a mass regulator R_k = k², we derive the closed-form Green's function and establish three main results: a complete integration-by-parts error bound for the monomial truncation basis yielding cumulative truncation error E_N = O(N⁻¹); a rank-one collapse theorem showing the regularised propagator reduces to a rank-one operator in the large-m limit; and a closed ODE system for propagator values and adaptive Nyström nodes via the Dirac–Frenkel–McLachlan variational principle, with exact self-consistency for a single node at the midpoint. All analytical estimates are confirmed by explicit numerical computations at M_k = 1.