Finite-Window Fourier–Stieltjes Recovery: Sharp Unrestricted Cyclic Leading Costs and Uniform Rotation Bounds

Description We study the minimum sum of total-variation norms of complex measures whose rotation-difference Fourier multipliers jointly reproduce the derivative multiplier on a complete symmetric finite frequency window. For fixed coprime integers a and b, with M = max(a, b), the principal result proves that the unrestricted recovery cost for phases $a/Q$ and $b/Q$ on frequencies 0 < k < Q satisfies I₃(Q-1; a/Q, b/Q) = Q²/2π M + Oa,b(Q3/2) as Q tends to infinity. The matching leading upper and lower bounds come from a positive cyclic Fourier-moment kernel, an exact signed-frequency repair on a second grid, and a single-mode obstruction. The manuscript also develops uniform one-rotation estimates, CRT–Riesz synthesis, support-restricted cyclic optima, and quantitative small-phase perturbation results. The universal all-window estimate for an arbitrary fixed rationally independent pair remains open. This is a research preprint and has not undergone journal peer review. Creator Lee Byoungwoo Independent Researcher, Daejeon, Republic of Korea Keywords Fourier multipliers; total-variation norm; derivative multiplier; rotation-difference; coprime integers; cyclic Fourier-moment; signed-frequency repair; CRT-Riesz synthesis; asymptotic recovery cost; small-phase perturbation; operator ideals; harmonic analysis