Frequency-separated Fourier block series ($f={\sum }_j{q}_j,{\sum }_j\Vert q_j{\Vert }_{\infty }<\infty$)

If trigonometric polynomials have spectra in increasing disjoint positive-frequency intervals, the displayed summability gives a continuous uniform sum. At any cutoff its Fourier partial sum contains complete earlier blocks, at most one active prefix and no later blocks. Differences across an active prefix therefore isolate that block exactly. Fixed-size excursions occurring in arbitrarily late blocks contradict the Cauchy criterion, even when the complete blocks vanish at the point under consideration.