arXiv:2610.19797v1 Announce Type: new Abstract: Non-periodic functions on closed intervals are extended into sufficiently smooth periodic functions through a continuous Hermite interpolation framework with necessary and sufficient endpoint conditions, using smooth blend-to-zero operators and short extrapolation recurrences. A matrix-free method is introduced for generating these smooth periodic extensions on the fly directly from function values at equidistant points. No precomputed continuation operator or auxiliary basis construction is required. The resulting extensions can then be used for FFT-based differentiation of non-periodic functions. Numerical tests are provided to show the effectiveness of the method, including for the highly accurate solution of a 1D linear advection equation.