Smooth periodic extensions by extrapolation and blending for FFT-based differentiation

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.

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BibTeX RIS

BibTeX

@misc{mendezcruz2026smooth,
  title = {{Smooth periodic extensions by extrapolation and blending for FFT-based differentiation}},
  author = {Ivan Méndez-Cruz and Faisal Amlani},
  howpublished = {HAL},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.19797},
  eprint = {2610.19797},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05798470v1},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Smooth periodic extensions by extrapolation and blending for FFT-based differentiation
AU  - Ivan Méndez-Cruz
AU  - Faisal Amlani
PY  - 2026
DA  - 2026/10/13
PB  - HAL
DO  - 10.48550/arxiv.2610.19797
UR  - https://hal.science/hal-05798470v1
AB  - 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.
ER  -

Abstract

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