Abstract
For each f ∈ Lp(ℝ) (1 ⩽ p < ∞) it is shown that the Fourier transform is the distributional derivative of a Hölder continuous function. For each p, a norm is defined so that the space of Fourier transforms is isometrically isomorphic to Lp(ℝ). There is an exchange theorem and inversion in norm.
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Dedicated to the memory of Jaroslav Kurzweil
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Talvila, E. The Fourier transform in Lebesgue spaces. Czech Math J (2024). https://doi.org/10.21136/CMJ.2024.0001-23
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DOI: https://doi.org/10.21136/CMJ.2024.0001-23
Keywords
- Fourier transform
- Lebesgue space
- tempered distribution
- generalised function
- Banach space
- continuous primitive integral