What question did this study set out to answer?

To analyze the non-compactness of the Fourier transform as an operator on Lp spaces, identifying its degree of non-compactness.

Notes on Non-Compactness of Fourier Transforms on Lᵖ Spaces

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To analyze the non-compactness of the Fourier transform as an operator on Lp spaces, identifying its degree of non-compactness.
Investigated Fourier transform behavior on Lp spaces for 1<p<2 and 2<p≤∞.
Utilized dichotomy results to demonstrate the nature of the Fourier transform regarding non-compactness and singularity.
Classified the Fourier transform based on the range of p values and derived implications concerning boundedness.
Demonstrated that for 1<p<2, the Fourier transform is bounded but not strictly singular at r=p.
Showed that for r>p, the operator is finitely strictly singular, while for 2<p≤∞, it maps to spaces of bounded functions.
Quantified the degree of non-compactness through the use of Bernstein numbers.

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Randomized trial quantifies non-compactness of Fourier transforms in Lp spaces, indicating key properties.

Lang et al. (2026) studied this question.

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