Revista Matemática Iberoamericana


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Volume 35, Issue 7, 2019, pp. 1997–2024
DOI: 10.4171/rmi/1107

Published online: 2019-06-28

Quantitative invertibility and approximation for the truncated Hilbert and Riesz transforms

Angkana Rüland[1]

(1) Max-Planck Institut für Mathematik in den Naturwissenschaften, Leipzig, Germany

In this article we derive quantitative uniqueness and approximation properties for (perturbations) of Riesz transforms. Seeking to provide robust arguments, we adopt a PDE point of view and realize our operators as harmonic extensions, which makes the problem accessible to PDE tools. In this context we then invoke quantitative propagation of smallness estimates in combination with qualitative Runge approximation results. These results can be viewed as quantifications of the approximation properties which have recently gained prominence in the context of nonlocal operators.

Keywords: Truncated Hilbert transform, perturbations of truncated Riesz transforms, stability, approximation, cost of approximation

Rüland Angkana: Quantitative invertibility and approximation for the truncated Hilbert and Riesz transforms. Rev. Mat. Iberoam. 35 (2019), 1997-2024. doi: 10.4171/rmi/1107