Revista Matemática Iberoamericana

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Volume 31, Issue 2, 2015, pp. 461–476
DOI: 10.4171/RMI/841

Published online: 2015-07-16

Exponentially sparse representations of Fourier integral operators

Elena Cordero[1], Fabio Nicola[2] and Luigi Rodino[3]

(1) Università degli Studi di Torino, Italy
(2) Politecnico di Torino, Italy
(3) Università degli Studi di Torino, Italy

We investigate the sparsity of the Gabor-matrix representation of Fourier integral operators with a phase having quadratic growth. It is known that such an infinite matrix is sparse and well organized, being in fact concentrated along the graph of the corresponding canonical transformation. Here we show that, if the phase and symbol have a regularity of Gevrey type of order $s>1$ or analytic ($s=1$), the above decay is in fact sub-exponential or exponential, respectively. We also show by a counterexample that ultra-analytic regularity ($s<1$) does not give super-exponential decay. This is in sharp contrast to the more favorable case of pseudodifferential operators, or even (generalized) metaplectic operators, which are treated as well.

Keywords: Fourier integral operators, Gelfand–Shilov spaces, short-time Fourier transform, Gabor frames, sparse representations, Schrödinger equations

Cordero Elena, Nicola Fabio, Rodino Luigi: Exponentially sparse representations of Fourier integral operators. Rev. Mat. Iberoam. 31 (2015), 461-476. doi: 10.4171/RMI/841