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The use of fractional differential equations is a key tool in modelingnon-local phenomena. Often, an efficient scheme for solving a linear systeminvolving the discretization of a fractional operator is evaluating the matrixfunction $x = \mathcal A^{-\alpha} c$, where $\mathcal A$ is a discretizationof the classical Laplacian, and $\alpha$ a fractional exponent between $0$ and$1$. In this work, we derive an exponential sum approximation for $f(z)=z^{-\alpha}$ that is accurate over $[1, \infty)$ and allows to efficientlyapproximate the action of bounded and unbounded operators of this kind ontensors stored in a variety of low-rank formats (CP, TT, Tucker). The resultsare relevant from a theoretical perspective as well, as they predict thelow-rank approximability of the solutions of these linear systems in low-ranktensor formats.

Low-rank tensor structure preservation in fractional operators by means of exponential sums