L∞ estimation of tensor truncations
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Submission date: 19. Mar. 2012 (revised version: April 2012)
published in: Numerische Mathematik, 125 (2013) 3, p. 419-440
DOI number (of the published article): 10.1007/s00211-013-0544-6
MSC-Numbers: 15A69, 15A18, 35J08, 46B70
Keywords and phrases: tensor calculus, tensor truncation, higher-order singular value decomposition (HOSVD), approximation, Gagliardo-Nirenberg inequality, Green function
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Tensor truncation techniques are based on singular value decompositions. Therefore, the direct error control is restricted to ℓ2 or L2 norms. On the other hand, one wants to approximate multivariate (grid) functions in appropriate tensor formats in order to perform cheap pointwise evaluations, which require ℓ∞ or L∞ error estimates. Due to the huge dimensions of the tensor spaces, a direct estimate of ∞ by 2 is hopeless. In the paper we prove that, nevertheless, in cases where the function to be approximated is smooth, reasonable error estimates with respect to ∞ can be derived from the Gagliardo-Nirenberg inequality because of the special nature of the SVD truncation.