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On N-termed approximations in H s-norms with respect to the Haar system

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Abstract

In [9], we proved numerically that spaces generated by linear combinations of some two-dimensional Haar functions exhibit unexpectedly nice orders of approximation for solutions of the single-layer potential equation in a rectangle. This phenomenon is closely related, on the one hand, to the properties of the approximation method of hyperbolic crosses and on the other to the existence of a strong singularity for solutions of such boundary integral equations. In the present paper, we establish several results on the approximation for the hyperbolic crosses and on the best N-term approximations by linear combinations of Haar functions in the H s-norms, −1 < s < 1/2; this provides a theoretical base for our numerical research. To the author's best knowledge, the negative smoothness case s < 0 was not studied earlier.

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  1. Bell Labs, Lucent Technologies, 600 Mountain Ave, Rm. 2C403, Murray Hill, NJ, 07974-0636, USA

    P. Oswald

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  1. P. Oswald
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Correspondence to P. Oswald.

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Translated from Sovremennaya Matematika. Fundamental'nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 25, Theory of Functions, 2007.

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Oswald, P. On N-termed approximations in H s-norms with respect to the Haar system. J Math Sci 155, 109–128 (2008). https://doi.org/10.1007/s10958-008-9213-1

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