Abstract:
In the paper [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 hyperbolic crosses approximation method 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 NN-term approximations by linear combinations of Haar functions in the HsHs-norms, −1<s<1/2−1<s<1/2; this provides a theoretical base for our numerical research. To the author best knowledge, the negative smoothness case s<0s<0 was not studied earlier.
Citation:
P. Oswald, “On NN-Termed Approximations in HsHs-Norms with Respect to the Haar System”, Theory of functions, CMFD, 25, PFUR, M., 2007, 106–125; Journal of Mathematical Sciences, 155:1 (2008), 109–128
\Bibitem{Osw07}
\by P.~Oswald
\paper On $N$-Termed Approximations in $H^s$-Norms with Respect to the Haar System
\inbook Theory of functions
\serial CMFD
\yr 2007
\vol 25
\pages 106--125
\publ PFUR
\publaddr M.
\mathnet{http://mi.mathnet.ru/cmfd110}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2342542}
\zmath{https://zbmath.org/?q=an:1160.41007}
\elib{https://elibrary.ru/item.asp?id=13955390}
\transl
\jour Journal of Mathematical Sciences
\yr 2008
\vol 155
\issue 1
\pages 109--128
\crossref{https://doi.org/10.1007/s10958-008-9213-1}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-55749109173}
Linking options:
https://www.mathnet.ru/eng/cmfd110
https://www.mathnet.ru/eng/cmfd/v25/p106
This publication is cited in the following 4 articles:
Chernov A., Reinarz A., “Sparse Grid Approximation Spaces For Space-Time Boundary Integral Formulations of the Heat Equation”, Comput. Math. Appl., 78:11 (2019), 3605–3619
Wang H., “Widths between the anisotropic spaces and the spaces of functions with mixed smoothness”, J. Approx. Theory, 164:3 (2012), 406–430
Zeiser A., “Wavelet Approximation in Weighted Sobolev Spaces of Mixed Order with Applications to the Electronic Schrodinger Equation”, Constr. Approx., 35:3 (2012), 293–322
Stasyuk S.A., “Best mm-term approximation of the classes Br∞,θBr∞,θ of functions of many variables by polynomials in the Haar system”, Ukrainian Math. J., 63:4 (2011), 638–645
Citation in format AMSBIB
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