Geodesic
On $N$-Termed Approximations in $H^s$-Norms with Respect to the Haar System
Contemporary Mathematics. Fundamental Directions, Theory of functions, Tome 25 (2007), pp. 106-125

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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 $N$-term approximations by linear combinations of Haar functions in the $H^s$-norms, $-1$; this provides a theoretical base for our numerical research. To the author best knowledge, the negative smoothness case $s0$ was not studied earlier. 
@article{CMFD_2007_25_a9,
     author = {P. Oswald},
     title = {On $N${-Termed} {Approximations} in $H^s${-Norms} with {Respect} to the {Haar} {System}},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {106--125},
     publisher = {mathdoc},
     volume = {25},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2007_25_a9/}
}
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P. Oswald. On $N$-Termed Approximations in $H^s$-Norms with Respect to the Haar System. Contemporary Mathematics. Fundamental Directions, Theory of functions, Tome 25 (2007), pp. 106-125. http://geodesic.mathdoc.fr/item/CMFD_2007_25_a9/