On $N$-Termed Approximations in $H^s$-Norms with Respect to the Haar System

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, $-1<s<1/2$; this provides a theoretical base for our numerical research. To the author best knowledge, the negative smoothness case $s<0$ was not studied earlier.