Geodesic
Fast computation of linear combinations of Sobolev functions generated by the Haar functions
Daghestan Electronic Mathematical Reports, Tome 9 (2018), pp. 7-14

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Some properties of orthogonal with respect to Sobolev inner product functions $\chi_{1,n}(x)$, generated by Haar functions, are considered. In particular, recurrent formulas for $\chi_{1,n}(x)$ are obtained. It is developed an algorithm for calculation of the linear combinations of $N$ functions $\chi_{1,n}(x)$ using $O(\log N)$ operations.
Keywords: Haar system, numerical method, Sobolev type inner product, fast algorithm. 
@article{DEMR_2018_9_a1,
     author = {M. G. Magomed-Kasumov and S. R. Magomedov},
     title = {Fast computation of linear combinations of {Sobolev} functions generated by the {Haar} functions},
     journal = {Daghestan Electronic Mathematical Reports},
     pages = {7--14},
     publisher = {mathdoc},
     volume = {9},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DEMR_2018_9_a1/}
}
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M. G. Magomed-Kasumov; S. R. Magomedov. Fast computation of linear combinations of Sobolev functions generated by the Haar functions. Daghestan Electronic Mathematical Reports, Tome 9 (2018), pp. 7-14. http://geodesic.mathdoc.fr/item/DEMR_2018_9_a1/