Geodesic
Systems of functions orthogonal in the sense of Sobolev associated with Haar functions and the Cauchy problem for ODEs
Daghestan Electronic Mathematical Reports, Tome 7 (2017), pp. 1-15

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We consider systems of functions ${\mathcal{X}}_{r,n}(x)$ $(r=1,2,\ldots, n=0,1,\ldots)$, generated by Haar functions $\chi_{n}(x)$ $(n=1,2,\ldots)$, that form the Sobolev orthonormal system with respect to the scalar product of the following form $$. It is shown that the Fourier series and sums with respect to the system ${\mathcal{X}}_{r,n}(x)$ $(n=0,1,\ldots)$ are a convenient and very effective tool for the approximate solution of the Cauchy problem for ordinary differential equations (ODEs).
Keywords: systems of functions orthogonal in the sense of Sobolev, Haar functions, the Cauchy problem for an ODE. 
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     author = {I. I. Sharapudinov and S. R. Magomedov},
     title = {Systems of functions orthogonal in the sense of {Sobolev} associated with {Haar} functions and the {Cauchy} problem for {ODEs}},
     journal = {Daghestan Electronic Mathematical Reports},
     pages = {1--15},
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     year = {2017},
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     url = {http://geodesic.mathdoc.fr/item/DEMR_2017_7_a0/}
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I. I. Sharapudinov; S. R. Magomedov. Systems of functions orthogonal in the sense of Sobolev associated with Haar functions and the Cauchy problem for ODEs. Daghestan Electronic Mathematical Reports, Tome 7 (2017), pp. 1-15. http://geodesic.mathdoc.fr/item/DEMR_2017_7_a0/