Geodesic
Approximation by step functions of functions belonging to Sobolev spaces
Izvestiya. Mathematics , Tome 71 (2007) no. 1, pp. 149-180

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The paper deals with the approximation of functions belonging to the Sobolev spaces $W^1_\infty$ and $W^1_2$ by functions of the form $\varphi=\sum_{k=1}^n a_k \chi_{[x_k,x_k+d]}$. The results obtained are applied to the study of the stability of solutions of non-linear second-order differential equations of a special form. We consider the problem of whether two solutions can coincide given supplementary information in terms of the values of the functionals $l_{x_k}(u)=\frac{1}{d}\int_{x_k}^{x_k+d}u(t)\,dt$, $k=1,\dots,n$, defined on the solutions. 
@article{IM2_2007_71_1_a7,
     author = {T. Yu. Semenova},
     title = {Approximation by step functions of functions belonging to {Sobolev} spaces},
     journal = {Izvestiya. Mathematics },
     pages = {149--180},
     publisher = {mathdoc},
     volume = {71},
     number = {1},
     year = {2007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2007_71_1_a7/}
}
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T. Yu. Semenova. Approximation by step functions of functions belonging to Sobolev spaces. Izvestiya. Mathematics , Tome 71 (2007) no. 1, pp. 149-180. http://geodesic.mathdoc.fr/item/IM2_2007_71_1_a7/