Geodesic
Binary basic splines in MRA
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 21 (2021) no. 4, pp. 458-471

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$B$-splines were introduced by Carry and Schoenberg. Constructed on a uniform mesh and defined in terms of convolutions, such splines generate a Riesz MRA. We constructed splines $\varphi_n$, where $n$ is the order of integration of the Walsh function with the number $2^n - 1$. We called these splines binary basic splines. We know that binary basic splines form a basis in the space of functions that are continuous on the segment $[0, 1]$ and $0$ outside of it. We proved that binary basic splines are a scaling function and generate an MRA of $(V_n)$ which is not a Riesz MRA. The order of approximation was determined by subspaces from Sobolev spaces. 
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     author = {S. A. Chumachenko},
     title = {Binary basic splines in {MRA}},
     journal = {Izvestiya of Saratov University. Mathematics. Mechanics. Informatics},
     pages = {458--471},
     publisher = {mathdoc},
     volume = {21},
     number = {4},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ISU_2021_21_4_a4/}
}
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S. A. Chumachenko. Binary basic splines in MRA. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 21 (2021) no. 4, pp. 458-471. http://geodesic.mathdoc.fr/item/ISU_2021_21_4_a4/