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Scientific Part
Mathematics
On binary B-splines of second order
S. F. Lukomskii, M. D. Mushko Saratov State University, 83, Astrakhanskaya Str., Saratov, 410012, Russia
Abstract:
The classical B-spline is defined recursively as the convolution $B_{n+1}=B_n*B_0$, where $B_0$ is the characteristic function of the unit interval. The classical B-spline is a refinable function and satisfies the Riesz inequality. Therefore any B-spline $B_n$ generates the Riesz multiresolution analysis (MRA). We define binary B-splines, obtained by double integration of the third Walsh function. We give an algorithm for constructing an interpolating spline of the second degree for a binary node system and find the approximation order of this interpolation process. We also prove that the system of dilations and shifts of the constructed B-spline generates an MRA $ (V_n) $ in De Boor sense. This MRA is not Riesz. But we can find the approximation order of functions from the Sobolev spaces $W_2^s, s>0$ by the subspaces $ (V_n) $.
Key words:
binary B-splines, multiresolution analysis, Sobolev spaces.
Citation:
S. F. Lukomskii, M. D. Mushko, “On binary B-splines of second order”, Izv. Saratov Univ. Math. Mech. Inform., 18:2 (2018), 172–182
Linking options:
https://www.mathnet.ru/eng/isu753 https://www.mathnet.ru/eng/isu/v18/i2/p172
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