For citation:
Lukomskii S. F., Mushko M. D. On Binary B-splines of Second Order. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2018, vol. 18, iss. 2, pp. 172-182. DOI: 10.18500/1816-9791-2018-18-2-172-182, EDN: XQFNRB
On Binary B-splines of Second Order
The classical B-spline is defined recursively as the convolution Bn+1 = Bn ∗ B0, where B0 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 Bn 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 (Vn) in De Boor sense. This MRA is not Riesz. But we can find the approximation order of functions from the Sobolev spaces Ws 2 , s > 0 by the subspaces (Vn).
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