A Method for the Construction of LocalParabolic Splines with Additional Knots
- PDF / 450,921 Bytes
- 16 Pages / 612 x 792 pts (letter) Page_size
- 42 Downloads / 145 Views
Method for the Construction of Local Parabolic Splines with Additional Knots Yu. N. Subbotin1,∗ and V. T. Shevaldin1,∗∗ Received February 8, 2019; revised March 26, 2019; accepted April 1, 2019
Abstract—We propose a general method for the construction of local parabolic splines with an arbitrary arrangement of knots for functions given on grid subsets of the number axis or its segment. Special cases of this scheme are Yu. N. Subbotin’s and B. I. Kvasov’s splines. For Kvasov’s splines, we consider boundary conditions different from those suggested by Kvasov. We study the approximating and smoothing properties of these splines in the case of uniform knots. In particular, we find two-sided estimates for the error of approximation of the function 2 3 and W∞ by these splines in the uniform metric and calculate the exact uniform classes W∞ 2 Lebesgue constants and the norms of the second derivatives on the class W∞ . These properties are compared with the corresponding properties of Subbotin’s splines. Keywords: local parabolic splines, approximation, interpolation, equally spaced knots.
DOI: 10.1134/S0081543820040173 INTRODUCTION Earlier the authors constructed (see [1, 2]) local parabolic splines (for functions defined on the axis or on a closed interval) that fix linear functions, have an arbitrary arrangement of knots, and possess nice approximation properties and the properties of local preservation of the sign, monotonicity, and convexity of the approximated functions (see, for example, [3]). These splines and their generalizations were widely applied in problems of computational mathematics (see, for example, [4–6]). In the present paper, we give a general scheme for the construction of such splines; a special case of this scheme was used for the construction of splines in [1, 2]. In addition, we give a detailed analysis of another special case of the general scheme; in this case, we also construct local parabolic splines both for functions defined on the axis and for functions defined on a closed interval (with a certain choice of boundary conditions at the ends of the interval). Yu. S. Volkov noted that the scheme of these splines, which is described below, has already been used in Kvasov’s paper [7, formulas (1.1) and (3.1)], but the splines considered there had different boundary conditions. In contrast to the widely known schemes of construction of local splines (see [8–10]), Kvasov’s parabolic splines interpolate the values of the approximated function at the main knots of the spline (and additional knots are chosen at the midpoints of the intervals between neighboring main knots). Finally, in the present paper, we study the approximation and smoothing properties of the constructed splines on classes of differentiable functions in the case where a spline has equidistant knots; we also compare the properties of the mentioned schemes. 1
Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia e-mail: ∗ [email protected], ∗∗ Valerii.Shevaldin@imm.
Data Loading...
