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 classes \(W_{\infty}^{2}\) and \(W_{\infty}^{3}\) by these splines in the uniform metric and calculate the exact uniform Lebesgue constants and the norms of the second derivatives on the class \(W_{\infty}^{2}\). These properties are compared with the corresponding properties of Subbotin’s splines.

中文翻译：

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带附加结的局部抛物线样条线的构造方法

我们提出了一种构造局部抛物线样条的通用方法，该方法可在数字轴或其分段的网格子集上给定函数的结的任意排列。此方案的特殊情况是Yu。N. Subbotin和B. I. Kvasov的样条曲线。对于Kvasov的样条曲线，我们认为边界条件与Kvasov建议的边界条件不同。我们研究在均匀结的情况下这些样条的逼近和平滑特性。特别是，我们在这些样条线中找到了函数类\（W _ {\ infty} ^ {2} \）和\（W _ {\ infty} ^ {3} \的近似误差的双面估计。统一度量，并计算类\（W _ {\ infty} ^ {2} \）上的确切统一Lebesgue常数和二阶导数的范数 。将这些属性与Subbotin样条的相应属性进行比较。