# V. T. Shevaldin. Uniform Lebesgue constants of local spline approximation ... P. 292-299.

Let a function $\varphi \in C^1[-h,h]$ be such that $\varphi (0)= \varphi '(0)=0$, $\varphi (-x)= \varphi (x)$ for $x\in [0;h])$, and $\varphi (x)$ is nondecreasing on $[0;h]$. For any function $f: \mathbb R\to \mathbb R$, we consider local splines of the form $$S(x)=S_{ \varphi }(f,x)=\sum_{j\in \mathbb Z} y_j B_{ \varphi }\Big( x+\frac{3h}{2}-jh\Big)\quad (x\in \mathbb R),$$ where $y_j=f(jh)$, $m(h)>0$, and $$B_{ \varphi }(x)=m(h)\left\{\begin{array}{cl} \varphi (x),& x\in [0;h],\\[1ex] 2 \varphi (h)- \varphi (x-h)- \varphi (2h-x),& x\in [h;2h],\\ \varphi (3h-x),& x\in [2h;3h],\\[1ex] 0, & x\not\in [0;3h]. \end{array} \right.$$ These splines become parabolic, exponential, trigonometric, etc., under the corresponding choice of the function $\varphi$. We study the uniform Lebesgue constants $L_{ \varphi }=\|S\|_C^C$ (the norms of linear operators from $C$ to $C$) of these splines as functions depending on $\varphi$ and $h$. In some cases, the constants are calculated exactly on the axis $\mathbb R$ and on a closed interval of the real line (under a certain choice of boundary conditions from the spline $S_{ \varphi }(f,x)$). Keywords: Lebesgue constants, local splines, three-point system.

The paper was received by the Editorial Office on June 2, 2017.

Valerii Trifonovich Shevaldin, Dr. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics Ural Branch of the Rassian Academy of Sciences, Yeraterinburg, 620990 Russia, e-mail: Valerii.Shevaldin@imm.uran.ru

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