On Integral Lebesgue Constants of Local Splines with Uniform Knots


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Abstract

We study the stability properties of generalized local splines of the form

\(S(x) = S(f,x) = \sum\limits_{j \in \mathbb{Z}} {{y_j}{B_\varphi }\left( {x + \frac{{3h}}{2} - jh} \right),}\;\;\; x \in \mathbb{R},\)
where ϕ ∈ C1[−h, h] for h > 0, ϕ(0) = ϕ′(0) = 0, ϕ(−x) = ϕ(x)for x ∈ [0; h], ϕ(x) is nondecreasing on [0; h], f is an arbitrary function from ℝ to ℝ, yj = f(jh) for j ∈ ℤ, and
\({B_\varphi }(x) = m(h)\left\{ {\begin{array}{*{20}{c}}
{\varphi (x),}&{x \in [0;h],} \\
{2\varphi (h) - \varphi (x - h) - \varphi (2h - x),}&{x \in [h;2h],} \\
{\varphi (3h - x),}&{x \in [2h;3h],} \\
{0,}&{x \notin [0;3h],}
\end{array}} \right.\)
with m(h) > 0. Such splines were constructed by the author earlier. In the present paper, we calculate the exact values of their integral Lebesgue constants (the norms of linear operators from l to L) on the axis ℝ and on any segment of the axis for a certain choice of the boundary conditions and the normalizing factor m(h) of the spline S.

About the authors

V. T. Shevaldin

Krasovskii Institute of Mathematics and Mechanics

Author for correspondence.
Email: Valerii.Shevaldin@imm.uran.ru
Russian Federation, Yekaterinburg, 620990

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