Abstract:
Even positive-definite splines with support in $[-1,1]$ that are equal to real algebraic polynomials on $[0,1]$ are investigated. Examples of such splines are presented. Under consideration are the $e$-splines, which have several extremal properties, and the positive-definite $A$-splines, which have the maximum possible smoothness on $\mathbb R$. An estimate of the approximation by a linear combination of shifts of an $A$-spline is indicated. New relations for the hypergeometric function ${_1F_2}$ are found.
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