Some Poised Lacunary Interpolation Polynomials

概要

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An interpolation similar to the (0,2)-polynomial is investigated. This interpolation is defined by a polynomial of degree at most k+1 whose value is prescribed at two end-points, x_<1k> and x_<kk>, together with its second derivatives at k points, x_<1k>, x_<2k>,. . ., x_<kk>. These points are arbitrary real numbers ordered as -1≤x_<1k><x_<2k>< . . .<x_<kk>≤1. In the present paper, such interpolation for any integer k≥2 is constructed in an explicit form and the uniform xonvergence of the interpolation is discussed for some classes of functions. Specifically, two convergence theorems are obtained when both end-points,x_<1k> and x_<kk> are -1 and 1, respectively. Namely, one theorem gives a sufficient condition under which the interpolation for every f∈C^2[-1,1] converges uniformly to f in the interval as the number of interpolation points increases infinitely. In the other theorem, another sufficient condition is also given. In particular, this condition is useful in the case where in. terpolation points consist of zeroes of the polynomial of the form,(x^2-1)p_<k-2>(x), where p_<k-2> is the (k-2)-nd orthogonal Polynomial over the open interval (-1,1). Here, a class of sufficiently smooth function is assumed. A computational algorithm for the interpolation polynomial is also constructed.
一般社団法人情報処理学会の論文
1982-03-31