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In the present paper we introduce the generalized alternatingpolynomials , the coefficients of which are defined together with the parameter Wn by the linear system
where Tn(x) = cos (n arc cos x), is a set of n+2 distinct points in the interval [1, 1], and fis a continuous function on [1, 1], For the set of nodesxk = cos [k/(n+1)] the wn-polynomials coincide with the polynomialsof equiamplitude alternation introduced by de La Vall?e-Poussinand discussed in the literature earlier (Eterman, Malozemov,Meinardus, Cheney & Rivlin, Phillips & Taylor, Brutman,and others). It is shown that the generalized alternating polynomials arerelated to the polynomials of interpolation through the Lanczoseconomization process. Some approximation properties of wn -polynomialsand Wn -parameters are studied. The application of wn -polynomialsto function approximation and to the estimate of remainder termsfor quadrature formulas is discussed. 相似文献
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A polynomial of degree n in z1 and n1 in z isdefined by an interpolation projection from the space of functionsanalytic in the annulus r|z|R and continuous on its boundary.The points of interpolation are chosen to coincide with then roots of zn=Rnein (0<<2/n) and the n roots of zn=rn.The behaviour of the corresponding Lebesgue function is studied,and an estimate for the operator norm is obtained. The resultsof the present paper give a partial affirmative answer to twoconjectures suggested earlier by Mason on the basis of numericalcomputations. 相似文献
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