Let υ be a weight on (−1, 1), i.e., a measurable integrable nonnegative function nonzero almost everywhere on (−1, 1). Denote by L^υ(−1, 1) the space of real-valued functions f integrable with weight υ on (−1, 1) with the norm \(f = \int\limits_{ - 1}^1 {f(x)v(x)dx.} \). We consider the problems of the best one-sided approximation (from below and from above) in the space Lυ(−1, 1) to the characteristic function of an interval (a, b), −1 < a < b < 1, by the set of algebraic polynomials of degree not exceeding a given number. We solve the problems in the case where a and b are nodes of a positive quadrature formula under some conditions on the degree of its precision as well as in the case of a symmetric interval (−h, h), 0 < h < 1, for an even weight υ.

中文翻译：

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区间特征函数的代数多项式的最佳单边逼近

令υ为（-1，1）的权重，即，几乎在（-1，1）的任何地方都可测量的可积非负函数非零。表示由大号^υ（-1,1）的实值函数的空间˚F积与重量υ上（-1,1）与所述规范\（F = \ INT \ limits_ { - 1} ^ 1 {F（X ）v（x）dx。} \）。我们考虑在空间Lυ（-1，1）中对间隔（a，b）的特征函数进行最佳单侧逼近的问题（-1，1 ），-1 <a <b <1，由度数不超过给定数的一组代数多项式组成。我们解决了a和b的情况在某些条件下，在其精确度以及对称间隔（-h，h）为0 <h <1的情况下，对于均等的权重υ，它们是正整数公式的节点。