Chebyshev rank in one-sided L1-approximation

In this paper we characterize those subspaces U_n of differentiable functions which have Chebyshev rank at most k in one-sided L¹-approximation to differentiable functions with respect to all continuous positive weights. This result is applied to determine the Chebyshev rank of several classes of subspaces.     

On the best L 1-approximation of truncated powers by algebraic polynomials

We determine the asymptotic behavior of the best L₁-approximations of the kernels (x - a) ₊^r- 1, 1 < r < 2, and the classes W₁^r by algebraic polynomials. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 4, pp. 593–598, April, 1998.     

勒让德多项式的性质与契贝谢夫多项式间的关系

主要讨论了著名的勒让德多项式的一些性质,同时得到勒让德多项式与契贝谢夫多项式之间的一些关系     

Some results on L1-approximation

The new inequalities for L¹-Approximation are obtained; J. W. Garrett, C. S. Ress and Č. V. Stanojevic's Theorem^[2] and T. Xie's Theorem^[3] are modified.     

Generalization of Chebyshev polynomials

Moscow. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 31, No. 1, pp. 217–218, January–February, 1990.     

Sign changes in rational L w 1 -approximation

Let f ∈ L _w ¹ [−1, 1], let r _n,m(f) be the best rational L _w ¹ -approximation for f with respect to real rational functions of degree at most n in the numerator and of degree at most m in the denominator, let m = m(n), and let lim _{n → ∞} (n-m(n)) = ∞. In this case, we show that the counting measures of certain subsets of sign changes of f-r _n,m (f) converge weakly to the equilibrium measure on [−1, 1] as n → ∞. Moreover, we prove estimates for discrepancy between these counting measures and the equilibrium measure. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 283–287, February, 2006.     

L 1-approximation rate of certain trigonometric series

By employing new ideas and techniques, we will refigure out the whole frame of L ¹-approximation. First, except generalizing the coefficients from monotonicity to a wider condition, Logarithm Rest Bounded Variation condition, we will also drop the prior requirement f∈L _2π but directly consider the sine or cosine series. Secondly, to achieve nontrivial generalizations in complex spaces, we use a one-sided condition with some kind of balance conditions. In addition, a conjecture raised in [9] is disproved in Section?3.     

关于勒让德多项式与契贝谢夫多项式间的关系

主要研究勒让德多项式与契贝谢夫多项式之间的关系的性质,利用生成函数和函数级数展开的方法,得出了勒让德多项式与契贝谢夫多项式之间的一个重要关系,这对勒让德多项式与契贝谢夫多项式的研究有一定的推动作用．     

Gaussian quadrature of Chebyshev polynomials

We investigate the behaviour of the maximum error in applying Gaussian quadrature to the Chebyshev polynomials T_m. This quantity has applications in determining error bounds for Gaussian quadrature of analytic functions.     

Best one-sided L 1-approximation by harmonic functions

Let , where B is the open unit ball in (), and let denote the collection of functions h in which are harmonic on B and satisfy on . A function h ^* in is called a best harmonic one-sided L ¹-approximant to f if for all h in . This paper characterizes such approximants and discusses questions of existence and uniqueness. Corresponding results for approximation on the cylinder are also established, but the proofs in this case are more difficult and rely on recent work concerning tangential harmonic approximation. The characterizations are quite different in nature from those recently obtained for harmonic L ¹-approximation without a one-sidedness condition. Received: 25 September 1997