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In this paper we consider rational approximation of functions in the Hardy spaces on the strip with . Observing that can be regarded as the direct sum of and , we give three kinds of adaptive rational approximation in enhancing the corresponding approximations in and where and We also obtain a type of approximation for subspaces in by making use of Riesz bases in these subspaces. [ABSTRACT FROM AUTHOR]

Let X be a normed space, V be an open convex subset of X and let be a given function. A function is called -midconvex if By the result of Tabor and Tabor, we know that under respective conditions on , if f is -midconvex and locally bounded above at a point then there exists a continuous function such that In this paper we determine the smallest function satisfying the above inequality. The required conditions on are such that the functions , satisfy them. As the main tool we use de Rham theorem. [ABSTRACT FROM AUTHOR]

In this paper, the study of best simultaneous Lp-approximation by algebraic polynomials on a union of intervals, I, is considered. We obtain convergence of the best simultaneous approximations when the measure of I tends to zero. We also get an interpolation result. [ABSTRACT FROM AUTHOR]

In this paper, we propose a three-term conjugate gradient method which can produce sufficient descent condition, that is, [image omitted] . This property is independent of any line search used. When an exact line search is used, this method reduces to the standard Hestenes-Stiefel conjugate gradient method. We also introduce two variants of the proposed method which still preserve the sufficient descent property, and prove that these two methods converge globally with standard Wolfe line search even if the minimization function is nonconvex. We also report some numerical experiment to show the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

This paper considers the problem of approximating an integrable function by piecewise linear functions that keep the integral and positivity of the original function. [ABSTRACT FROM AUTHOR]

A novel technique for the construction of positive weight high-precision rational approximations to a class of transcendental curves is presented. The approximations are induced from the rational parametrisations of the circle. The previously published rational parametrisations of the circle are not suited to the induction process and the new parametrisations are constructed in the paper for the purpose. Explicit rational approximations of a number of transcendental curves are then given. The work is a development of the authors' previous work on induced rational parametrisations of special algebraic curves. [ABSTRACT FROM AUTHOR]

One of the most popular algorithms for solving minimax approximation problems is the Barrodale and Phillips simplex-based method [I. Barrodale and C. Phillips (1974). An improved algorithm for discrete Chebyshev linear approximation. In: Proceedings of the Fourth Manitoba Conference on Numerical Mathematics , pp. 177-190, University of Manitoba, Winnipeg, Canada.]. Our new research has found that the modified pivoting strategy in the first stage of the Barrodale and Phillips algorithm could result in an invalid starting point and system failure in later stages [D. Lei (2002). Robust and efficient algorithms for ℓ 1 and ℓ ∞ approximations. PhD Thesis, School of Computing and Engineering, University of Huddersfield, Huddersfield, UK.]. In contrast, a primal method that we propose in this paper is numerically more stable, and the approximating function does not suffer the effects of wild oscillation. However, on average, the primal method requires more searching steps to find an optimal solution. To improve the efficiency of the primal method, we have considered using the least squares solution to provide a better starting point. Experimental results show that if the least squares estimator is available, then it can speed up the convergence rate significantly in most cases. The number of steps required by using the least squares solution for the primal method also compares favourably with that of the dual method. [ABSTRACT FROM AUTHOR]