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In the present paper, we investigate monotone rational approximation. We prove that if fC [0, 1] is an increasing function on the interval [0, 1] then R n * ( f ) ≦ C log 2 µ/ nf ‖, where R n * ( f ) is the best approximation of f by incrasing rational functions of order ( n, n ), µ > 1 matches n in the equation n = log 2 µ/ ω ( f . With some new techniques created, this result essentially generalizes and improves previous result appeared in Zhou [10].

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For periodic functions, sequences of trigonometric polynomialsP m (x) are constructed which provide close-to-best approximation on the whole period and such that if on a certain interval the function possesses better properties, thenP m (x) approximate it inside of this interval at a higher rate of convergence than on the whole period. The results of this article extend investigations by S. Bochner, T. Frey, and V. Ja. Janиak.

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This paper deals with the dynamical analysis of a crane model. Truss finite elements are used to discretize the suspending chains with the so called updated Lagrangian description. This nonlinear model is regarded to be the best approximation to which linear models are compared. The inertia and independent degrees of freedom are also taken into consideration by linear models. The goal is to find a linear model, which can be used as an observer in antisway control of a crane.

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In this paper the author studies classesH q Ω of periodic functions of several variables whose mixed moduli of continuity do not exceed a given modulus of continuity ω(t 1 ...,t d). Necessary and sufficient conditions of belonging of a functionf(x 1, ...,x d) to the classH q Ω are considered (Theorem 1). These necessary and sufficient conditions are proved under some additional assumptions on ω(t 1, ...,t d). It is shown that additional assumptions cannot be omitted (Theorem 3). Besides, the estimates of best approximations of classesH q Ω with some special ω(t 1, ...,t d) are given (Theorems 4 and 5).

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The following results are obtained for the metric of a sign sensitive weight: necessary and sufficient conditions imposed on the weight in order to ensure the fulfillment of the complete analogue of Jackson's theorem on the estimate of the best approximation of an arbitrary continuous function by means of its modulus of continuity; the analogue of Bernstein's inequality on the estimate of the derivative of a trigonometric polynomial; the analogue of Stechkin's theorem on the connection between the modulus of continuity of a function and the rate of its approximation by polynomials; the analogue of Dolzhenko's inequality on the estimate of the variation of a rational function; and the analogue of Dolzhenko's theorem on the estimate of the variation of a function by means of its best rational approximation.

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A subspaceY of a Banach spaceX is called a Chebyshev one if for everyxX there exists a unique elementP Y(x) inY of best approximation. In this paper, necessary and sufficient conditions are obtained in order that certain classes of subspacesY of the Hardy spaceH 1=H 1 (|z|<1) be Chebyshev ones, and also the properties of the operatorP Y are studied. These results show that the theory of Chebyshev subspaces inH 1 differs sharply from the corresponding theory inL 1(C) of complex-valued functions defined and integrable on the unit circleC:|z|=1. For example, it is proved that inH 1 there exist sufficiently many Chebyshev subspaces of finite dimension or co-dimension (while inL 1(C) there are no Chebyshev subspaces of finite dimension or co-dimension). Besides, it turned out that the collection of the Chebyshev subspacesY with a linear operatorP Y inH 1 (in contrast toL 1(C)) is exhausted by that minimum which is necessary for any Banach space.

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The paper deals with the order of best rational approximation of some classes of functions, depending on their differentiability properties. Improvements and generalizations of some results by P. P. Petrushev, V. A. Popov and the author are obtained. The proofs are based on the author's direct rational approximation theorems received recently. One of the results reads as follows. LetR n(f,L p) denote the value of the best approximation of a functionf inL p,f∈L p [0,1], by rational fractions of degree not exceedingn, n≧1. Suppose that 0<p≦∞,s∈NU{0}, andp≠∞ fors=0. Iff is thes-th primitive of some function of bounded variation on [0,1], then
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\sum\limits_{n = 1}^\infty {\frac{1}{n}(n^{s + 1} R_n (f,L_p ))^2< \infty }$$ \end{document}
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Blatter, J., Reflexivity and the existence of best approximations, Approximation Theory, Proc. Internat. Sympos. , Univ. Texas, Austin, Tex., Academic Press, New York (1976), 299-301. MR 54 #13423 Approximation

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The plant height and the height of the main ear were studied over two years in twelve single cross maize hybrids sown at three different plant densities (45, 65 and 85 thousand plants/ha) at five locations in Hungary (Keszthely, Gönc, Gyöngyös, Sopronhorpács, Martonvásár). The results revealed that plant height and the height of the main ear are important variety traits and are in close correlation with each other. It was found that the hybrids grew the tallest when the genetic distance between the parental components was greatest (Mv 4, Mv 5). The height of the main ear was also the greatest in these hybrids, and the degree of heterosis was highest (193% for plant height, 194% for the height of the main ear). The shortest hybrids were those developed between related lines (Mv 7, Mv 11). In this case the heterosis effect was the lowest for both plant height (128%) and the height of the main ear (144%). The ratio of the height of the main ear to the plant height was stable, showing little variation between the hybrids (37–44%). As maize is of tropical origin it grows best in a humid, warm, sunny climate. Among the locations tested, the Keszthely site gave the best approximation to these conditions, and it was here that the maize grew tallest. The dry, warm weather in Gyöngyös stunted the development of the plants, which were the shortest at this location. Plant density had an influence on the plant size. The plants were shortest when sown at a plant density of 45,000 plants/ha, and the main ears were situated the lowest in this case. At all the locations the plant and main ear height rose when the plant density was increased to 65,000 plants/ha. At two sites (Gönc and Sopronhorpács) the plants attained their maximum height at the greatest plant density (85,000 plants/ha). In Keszthely there was no significant difference between these two characters at plant densities of 65 and 85 thousand plants/ha, while in Gyöngyös and Martonvásár the greatest plant density led to a decrease in the plant and main ear height. The year had a considerable effect on the characters tested.

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.07.002 . [13] Singer , I. 1970 Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces Springer-Verlag Berlin

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