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

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. [12] Ditzian , Z. 1998 Fractional derivatives and best approximation Acta Math. Hungar. 81 323 – 348 10.1023/A:1006554907440

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Fourier Analysis and Approximation Birkhäuser Verlag Basel . [4] Chernykh , N. I. 1967 Best approximation of periodic functions by trigonometric polynomials in L 2

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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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] Ul'yanov , P. L. 1970 Imbedding theorems and relations between best approximation in different metrics Mat. Sb. 81 123 104 – 131 English transl. in

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References [1] V. V . Arestov . Approximation of unbounded operators by bounded operators and related extremal problems . Russian Math. Surveys , 51 ( 6 ): 1093 – 1126 , 1996 . [2] V. V . Arestov . On the best approximation of the

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Abstract  

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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