For periodic functions, sequences of trigonometric polynomialsPm(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, thenPm(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.
In the present paper, we investigate monotone rational approximation. We prove that if
is an increasing function on the interval [0, 1] then
) is the best approximation of
by incrasing rational functions of order (
), µ > 1 matches
in the equation
. With some new techniques created, this result essentially generalizes and improves previous result appeared in Zhou .
In this paper the author studies classesHqΩ of periodic functions of several variables whose mixed moduli of continuity do not exceed a given modulus of continuity ω(t1 ...,td). Necessary and sufficient conditions of belonging of a functionf(x1, ...,xd) to the classHqΩ are considered (Theorem 1). These necessary and sufficient conditions are proved under some additional assumptions on ω(t1, ...,td). It is shown that additional assumptions cannot be omitted (Theorem 3). Besides, the estimates of best approximations of classesHqΩ with some special ω(t1, ...,td) are given (Theorems 4 and 5).
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.
Authors:Vu Nhat Huy, Nguyen Ngoc Huy, and Chu Van Tiep
References  V. V . Arestov . Approximation of unbounded operators by bounded operators and related extremal problems . Russian Math. Surveys , 51 ( 6 ): 1093 – 1126 , 1996 .  V. V . Arestov . On the bestapproximation of the
A subspaceY of a Banach spaceX is called a Chebyshev one if for everyx∈X there exists a unique elementPY(x) inY of best approximation. In this paper, necessary and sufficient conditions are obtained in order that certain classes of subspacesY of the Hardy spaceH1=H1 (|z|<1) be Chebyshev ones, and also the properties of the operatorPY are studied. These results show that the theory of Chebyshev subspaces inH1 differs sharply from the corresponding theory inL1(C) of complex-valued functions defined and integrable on the unit circleC:|z|=1. For example, it is proved that inH1 there exist sufficiently many Chebyshev subspaces of finite dimension or co-dimension (while inL1(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 operatorPY inH1 (in contrast toL1(C)) is exhausted by that minimum which is necessary for any Banach space.