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1.
Suppose thatX andY are real Banach spaces,U ⊂X is an open bounded set star-shaped with respect to some point,n, k ∈ ℕ,k <n, andMn, k (U,Y) is the sharp constant in the Markov type inequality for derivatives of polynomial mappings. It is proved that for anyM ≥M
n,k
(U, Y) there exists a constantB > 0 such that for any functionf ∈C
n
(U, Y) the following inequality holds:
The constantM =M
n−1,k
(U, Y) is best possible in the sense thatM
n−1,k
(U, Y) = infM, where inf is taken over allM such that for someB > 0 the estimate holds for allf ∈C
n
(U, Y).
Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 332–342, March, 1998.
This research was partially supported by the International Science Foundation under grant No. U92000. and by the State Committee
of the Ukraine for Science and Technology. 相似文献
2.
Beatriz Margolis 《Numerical Functional Analysis & Optimization》2013,34(5-6):577-588
We develop a calculus structure in the Banach lattice introduced in a preceding paper, having in mind an approximation problem appearing in non-smooth optimization. We show that the essential results depend as much on the order structure as on the analytical one. 相似文献
3.
T. Rapcsák 《Journal of Global Optimization》1997,11(2):207-217
The Fenchel problem of level sets is solved under the conditions that theboundaries of the nested family of convex sets in Rn>+1 aregiven by C3 n-dimensional differentiable manifolds and theconvex sets determine an open or closed convex set inRn+1. 相似文献
4.
The differentiability properties of statistical functionals have several interesting applications. We are concerned with two of them. First, we prove a result on asymptotic validity for the so-called smoothed bootstrap (where the artificial samples are drawn from a density estimator instead of being resampled from the original data). Our result can be considered as a smoothed analog of that obtained by Parr (1985, Stat. Probab. Lett., 3, 97-100) for the standard, unsmoothed bootstrap. Second, we establish a result on asymptotic normality for estimators of type
generated by a density functional
being a density estimator. As an application, a quick and easy proof of the asymptotic normality of
, (the plug-in estimator of the integrated squared density
) is given. 相似文献
5.
Let () be the set of all continuous functions on which have a derivative ( , respectively) at least at one point . B. R. Hunt (1994) proved that is Haar null (in Christensen's sense) in .
In the present article it is proved that neither nor its complement is Haar null in . Moreover, the same assertion holds if we consider the approximate derivative (or the ``strong' preponderant derivative) instead of the ordinary derivative; these results are proved using a new result on typical (in the sense of category) continuous functions, which is of interest in its own right.
6.
Zai-jiu Shang 《Journal of Dynamics and Differential Equations》2000,12(2):357-383
The mapping version of Pöschel's theory on differentiable foliation structures of invariant tori is presented and the relevant estimates explicitly in terms of the diophantine constant and the nondegeneracy parameters of frequency maps are given. As a direct application of the main result, a generalization of Moser's small twist theorem to high dimensions is given. 相似文献
7.
WANG Xinghua~ 《中国科学A辑(英文版)》2005,48(1)
The best quadrature formula has been found in the following sense:for afunction whose norm of the second derivative is bounded by a given constant and thebest quadrature formula for the approximate evaluation of integration of that function canminimize the worst possible error if the values of the function and its derivative at certainnodes are known.The best interpolation formula used to get the quadrature formula aboveis also found.Moreover,we compare the best quadrature formula with the open compoundcorrected trapezoidal formula by theoretical analysis and stochastic experiments. 相似文献
8.
The note demonstrates that modeling a nonlinear minimax problem as a nonlinear programming problem and applying a classical differentiable penalty function to attempt to solve the problem can lead to convergence to a stationary point of the penalty function which is not a feasible point of the nonlinear programming problem. This occurred naturally in an application from statistical reliability theory. The note resolves the problem through modification of both the problem formulation and the iterative penalty function method. 相似文献
9.
Naoya Sumi 《Proceedings of the American Mathematical Society》1999,127(3):915-924
We show that on the 2-torus there exists a open set of regular maps such that every map belonging to is topologically mixing but is not Anosov. It was shown by Mañé that this property fails for the class of toral diffeomorphisms, but that the property does hold for the class of diffeomorphisms on the 3-torus . Recently Bonatti and Diaz proved that the second result of Mañé is also true for the class of diffeomorphisms on the -torus ().
10.
Ruchi Ruchi Nurhayat Ispir P. N. Agrawal 《Mathematical Methods in the Applied Sciences》2017,40(16):5687-5706
Ren and Zeng (2013) introduced a new kind of q‐Bernstein–Schurer operators and studied some approximation properties. Acu et al. (2016) defined the Durrmeyer modification of these operators and studied the rate of convergence and statistical approximation. The purpose of this paper is to introduce a Kantorovich modification of these operators by using q‐Riemann integral and investigate the rate of convergence by means of the Lipschitz class and the Peetre's K‐functional. Next, we introduce the bivariate case of q‐Bernstein–Schurer–Kantorovich operators and study the degree of approximation with the aid of the partial modulus continuity, Lipschitz space, and the Peetre's K‐functional. Finally, we define the generalized Boolean sum operators of the q‐Bernstein–Schurer–Kantorovich type and investigate the approximation of the Bögel continuous and Bögel differentiable functions by using the mixed modulus of smoothness. Furthermore, we illustrate the convergence of the operators considered in the paper for the univariate case and the associated generalized Boolean sum operators to certain functions by means of graphics using Maple algorithms. Copyright © 2017 John Wiley & Sons, Ltd. 相似文献