ON THE LEAST DOMINANT CONTINUOUS MODULUS AND ITS APPLICATION

o.IntroductionInthispaperwedealwithqualltitativeKorovkintypetheoremsfortheaPproalmationbyboundedlinearoperatorsdefinedonC(X),andinparticularbypositiveones.HereC(X)=CR(X,d)denotestheBanachlatticeofreal-valuedcontinuousfunctionsdefinedonthecompactmetricspace(X,d)withnormgivenbylIfIlx=ma-xlf(x)I,xEX.WealsoassumethatXhasdiameterd(X)>o.ThefirstsuchtheoremforgeneralpositivelinearoperatorsandX=[a,b]equippedwiththeeuclidiandistanceisduetoR.Mamedovl4].Forspaces(X,d)beingmetricallyconvexinthes…     

On the order of magnitude of the uniform convergence of multiple trigonometric Fourier series with respect to cubes on the function classes $ H^l_{p,m} [\omega]$

We present a solution for the order problem of magnitude of the deviation in uniform metric of the cubic partial sums of the multiple trigonometric Fourier series on certain classes of functions with a given majorant of their $L_p$-modulus of smoothness of order $l$, where $l\in \N$ and $1相似文献   

一般赋范空间上连续函数的Korovkin型定理

本文引入一种连续模的新的控制泛函，利用直接方法（避免使用Ｋ－泛函）得到了一般赋范空间上连续函数的Ｋｏｒｏｖｋｉｎ型定理，得到的结果与Ｈ．Ｇｏｎｓｋａ用Ｋ－泛函得到的结果相比各有优点。     

Functional a posteriori error estimates for incremental models in elasto-plasticity

We consider incremental problem arising in elasto-plastic models with isotropic hardening. Our goal is to derive computable and guaranteed bounds of the difference between the exact solution and any function in the admissible (energy) class of the problem considered. Such estimates are obtained by an advanced version of the variational approach earlier used for linear boundary-value problems and nonlinear variational problems with convex functionals [24, 30]. They do no contain mesh-dependent constants and are valid for any conforming approximations regardless of the method used for their derivation. It is shown that the structure of error majorant reflects properties of the exact solution so that the majorant vanishes only if an approximate solution coincides with the exact one. Moreover, it possesses necessary continuity properties, so that any sequence of approximations converging to the exact solution in the energy space generates a sequence of positive numbers (explicitly computable by the majorant functional) that tends to zero.      

On the Order of Decrease of Uniform Moduli of Smoothness for the Classes of Functions E p,m [ɛ]

In this paper, we solve the problem of the exact order of decrease of uniform moduli of smoothness for the classes of 2π-periodic functions of several variables with a given majorant of the sequence of total best approximations in the metric of L _p , 1 ≤ p < ∞.     

Test function space for Wick power series

We derive a criterion that is convenient for applications and exactly characterizes the test function space on which the operator realization of a given series of Wick powers of a free field is possible. The suggested derivation does not use the assumption that the metric of the state space is positive and can therefore be used in a gauge theory. It is based on the systematic use of the analytic properties of the Hilbert majorant of the indefinite metric and on the application of a suitable theorem on the unconditional convergence of series of boundary values of analytic functions. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 3, pp. 355–373, June, 2000.     

Best approximations of classes of functions defined by means of the modulus of continuity

This paper is devoted to an exact solution of problems of best approximation in the uniform and integral metrics of classes of periodic functions representable as a convolution of a kernel not increasing the oscillation with functions having a given convex upwards majorant of the modulus of continuity. The approximating sets are taken to be the trigonometric polynomials in the case of the uniform and integral metrics, and convolutions of the kernel defining the class with polynomial splines in the case of the integral metric.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 5, pp. 579–589, May, 1992.     

The asymptotic manifold of a perturbed linear differential equation as determined by the comparison principle

The asymptotic properties of the solutions of a linear homogeneous system of differential equations determine, under suitable restrictions, the asymptotic properties of a set of solutions of a nonlinear perturbation of this linear equation. The comparison principle is used here to generate an asymptotic manifold of the perturbed equation. The majorant function that is used in connection with the comparison technique is usually assumed to be nondecreasing in the dependent variable. However, properties of the asymptotic manifold are discussed here under the opposite monotonicity assumption, namely, that the majorant function is nonincreasing in the dependent variable. This type of majorant, function arises, for example, in certain gravitation problems. The main result on the structure of asymptotic manifolds which have an asymptotic uniformity is that solutions close to the manifold are either in the manifold or do not exist in the future. This research was supported in part by the National Science Foundation under grant GP-11543. Entrata in Redazione il 6 giugno 1970.     

Estimates of Deviations for Generalized Newtonian Fluids

The paper is concerned with estimates of deviations from exact solutions for stationary models of viscous incompressible fluids. It is shown that if a function compared with exact solution is subject to the incompressibility condition, then the deviation majorant consists of terms that penalize the inaccuracy in the equilibrium equation and the rheological relation defined by a ceratin dissipative potential. If such a function does not satisfy the incompressibility condition, then the majorant includes an additional term. The factor of this term depends on the constant in the Ladyzhenskaya–Babuka–Brezzi condition. Bibliography: 27 titles.     

Approximations by V. A. Steklov functions in Hausdorff metric

The paper studies approximations of functions defined on the entire number axis and bounded on any finite segment, the approximations being by V. A. Steklov functions in Hausdorff metric. We obtain the value of the exact upper bound of the approximation on classes of functions of given majorant of their moduli of nonmonotonicity.Translated from Matematicheskie Zametki, Vol. 5, No. 1, pp. 21–30, January 1969.The author wishes to express his gratitude to N. P. Korneichuk under whose direction this paper was prepared.     

Estimation of Deviations from the Exact Solution for the Reissner-Mindlin Plate Problem

In this paper, we obtain a majorant of the difference between the exact solution and any conforming approximate solution of the Reissner-Mindlin plate problem. This majorant is explicitly computable and involves constants that depend only on given data of the problem. The majorant allows us to compute guaranteed upper bounds of errors with any desired accuracy and vanishes if and only if the approximate solution coincides with the exact one. Bibliography: 12 titles. To N. N. Uraltseva with gratitude __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 310, 2004, pp. 145–157.     

On the Hardy-Littlewood majorant problem for random sets

We study the classical Hardy-Littlewood majorant problem for trigonometric polynomials. We show that the constant in the majorant inequality grows at most like an arbitrary small power of the degree provided the spectrum is chosen at random. We also give an example of a deterministic set where the majorant property fails, i.e., the constant grows like a fixed small power in the degree.     

Method of Linear Majorant in the Theory of Monotonic Entropy Operators

A method for constructing a linear majorant for a monotone entropy operator is developed, and the majorant characteristics are determined.     

Zygmund-type estimates for fractional integration and differentiation operators of variable order

We consider non-standard generalized Hölder spaces of functions defined on a segment of the real axis, whose local continuity modulus has a majorant varying from point to point. We establish some properties of fractional integration operators of variable order acting from variable generalized Hölder spaces to those with a “better” majorant, as well as properties of fractional differentiation operators of variable order acting from the same spaces to those with a “worse” majorant.     

Increasing the applicability of Steffensen's method

We present an original alternative to the majorant principle of Kantorovich to study the semilocal convergence of Steffensen's method when it is applied to solve nonlinear systems which are differentiable. This alternative allows choosing starting points from which the convergence of Steffensen's method is guaranteed, but it is not from the majorant principle. Moreover, this study extends the applicability of Steffensen's method to the solution of nonlinear systems which are nondifferentiable and improves a previous result given by the authors.     

Estimates with Sharp Constants of the Sums of Sine Series with Monotone Coefficients of Certain Classes in Terms of the Salem Majorant

New estimates of the sums of sine series with monotone coefficients of special classes in terms of the Salem majorant are obtained. The asymptotic sharpness of the obtained estimates for sequences of coefficients from the classes under consideration is proved.     

A majorant-minorant criterion for the total preservation of global solvability of a functional operator equation

We study a nonlinear controlled functional operator equation in an ideal Banach space. We establish sufficient conditions for the global solvability for all controls from a given set, and obtain a pointwise estimate for solutions. Using upper and lower estimates of the functional component in the right-hand side of the initial equation (with a fixed operator component), we obtain majorant and minorant equations. We prove the stated theorem, assuming the monotonicity of the operator component in the right-hand side and the global solvability of both majorant andminorant equations. We give examples of the reduction of controlled initial boundary value problems to the equation under consideration.     

Characterization of the least concave majorant of brownian motion,conditional on a vertex point,with application to construction

The characterization of the least concave majorant of brownian motion by Pitman (1983,Seminar on Stochastic Processes, 1982 (eds. E. Cinlar, K. L. Chung and R. K. Getoor), 219–228, Birkhäuser, Boston) is tweaked, conditional on a vertex point. The joint distribution of this vertex point is derived and is shown to be generated with extreme ease. A procedure is then outlined by which one can construct the least concave majorant of a standard Brownian motion path over any finite, closed subinterval of (0, ∞). This construction is exact in distribution. One can also construct a linearly interpolated version of the Brownian motion path (i.e. we construct the Brownian motion path over a grid of points and linearly interpolate) corresponding to this least concave majorant over the same finite interval. A discussion of how to translate the aforementioned construction to the least concave majorant of a Brownian bridge is also presented.     

Harmonic and Superharmonic Majorants on the Disk

A proof is given to show that a positive function on the unitdisk admits a harmonic majorant if and only if it has a certainexplicit upper envelope that admits a superharmonic majorant.The (logarithmic) Lipschitz regularity of this superharmonicmajorant is discussed. 2000 Mathematics Subject Classification31A05.     

Splitting superfluid hydrodynamic equations and instability of solutions in the case of degenerate bosons

Properties of solutions to superfluid hydrodynamic equations as applied to the degenerate Bose gas are considered. The equations are split into two independent pairs of equations. One pair is written for the normal component implies the instability of solutions, which manifests itself in the majorant catastrophe with respect to the total density. The case when the thermodynamic functions depend on the difference of the normal and superfluid velocities is also considered. In that case, the system is not split; however, the instability and the majorant catastrophe occur when the initial temperature tends to absolute zero.