Transformation of the second order modulus by positive linear operators
Abstract We obtain estimates for the transformation of the second order modulus of continuity by positive linear operators which satisfy certain conditions, which are common to a large class of approximation operators.
Estimates with Second Order Moduli
In this chapter we continue the study of estimating the degree of an approximation using general linear positive operators by considering combinations of first and second order moduli, in terms of the moments of order 0, 1, and 2, see Remark 1.2.4. Estimates with such combinations of first and second order modulus, (and also with the absolute value of the function, which can be regarded as a modulus of order 0) are more refined then estimates using only the first modulus. A first observation is that, from estimates with the second order modules, one can derive estimates with the first order modulus. A second observation is the fact that such combinations decompose the error of approximation in three components, corresponding to three specific features of the functions that affect the error: amplitude, deviation from the linear functions, and deviation from the polynomials of degree 2. Roughly speaking, these moduli measure the deviation from the test functions of the algebraic Chebychev system.
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air at VA facilities, one of the many pieces of unfinished business beyond those major issues that we discussed in our Commentary. We agree that all facilities where current and past military personnel serve and receive treatment should be 100% smoke-free and note the recent Institute of Medicine report Combating Tobacco in Military and Veteran Populations. The report, which was commissioned by the Department of Defense (DOD) in 2007, calls for the DOD to set a timeline to eliminate smoking on military installations. The report also recommends that the DOD stop selling tobacco products in military commissaries and exchanges, prohibit tobacco use anywhere on military installations, and treat tobacco use in the same way as other health-related behaviors, such as alcohol abuse and poor physical fitness. We join Matthews and the Institute of Medicine in urging the VA and the DOD to take more immediate steps to protect the health of the men and women who serve in the armed services, as well as the civilians (such as Dr Matthews) working to serve them.