On approximation by linear operators: improved estimates
Keywords:
quantitative approximation, direct estimates, (positive) linear operators, almost lattice homomorphisms, moduli of continuity of order 1 and 2, Bernstein operators and the associated semigroup, Meyer-König operator, Zeller operator, Hermite-Fejér operatorAbstract
The present paper describes a unified approach to quantitative approximation theorems for certain linear operators L including positive linear ones. It is shown for so-called almost lattice homomorphisms A that the difference \((L-A)(f,x)\) can be estimated in terms of a certain three parameter functional \(\Omega\). This functional is in turn bounded from above by various classical seminorms such as (modifications of) moduli of continuity of order 1 and 2. There is a large variety of opportunities to combine results of this paper in order to arrive at direct quantitative assertions. Several examples show that the general theory implies a number of results which improve those known so far.
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