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Estimates with Second Order Moduli

In: Approximation Theory Using Positive Linear Operators

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  • Radu Păltănea

    (Transilvania University, Department of Mathematics)

Abstract

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.

Suggested Citation

  • Radu Păltănea, 2004. "Estimates with Second Order Moduli," Springer Books, in: Approximation Theory Using Positive Linear Operators, chapter 2, pages 15-68, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-2058-9_2
    DOI: 10.1007/978-1-4612-2058-9_2
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