Mixed K-functionals: A measure of smoothness for blending-type approximation. (English) Zbl 0676.41022
Summary: The K-functionals of J. Peetre play an important role in the derivation of quantitative estimates for the degree of approximation by certain approximants of univariate functions. One reason for this is the fact that they are equivalent to the standard moduli of smoothness.
In the case of blending-type approximation (e.g., approximation by certain operators of Boolean sum type or by pseudo-polynomials), so- called mixed moduli of smoothness have been used for measuring the smoothness of bivariate functions. We introduce mixed K-functionals as an analogue to the Peetre K-functionals in the context of uniform blending- type approximation, and state an equivalence relation between mixed K- functionals and mixed moduli of smoothness.
In a similar way as the Peetre K-functionals, the mixed K-functionals can be used to formulate certain smoothing principles. As applications we prove a Korovkin theorem for approximation by a class of blending-type operators and give an optimal estimate for the degree of approximation by trigonometric pseudo-polynomials. Moreover, we use mixed K-functionals as concave majorants of mixed moduli of smoothness.
In the case of blending-type approximation (e.g., approximation by certain operators of Boolean sum type or by pseudo-polynomials), so- called mixed moduli of smoothness have been used for measuring the smoothness of bivariate functions. We introduce mixed K-functionals as an analogue to the Peetre K-functionals in the context of uniform blending- type approximation, and state an equivalence relation between mixed K- functionals and mixed moduli of smoothness.
In a similar way as the Peetre K-functionals, the mixed K-functionals can be used to formulate certain smoothing principles. As applications we prove a Korovkin theorem for approximation by a class of blending-type operators and give an optimal estimate for the degree of approximation by trigonometric pseudo-polynomials. Moreover, we use mixed K-functionals as concave majorants of mixed moduli of smoothness.
MSC:
| 41A25 | Rate of convergence, degree of approximation |
| 41A35 | Approximation by operators (in particular, by integral operators) |
| 41A36 | Approximation by positive operators |
| 41A44 | Best constants in approximation theory |
| 42B99 | Harmonic analysis in several variables |
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