Shift Invariant Multivariate Integral Operators

George A. Anastassiou and Sorin G. Gal
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George A. Anastassiou: University of Memphis, Department of Mathematical Sciences
Sorin G. Gal: University of Oradea, Department of Mathematics

Chapter 11 in Approximation Theory, 2000, pp 297-323 from Springer

Abstract: Abstract Here among other topics we further study global smoothness preservation over Rd,d ≥ 1. This is a generalization of Chapter 10 in the multivariate case. Namely, a general positive linear multivariate integral type operator is given through a convolution-like iteration of another general positive linear multivariate operator with a multivariate scaling type function. For this sufficient conditions are given for shift invariance, global smoothness preservation and its sharpness, convergence to the unit with rates, shape preserving and preservation of continuous probabilistic distribution functions. Additionally, four examples of general specialized multivariate operators are given fulfilling all the above properties; especially, the inequalities for global smoothness preservation are sharp. In this chapter global smoothness preservation and convergence to the unit with rates involve a naturally arising suitable multivariate modulus of continuity. Here we follow the basic study done by the first author and Gonska [29]. Chapter 8 is also related here.

Keywords: Probabilistic Distribution Function; Dominate Convergence Theorem; Positive Linear Operator; Continuous Distribution Function; Shape Preserve (search for similar items in EconPapers)
Date: 2000
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4612-1360-4_11

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DOI: 10.1007/978-1-4612-1360-4_11

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