In this paper we introduce and study a sequence of integral-type positive linear operators acting on a sufficiently large continuous function space which contains wide classes of weighted spaces of continuous functions on the real line. These operators depend on three given functions \alpha ,\beta, and \gamma\in C(R), \gamma bounded, and generalize the classical Gauss-Weierstrass convolution operators The main motivation to introduce these operators rests on the aim to construct a sequence of positive linear operators which satisfies an asymptotic formula (with respect to a given weighted norm) whose limit operator is a second order elliptic differential operator of the form% Lu=\alpha u’’+\beta u’+\gamma u. The operators which we introduce in this paper satisfy, indeed, such an asymptotic formula opening the way to a possible investigation to find suitable domains on which the differential operator L generates a C0 -semigroup of positive operators which can be represented in terms of iterates of these operators. However this aspect will be developed in a forthcoming paper. Besides the connection with semigroup theory, our operators seems to have a possible interest in the weighted approximation of continuous functions on the real line for a wide class of weights. We start our analysis by first introducing a sequence of integral-type positive linear operators depending only on the functions \alpha and \beta . We discuss their approximation properties in several spaces of continuous functions and we give some estimates of the rate of convergence. We also establish an asymptotic formula together with a saturation result. Subsequently we study some shape preserving properties. Finally, in the last section, by modifying these operators, we obtain a further approximation process which involves the function \gamma and which verifies analogous qualitative properties, including the general above mentioned asymptotic formula.

Integral-type operators on continuous function spaces on the real line,

ALTOMARE, Francesco;
2008

Abstract

In this paper we introduce and study a sequence of integral-type positive linear operators acting on a sufficiently large continuous function space which contains wide classes of weighted spaces of continuous functions on the real line. These operators depend on three given functions \alpha ,\beta, and \gamma\in C(R), \gamma bounded, and generalize the classical Gauss-Weierstrass convolution operators The main motivation to introduce these operators rests on the aim to construct a sequence of positive linear operators which satisfies an asymptotic formula (with respect to a given weighted norm) whose limit operator is a second order elliptic differential operator of the form% Lu=\alpha u’’+\beta u’+\gamma u. The operators which we introduce in this paper satisfy, indeed, such an asymptotic formula opening the way to a possible investigation to find suitable domains on which the differential operator L generates a C0 -semigroup of positive operators which can be represented in terms of iterates of these operators. However this aspect will be developed in a forthcoming paper. Besides the connection with semigroup theory, our operators seems to have a possible interest in the weighted approximation of continuous functions on the real line for a wide class of weights. We start our analysis by first introducing a sequence of integral-type positive linear operators depending only on the functions \alpha and \beta . We discuss their approximation properties in several spaces of continuous functions and we give some estimates of the rate of convergence. We also establish an asymptotic formula together with a saturation result. Subsequently we study some shape preserving properties. Finally, in the last section, by modifying these operators, we obtain a further approximation process which involves the function \gamma and which verifies analogous qualitative properties, including the general above mentioned asymptotic formula.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11586/16106
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