On the Iterates of Mellin-Fejer Convolution Operators
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On the Iterates of Mellin-Fejer Convolution Operators Carlo Bardaro · Ilaria Mantellini
Received: 19 January 2012 / Accepted: 24 February 2012 / Published online: 10 March 2012 © Springer Science+Business Media B.V. 2012
Abstract We study the behaviour of iterates of Mellin-Fejer type operators with respect to pointwise and uniform convergence. We introduce a new method in the construction of linear combinations of Mellin type operators using the iterated kernels. In some cases this provides a better order of approximation. Keywords Mellin-Fejer convolution operators · Mellin derivatives · Moments · Iterates Mathematics Subject Classification 41A35 · 41A25 · 47G10
1 Introduction As it is well known, the optimal rate of (pointwise) convergence for nets or sequences of positive linear operators Lw f , acting on a function f which is not smoother than C 2 , does not exceed O(w−2 ) and this bound remains also for more regular functions. This is due essentially to the Korovkin theorem for positive linear operators [14], which states that the optimal rate of convergence cannot be faster than C 2 -functions (see e.g. [1, 2, 16]). Thus, in order to obtain higher order of approximation, it is necessary to consider linear operators not necessarily positive. A classical approach is based on the construction of suitable linear combinations of positive linear operators. This idea comes from the classical work of P.L. Butzer for Bernstein polynomials [7] and then developed by several authors (see e.g. [3, 8, 16–18]). In some recent papers (see e.g. [4, 6]) we have developed the pointwise approximation theory for Mellin convolution operators, acting on functions defined on the multiplicative group R+ . Mellin convolution operators represent an important tool in the Mellin transform C. Bardaro · I. Mantellini () Department of Mathematics and Informatics, University of Perugia, Via Vanvitelli 1, 06123 Perugia, Italy e-mail: [email protected] C. Bardaro e-mail: [email protected]
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C. Bardaro, I. Mantellini
theory, playing the same role of the classical convolution operators in Fourier analysis (see e.g. [9, 10]). The study of the approximation properties of such convolution integrals is strongly motivated by the behaviour of solutions of certain boundary value problems in wedge-shaped region [13, 19]. In [5] and [6] we have presented a Voronovskaja formula for suitable linear combinations of moment or Mellin-Gauss-Weierstrass operators of type +∞ r dt αj Kj w (t)f (st) , s ∈ R+ , w > 0 (Gw,r f )(s) = t 0 j =1 where αj , j = 1, . . . , r are real numbers with rj =1 αj = 1 and Kw is the moment or the Mellin-Gauss-Weierstrass kernel. These operators are not positive in general and this enables one to get a better order of approximation. Here we develop a different method in the construction of linear combinations, using iterated kernels instead of the basic kernels. More precisely, we consider linear operators of the form +∞ n,h,j,ν dt f (s) = Hwn,h,j,ν (s, t)f (t) , s ∈ R+ , w > 0 Tw t 0 where h (s, t), Hwn,h,j,ν
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