Quantitative estimates for Durrmeyer-sampling series in Orlicz spaces
Danilo Costarellı, Michele Piconi, Gianluca Vıntı
Abstract
Abstract In this paper, we establish a quantitative estimate for Durrmeyer-sampling type operators in the general framework of Orlicz spaces, using a suitable modulus of smoothness defined by the involved modular functional. As a consequence of the above result, we can deduce quantitative estimates in several instances of Orlicz spaces, such as $$L^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> -spaces, Zygmund spaces and the exponential spaces. By using a direct approach, we also provide a further estimate in the particular case of $$L^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> -spaces, with $$1\le p <+\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>p</mml:mi> <mml:mo><</mml:mo> <mml:mo>+</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> , that turns out to be sharper than the previous general one. Moreover, we deduce the qualitative order of convergence, when functions belonging to suitable Lipschitz classes are considered.