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 $$L^p$$
L
p
-spaces, Zygmund spaces and the exponential spaces. By using a direct approach, we also provide a further estimate in the particular case of $$L^p$$
L
p
-spaces, with $$1\le p <+\infty $$
1
≤
p
<
+
∞
, 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.
Here, we provide a unifying treatment of the convergence of a general form of sampling-type operators, given by the so-called sampling Durrmeyer-type series. The main result consists of the study of a modular convergence theorem in the general setting of Orlicz spaces 𝐿 𝜑 (ℝ). From the latter theorem, the convergence in 𝐿 𝑝 (ℝ), in 𝐿 𝛼 log 𝛽 𝐿, and in the exponential spaces can be obtained as particular cases. For the completeness of the theory, we provide a pointwise and uniform convergence theorem on ℝ, in case of bounded continuous and bounded uniformly continuous functions; in this context, we also furnish a quantitative estimate for the order of approximation, using the modulus of continuity of the function to be approximated. Finally, applications and examples with graphical representations are given for several sampling series with special kernels.
Here we provide a unifying treatment of the convergence of a general form of sampling type operators, given by the so-called Durrmeyer sampling type series. In particular we provide a pointwise and uniform convergence theorem on R, and in this context we also furnish a quantitative estimate for the order of approximation, using the modulus of continuity of the function to be approximated. Then we obtain a modular convergence theorem in the general setting of Orlicz spaces L ϕ (R). From the latter result, the convergence in L p (R)-space, L α log β L, and the exponential spaces follow as particular cases. Finally, applications and examples with graphical representations are given for several sampling series with special kernels.
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