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Approximation results for nonlinear integral operators in modular spaces and applications

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Mantellini I., Vinti G.

Annales Polonici Mathematici

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2003

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tom 81

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nr 1

55-71

EN

We obtain modular convergence theorems in modular spaces for nets of operators of the form $(T_wf)(s) = ∫_{H} K_w (s - h_w(t),f(h_w(t))) dμ_H(t)$, w > 0, s ∈ G, where G and H are topological groups and ${h_w}_{w>0}$ is a family of homeomorphisms $h_w :H → h_w (H) ⊂ G.$ Such operators contain, in particular, a nonlinear version of the generalized sampling operators, which have many applications in the theory of signal processing.

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Order of approximation for nonlinear sampling Kantorovich operators in Orlicz spaces

100%

Costarelli D., Vinti G.

Commentationes Mathematicae

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2013

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tom 53

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nr 2

EN

In this paper, we study the rate of approximation for the nonlinear sampling Kantorovich operators. We consider the case of uniformly continuous and bounded functions belonging to Lipschitz classes of the Zygmund-type, as well as the case of functions in Orlicz spaces. We estimate the aliasing errors with respect to the uniform norm and to the modular functional of the Orlicz spaces, respectively. The general setting of Orlicz spaces allows to deduce directly the results concerning the rate of convergence in $L^p$-spaces, $1 \miu p$ < $\infty$, very useful in the applications to Signal Processing. Others examples of Orlicz spaces as interpolation spaces and exponential spaces are discussed and the particular cases of the nonlinear sampling Kantorovich series constructed using Fej\'er and B-spline kernels are also considered.

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Approximation by nonlinear integral operators in some modular function spaces

81%

Bardaro C., Musielak J., Vinti G.

Annales Polonici Mathematici

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1996

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tom 63

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nr 2

173-182

EN

Let G be a locally compact Hausdorff group with Haar measure, and let L⁰(G) be the space of extended real-valued measurable functions on G, finite a.e. Let ϱ and η be modulars on L⁰(G). The error of approximation ϱ(a(Tf - f)) of a function $f ∈ (L⁰(G))_{ϱ+η} ∩ Dom T$ is estimated, where $(Tf)(s) = ∫_G K(t-s,f(t))dt$ and K satisfies a generalized Lipschitz condition with respect to the second variable.

Ograniczanie wyników

2 Annales Polonici Mathematici

1 Commentationes Mathematicae

3 Vinti G.

1 Bardaro C.

1 Costarelli D.

1 Mantellini I.

1 Musielak J.

1 2013

1 2003

1 1996

Approximation results for nonlinear integral operators in modular spaces and applications

Order of approximation for nonlinear sampling Kantorovich operators in Orlicz spaces

Approximation by nonlinear integral operators in some modular function spaces