Functiones et Approximatio Commentarii Mathematici

Convergence in $BV_\varphi$ by nonlinear Mellin-type convolution operators

Carlo Bardaro, Sarah Sciamannini, and Gianluca Vinti

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In this paper we establish convergence results for a family $\mathbb{T}$ of nonlinear integral operators of the form: $$(T_{w}f)(s) = \int_0^{+\infty}K_{w}(t,f(st))dt = \int_0^{+\infty} L_{w}(t)H_{w}(f(st))dt, \ s \in \mathbb{R}_0^+,$$ where $f \in Dom \mathbb{T}$, $Dom \mathbb{T}$ being the class of all the measurable functions $f:\mathbb{R}_0^+ \rightarrow \mathbb{R}$ such that $T_{w}f$ is well defined as Lebesgue integral for every $s \in \mathbb{R}_0^+$. For the above family of nonlinear Mellin type operators, under suitable singularity assumptions on the kernels $\mathbb{K} = \{K_{w}\}$, we state a convergence result of type $\mathrm{lim}_{w \rightarrow + \infty}V_{\varphi}[\mu(T_{w}f - f)] = 0$, for some constant $\mu > 0$ and for every $f$ belonging to a suitable subspace of $BV_\varphi$-functions.

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Funct. Approx. Comment. Math., Volume 29 (2001), 17-28.

First available in Project Euclid: 29 September 2018

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Primary: 26D15: Inequalities for sums, series and integrals 47H30: Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) [See also 45Gxx, 45P05]
Secondary: 41A17: Inequalities in approximation (Bernstein, Jackson, Nikol s kii-type inequalities) 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 47B38: Operators on function spaces (general) 47G10: Integral operators [See also 45P05]

Musielak-Orlicz $\varphi$-variation $V_\varphi$-convergence locally $\varphi,\eta$-absolutely continuous functions nonlinear Mellin type convolution operators


Bardaro, Carlo; Sciamannini, Sarah; Vinti, Gianluca. Convergence in $BV_\varphi$ by nonlinear Mellin-type convolution operators. Funct. Approx. Comment. Math. 29 (2001), 17--28. doi:10.7169/facm/1538186713.

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