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Generalized sampling operators in modular spaces. (English) Zbl 0984.47025
Summary: Modular convergence theorems in modular spaces for nets of operators of the form $(T_wf)(s)= \int_H K_w(s- h_w(t)) f(h_w(t)) d\mu_H(t),\quad w> 0,\quad s\in G,$ where $$G$$ and $$H$$ are topological groups and $$\{h_w\}$$ a family of homeomorphism $$h_w: H\to h_w(H)\subset G$$, are given. As applications, in the particular case of $$G= (\mathbb{R},+)$$, $$H= (Z,+)$$, $$h_w(k)= {k\over w}$$, $$K_w(z)= k(wz)$$, we obtain modular convergence theorems for the generalized sampling series of $f: (T_wf)(s)= \sum_{k\in Z} K(sw- k)f\Biggl({k\over w}\Biggr),\quad s\in \mathbb{R},\quad w>0.$

##### MSC:
 47B34 Kernel operators 47B38 Linear operators on function spaces (general) 47G10 Integral operators