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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
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