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Abstract Korovkin-type theorems in modular spaces and applications

Carlo Bardaro, Antonio Boccuto, Xenofon Dimitriou, Ilaria Mantellini (2013)

Open Mathematics

We prove some versions of abstract Korovkin-type theorems in modular function spaces, with respect to filter convergence for linear positive operators, by considering several kinds of test functions. We give some results with respect to an axiomatic convergence, including almost convergence. An extension to non positive operators is also studied. Finally, we give some examples and applications to moment and bivariate Kantorovich-type operators, showing that our results are proper extensions of the...

Adaptive convex optimization in Banach spaces: a multilevel approach

Claudio Canuto (2003)

Bollettino dell'Unione Matematica Italiana

This is mainly a review paper, concerned with some applications of the concept of Nonlinear Approximation to adaptive convex minimization. At first, we recall the basic ideas and we compare linear to nonlinear approximation for three relevant families of bases used in practice: Fourier bases, finite element bases, wavelet bases. Next, we show how nonlinear approximation can be used to design rigorously justified and optimally efficient adaptive methods to solve abstract minimization problems in...

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