Discrete operators of sampling type and approximation in φ-variation

The paper deals with approximation results with respect to the phi-variation by means of a family of discrete operators for phi-absolutely continuous functions. In particular, for the considered family of operators and for the error of approximation, we first obtain some estimates which are important in order to prove the main result of convergence in phi-variation. The problem of the rate of approximation is also studied. The discrete operators that we consider are deeply connected to some problems of linear prediction from samples in the past, and therefore have important applications in several fields, such as, for example, in speech processing. Moreover such family of operators coincides, in a particular case, with the generalized sampling-type series on a subset of the space of the phi-absolutely continuous functions: therefore we are able to obtain a result of convergence in variation also for the generalized sampling-type series. Some examples are also discussed.