Superdense a.e. Unbounded Devergence in Some Approximation Processes of Analysis

Abstract

The paper deals with divergence phenomena for various approximation processes of analysis such as Fourier series, Lagrange interpolation, Walsh-Fourier series. We prove the existence of superdense (meaning residual, dense and uncountable) families of functions in appropriate function spaces over an interval $T\subset \mathbb R.$ One proves that for each function in the family, the corresponding approximation process is unboundedly divergent on a superdense subset of $T$ of full measure.