Some 2-norm-based deviation estimations associated with the Fourier partial sum

The analysis theory associated with linear 2-normed spaces has interested some mathematicians in recent decades. In contrast to the classical norm, 2-norm is an area-like metric instead of a length metric of vectors. This paper addresses the estimation of area-like deviation under 2-norm using Fourier partial sums. In the Hilbert space setting, we present an estimation of the area-like deviation and apply it to the Sobolev class, principle shift-invariant subspaces of [Formula: see text] and the Paley–Wiener space. We introduce the concepts of generalized Sobolev class and generalized Hölder class which are more general than the classical Sobolev class and classical Hölder class, and estimate the area-like deviation associated with them.