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

The analysis theory associated linear 2-normed spaces has interested some mathematicians in last 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 shiftinvariant subspaces of L 2 (ℝ) and the Paley-Winner 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.