On the boundedness of general partial sums

From S. Banach’s results it follows that even for the function f ( x ) = 1 ( x ∈ [ 0 , 1 ] ) the general partial sums of its general Fourier series are not bounded a.e. on [0, 1]. In the present paper, we find conditions for the functions φ n of an orthonormal system ( φ n ) under which the partial sums of functions from some differentiable class are bounded. We prove that the obtained results are best possible. We also investigate the properties of subsequences of general orthonormal systems.